holomorphic extension of cr functions, envelopes of ...merker/... · propagational aspects of cr...

HOLOMORPHIC EXTENSION OF CR FUNCTIONS,

ENVELOPES OF HOLOMORPHY,

AND REMOVABLE SINGULARITIES

JOEL MERKER AND EGMONT PORTEN

Departement de Mathematiques et Applications, UMR 8553 du CNRS, Ecole Normale Superieure, 45 rued’Ulm, F-75230 Paris Cedex 05, [email protected] http://www.cmi.univ-mrs.fr/ � merker/index.htmlDepartment of Engineering, Physics and Mathematics, Mid Sweden University, Campus Sundsvall, S-85170Sundsvall, Sweden [email protected]

Table of contents (7 main parts)

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.II. Analytic vector field systems, formal CR mappings and local CR automorphism groups . . . . . . . . . . II.III. Sussman’s orbit theorem, locally integrable systems of vector fields and CR functions . . . . . . . . . . . III.IV. Hilbert transform and Bishop’s equation in Holder spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.V. Holomorphic extension of CR functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.VI. Removable singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI.

[0+3+7+1+19+7 diagrams]

IMRS International Mathematics Research Surveys

Volume 2006, Article ID 28295, 287 pages

www.hindawi.com/journals/imrs

Date: 2007-1-7.2000 Mathematics Subject Classification. Primary: 32-01, 32-02. Secondary: 32Bxx,

32Dxx, 32D10, 32D15, 32D20, 32D26, 32H12, 32F40, 32T15, 32T25, 32T27, 32Vxx,32V05, 32V10, 32V15, 32V25, 32V35, 32V40.

1

2 JOEL MERKER AND EGMONT PORTEN

I: Introduction

1.1. CR extension theory. In the past decades, remarkable progress hasbeen accomplished towards the understanding of compulsory extendabilityof holomorphic functions, of CR functions and of differential forms. Thesephenomena, whose exploration is still active in current research, originatefrom the seminal Hartogs-Bochner extension theorem.

In local CR extension theory, the most satisfactory achievement was thediscovery that, on a smooth embedded generic submanifold

� ��, there

is a precise correspondence between CR orbits of�

and families of smallBishop discs attached to

�. Such discs cover a substantial part of the poly-

nomial hull of�

, and in most cases, this part may be shown to constitutea global one-sided neighborhood �� of

�, if

�is a hypersurface, or

else a wedgelike domain � attached to�

, if�

has codimension �� .A local polynomial approximation theorem, or a CR version of the Konti-nuitatssatz (continuity principle) assures that CR functions automaticallyextend holomorphically to such domains � , which are in addition con-tained in the envelope of holomorphy of arbitrarily thin neighborhoods of�

in� �

.Trepreau in the hypersurface (1986) case and slightly after Tumanov in

arbitrary codimension (1988) established a nowadays celebrated extensiontheorem: if

� ��is a sufficiently smooth ( �� or �� suffices) generic

submanifold, then at every point �� whose local CR orbit ��! #"$&% �(' � �has maximal dimension equal to )+*-, � , there exists a local wedge �/. ofedge

�at � to which continuous CR functions extend holomorphically.

Several reconstructions and applications of this groundbreaking result, to-gether with surveys about the local Bishop equation have already appearedin the literature.

Propagational aspects of CR extension theory are less known by con-temporary experts of several complex variables, but they lie deeper in thetheory. Using FBI transform and concepts of microlocal analysis, Trepreaushowed in 1990 that holomorphic extension to a wedge propagates alongcurves whose velocity vector is complex-tangential to

�. His conjecture

that extension to a wedge should hold at every point of a generic subman-ifold

� �0�1�consisting of a single global CR orbit has been answered

independently by Joricke and by the first author in 1994, using tools intro-duced previously by Tumanov. To the knowledge of the two authors, thereis no survey of these global aspects in the literature.

The first main objective of the present survey is to expose the techniquesunderlying these results in a comprehensive and unified way, emphasizing

HOLOMORPHIC EXTENSIONS AND REMOVABLE SINGULARITIES 3

propagational aspects of embedded CR geometry and discussing optimalsmoothness assumptions. Thus, topics that are necessary to build the the-ory from scratch will be selected and accompanied with thorough proofs,whereas other results that are nevertheless central in CR geometry will bepresented in concise survey style, without any proof.

The theory of CR extension by means of analytic discs combines var-ious concepts emanating mainly from three (wide) mathematical areas:Harmonic analysis, Partial differential equations and Complex analysis inseveral variables. As the project evolved, we felt the necessity of beingconceptional, extensive and systematic in the restitution of (semi)knownresults, so that various contributions to the subject would recover a cer-tain coherence and a certain unity. With the objective of adressing to ayounger audience, we decided to adopt a style accessible to doctoral can-didates working on a dissertation. Parts III, IV and V present elementarilygeneral CR extension theory. Also, most sections of the text may be readindependently by experts, as quanta of mathematical information.

1.2. Concise presentation of the contents. The survey text is organized insix main parts. Actually, the present brief introduction constitutes the firstand shortest one. Although the reader will find a “conceptional summary-introduction” at the beginning of each part, a few descriptive words ex-plaining some of our options governing the reconstruction of CR extensiontheory (Parts III, IV and V) are welcome.

The next Part II is independent of the others and can be skipped in afirst reading. It opens the text, because it is concerned with propagationalaspects of analytic CR structures, better understood than the smooth ones.2 In Part III, exclusively concerned with the smooth category, Sussmann’sorbit theorem and its consequences are first explained in length. Involu-tive structures and embedded CR manifolds, together with their elementaryproperties, are introduced. Structural properties of finite type structures, ofCR orbits and of CR functions are presented without proofs. As a collec-tion of background material, this part should be consulted first.2 In Part IV, fundamental results about singular integral operators in thecomplex plane are first surveyed. Explicit estimates of the norms ofthe Cauchy, of the Schwarz and of the Hilbert transforms in the Holderspaces ��3 � � are provided. They are useful to reconstruct the main Theo-rem 3.7(IV), due to Tumanov, which asserts the existence of unique so-lutions to a parametrized Bishop-type equation with an optimal loss ofsmoothness with respect to parameters. Following Bishop’s constructivephilosophy, the smallness of the constants insuring existence is precised


explicitly, thanks to sharp norm inequalities in Holder spaces. This partis meant to introduce interested readers to further reading of Tumanov’srecent works about extremal (pseudoholomorphic) discs in higher codi-mension.2 In Part V, CR extension theory is first discussed in the hypersurface case.A simplified proof of wedge extendability that treats both locally minimaland globally minimal generic submanifolds on the same footing constitutesthe main Theorem 4.12(V): If

�is a globally minimal �4�� 57698;:<8= �

generic submanifold of��

of codimension � =and of CR dimension� =

, there exists a wedgelike domain � attached to�

such that everycontinuous CR function >?�@��A$&% �B� possesses a holomorphic extensionC �D�EF� �HG � A ��I � �

withCKJMLON > . The figures are intended to share

the geometric insight of experts in higher codimensional geometry.

In fact, throughout the text, diagrams (33 in sum) facilitating readabil-ity (especially of Part V) are included. Selected open questions and openproblems (16 in sum) are formulated. They are systematically inserted inthe right place of the architecture. The sign “[ P ]” added after one or sev-eral bibliographical references in a statement (Problem, Definition, Theo-rem, Proposition, Lemma, Corollary, Example, Open question and Openproblem, e.g. Theorem 1.11(I)) indicates that, compared to the existingliterature, a slight modification or a slight improvement has been broughtby the two authors. Statements containing no bibliographical reference areoriginal and appear here for the first time.

We apologize for having not treated some central topics of CR geometrythat also involve propagation of holomorphicity, exempli gratia the geo-metric reflection principle, in the sense of Pinchuk, Webster, Diederich,Fornæss, Shafikov and Verma. By lack of space, embeddability of abstractCR structures, polynomial hulls, Bishop discs growing at elliptic complextangencies, filling by Levi-flat surfaces, Riemann-Hilbert boundary valueproblems, complex Plateau problem in Kahler manifolds, partial indicesof analytic discs, pseudoholomorphic discs, etc. are not reviewed either.Certainly, better experts will fill this gap in the near future.

To conclude this introductory presentation, we believe that, although un-easy to build, surveys and syntheses play a decisive role in the evolution ofmathematical subjects. For instance, in the last decades, the remarkable de-velopment of Q techniques and of RS� estimates has been regularly accom-panied by monographs and panoramas, some of which became landmarksin the field. Certainly, the (local) method of analytic discs deserves to be


known by a wider audience; in fact, its main contributors have brought itto the degree of achievement that opened the way to the present survey.

1.3. Further readings. Using the tools exposed and reconstructed in thissurvey, the research article [MP2006a] studies removable singularities onCR manifolds of CR dimension equal to 1 and solves a delicate remain-ing open problem in the field (see the Introduction there for motivations).Recently also, the authors built in [MP2006c] a new, rigorous proof of theclassical Hartogs extension theorem which relies only on the basic localLevi argument along analytic discs, hence avoids both multidimensionalintegral representation formulas and the Serre-Ehrenpreis argument aboutvanishing of Q cohomology with compact support.


II: Analytic vector field systemsand formal CR mappings

Table of contents

1. Analytic vector field systems and Nagano’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Analytic CR manifolds, Segre chains and minimality . . . . . . . . . . . . . . . . . . . . . . 19.3. Formal CR mappings, jets of Segre varieties and CR reflection mapping . . . 28.

[3 diagrams]

According to the theorem of Frobenius, a system T of local vector fields hav-ing real or complex analytic coefficients enjoys the integral manifolds property,provided it is closed under Lie bracket. If the Lie brackets exceed T , consideringthe smallest analytic system TVUMWMX containing T which is closed under Lie bracket,Nagano showed that through every point, there passes a submanifold whose tan-gent space is spanned by TYUMWMX . Without considering Lie brackets, these submani-folds may also be constructed by means of compositions of local flows of elementsof T . Such a construction has applications in real analytic Cauchy-Riemann ge-ometry, in the reflection principle, in formal CR mappings, in analytic hypoellip-ticity theorems and in the problem of local solvability and of local uniqueness forsystems of first order linear partial differential operators (Part III).

For a generic set of Z\[B] vector fields having analytic coefficients, T UMWMX hasmaximal rank equal to the dimension of the ambient space.

The extrinsic complexification ^ of a real algebraic or analytic Cauchy-Riemann submanifold _ of ` � carries two pairs of intrinsic foliations, obtainedby complexifying the classical Segre varieties together with their conjugates. TheNagano leaves of this pair of foliations coincide with the extrinsic complexifica-tions of local CR orbits. If _ is (Nash) algebraic, its CR orbits are algebraic too,because they are projections of complexified algebraic Nagano leaves.

A complexified formal CR mapping between two complexified generic sub-manifolds must respect the two pairs of intrinsic foliations that lie in the sourceand in the target. This constraint imposes strong rigidity properties, as for in-stance: convergence, analyticity or algebraicity of the formal CR mapping, ac-cording to the smoothness of the target and of the source. There is a combina-torics of various nondegeneracy conditions that entail versions of the so-calledanalytic reflection principle. The concept of CR reflection mapping provides aunified synthesis of recent results of the literature.

HOLOMORPHIC EXTENSIONS AND REMOVABLE SINGULARITIES 7a1. ANALYTIC VECTOR FIELD SYSTEMS AND NAGANO’S THEOREM

1.1. Formal, analytic and (Nash) algebraic power series. Let bc�edwith bf� =

and let g N hgji 'lklklkm' g � � �@n � , where n Npoor�

. Let nrq qsgut tbe the ring of formal power series in hgvi 'lklklkm' g � � . An element wxhg � �ynrq qsgut twrites wzhg ��NO{ �H|m}�~ w � g � , with g �� N g �H�i�� g � ~� and with w � �5n , forevery multiindex : � N 7:�i 'lklk�k�' : � � ��d � . We put

J : J � N :�i�� f: � .On the vector space n � , we choose once for all the maximum normJ g J � N ,��u�&i�� J gu� J and, for any “radius” �+i satisfying 6\8��+i��e� , we

define the open cube � �� N�� g��yn � � J ��J 8?��i��as a fundamental, concrete open set. For ��i N � , we identify of course� � �

with n � .If the coefficients w � satisfy a Cauchy estimate of the form

J w � J �� Y¡ � ¡� ,� ¢ 6 , for every � � satisfying 6£8 � � 8 ��i , the formal

power series is n -analytic ( �¥¤ ) in

� �� . It then defines a true point mapw � � �� §¦ n . Such a n -analytic function w is called (Nash) n -algebraicif there exists a nonzero polynomial ¨�F© '�ª«� �yn¬q © '�ª t in Fb�� = � variablessuch that the relation ¨�hg ' wz�g ��® 6 holds in nrq qsgut t , hence for all g in

� �� .The category of n -algebraic functions and maps is stable under elementaryalgebraic operations, under differentiation and under composition. Implicitsolutions of n -algebraic equations are n -algebraic ([BER1999]).

1.2. Analytic vector field systems and their integral manifolds. Let¯ A � NO� R�°±�²i��j°³��´ '¶µ ��d ';µ � = 'be a finite set of vector fields R·° N¸{ ��º¹»i w®° � �¼hg �5½½�¾¼¿ , whose coefficientsw®° � � are algebraic or analytic in

� �� . Let À � � denote the ring of algebraicor analytic functions in

� �� . The set of linear combinations of elements of¯ A with coefficients in À � � will be denoted by¯

(or¯ i ) and will be called

the À � � -linear hull of¯ A .

If � is a point of

� �� , denote by R·°²Á� � the vector{ ��º¹»i w4° � ��º� �D½½�¾�¿ÃÂÂ . . It

is an element of Ä+. � �� ·Å n � . Define the linear subspace¯ º� � � NBÆvÇ �ÃÈ�É � R�°Hº� � � = ��Ê�� µ � NË� RÌÁ� � � R?� ¯ � kNo constancy of dimension, no linear independency assumption are made.

Problem 1.3. Find local submanifolds Í passing through the origin satis-fying ÄVÎÏÍ9Ð ¯ FÑ � for every Ñ¬��Í .


By the theorem of Frobenius ([Stk2000]; original article: [Fr1877]), if theR·° are linearly independent at every point of

� �� and if the Lie bracketsq!R�° ' R·°¼ÒÓt belong to¯

, for all Ê ' ÊÕÔ N = 'lklk�km'Öµ, then

� �� is foliated byµ-

dimensional submanifolds × satisfying Ä�Î�× N9¯ ØÑ � for every Ñ��× .

Lemma 1.4. If there exists a local submanifold Í passing through the ori-gin and satisfying ÄVÎ�ÍËÐ ¯ ØÑ � for every ÑK�DÍ , then for every two vectorfields R ' R�ÔV� ¯ , the restriction to Í of the Lie bracket qÙR ' R·Ôst is tangent toÍ .

Accordingly, set¯ i � N ¯

and for ÚÛ� � , define¯®Ü

to be the À � � -linear hull of

¯ Ü i � Ý ¯ i 'Ï¯ Ü i�Þ . Concretely,¯ Ü

is generated by À � � -linearcombinations of iterated Lie brackets qÙR«i ' qMR � 'lklklkm' qßR Ü i ' R Ü t k�klk tÓt , whereRzi ' R � '�klklk�' R Ü i ' R Ü � ¯ i . The Jacobi identity insures (by induction) thatÝ ¯ Ü � 'Ï¯ ÜÏà Þ �(¯ Ü �Ïá Ü�à . Define then

¯ UMWMX � NpI Ümâ i ¯ Ü . Clearly, qÙR ' R�Ôstã� ¯ UMWMX ,for every two vector fields R ' R·Ô»� ¯ UMWMX .Theorem 1.5. (NAGANO [Na1966, Trv1992, BER1999, BCH2005])There exists a unique local n -analytic submanifold Í of n � passingthrough the origin which satisfies

¯ ØÑ �ä� Ä�ÎÖÍ N9¯ UMWMX FÑ � , for every Ñ��yÍ .

A discussion about what happens in the algebraic category is postponedtoa1.12. In Frobenius’ theorem,

¯ UMWMX N¸¯and the dimension of

¯ UMWMX º� �is constant. In the above theorem, the dimension of

¯ UMWMX ØÑ � is constant forÑ belonging to Í , but in general, not constant for �(� � �� , the function�Kå¦ )+*�, É ¯ º� � being lower semi-continuous.Nagano’s theorem is stated at the origin; it also holds at every point�D� � �� . The associated local submanifold Í�. passing through � with the

property that ÄVÎÖÍ N0¯ UMWMX FÑ � for every Ñæ� Íã. is called a (local) Naganoleaf.

In the � � category, the consideration of¯ UMWMX is insufficient. Part III

handles smooth vector field systems, providing a different answer to thesearch of similar submanifolds Í1. .Example 1.6. In

o � , take¯ A N � Rzi ' R � � , where Rzi N Q ¾ � and R � Nç i7è ¾ à � Q ¾ à . Then

¯ UMWMX Ø6 � is the lineo Q ¾êé J A , while

¯ UMWMX º� �·N9o Q ¾ � J .�� o Q ¾ à J .at every point �Dë� oKìí� 6v� . Hence, there cannot exist a � � curve Í passingthrough 6 with Ä A Í N�o Q ¾êé J A and Ä»Î�Í N9¯ UMWMX ØÑ � for every Ñ��Í .

Proof of Theorem 1.5. (May be skipped in a first reading.) If b N =,

the statement is clear, depending on whether or not all vector fields in¯ UMWMX

vanish at the origin. Let b�� . Since¯ ØÑ �î�£¯ UMWMX ØÑ � , the condition

HOLOMORPHIC EXTENSIONS AND REMOVABLE SINGULARITIES 9ÄVÎÖÍ Nï¯ UMWMX FÑ � implies the inclusion¯ FÑ �ð� Ä¥ÎÏÍ . Replacing

¯by¯ UMWMX if

necessary, we may therefore assume that¯ UMWMX Nñ¯

and we then have toprove the existence of Í with Ä¥Î�Í N�¯ UMWMX ØÑ ��N�¯ FÑ � , for every Ñò��Í .

We reason by induction, supposing that, in dimension Øb�ó = � , for everyÀ � � -linear system¯ Ô N ¯ Ô � UMWMX of vector fields locally defined in a neighbor-

hood of the origin in n � i , there exists a local n -analytic submanifold Í Ôpassing through the origin and satisfying Ä�Î Ò Í�Ô N�¯ ÔÏFÑuÔ � , for every ÑuÔY�yÍSÔ .

If all vector fields in¯ N�¯ UMWMX vanish at 6 , we are done, trivially. Thus,

assume there exists Rzi\� ¯with Rzi�Ø6 � ëN 6 . After local straightening,Rzi N Q ¾ � . Every Rô� ¯

writes uniquely R N ÊYhg � Q ¾ � �ÛõR , for someÊ»�g � �ön � gj� , with õR Np{ � �� Ê²�¼�g � Q ¾�¿ . Introduce the space õ¯ � NÛ� õR �R�� ¯ � of such vector fields. As Q ¾ � belongs to¯

and as¯

is À � � -linear,õR N R/óîÊY�g � Q ¾ � belongs to¯

. Since q ¯�'Ï¯ t �p¯, we have Ý�õ¯Ì' õ¯ Þ �p¯

.On the other hand, we observe that the Lie bracket between two elementsof õ¯ does not involve Q ¾ � :(1.7)

Ý õRxi ' õR � Þ Nø÷fù� �� à � � Ê i� à Q ¾�¿ à '¸ù� �� Ê �� Q ¾¼¿ �ÖúN ù� �� Dû ù� �� à � � ÷ Ê i� à Q�ÊÕ�� Q � � à ó@Ê �� à Q�Ê i� �Q � � à úHü Q ¾¼¿ � kWe deduce that Ý-õ¯ä' õ¯ Þ � õ¯ . In other words, õ¯ UMWMX N õ¯ . Next, we define therestriction ¯ Ô � N�ý R Ô N õR ÂÂsþ ¾ � ¹ Aêÿ � õR?�5õ¯��+'and we claim that ¯ Ô � UMWMX Nñ¯ Ô also holds true. Indeed, restricting (1.7)above to

� gji N 6v� , we observe thatÝ õRxi ÂÂ þ ¾ � ¹ Aêÿ ' õR � ÂÂ þ ¾ � ¹ Aêÿ Þ N Ý õRxi ' õR � Þ ÂÂ þ ¾ � ¹ Aêÿ 'since neither õRxi nor õR � involves Q ¾ � . This shows that Ý ¯ Ô 'ê¯ Ô Þ ��¯ Ô , asclaimed.

Since ¯ Ô � UMWMX N�¯ Ô , the induction assumption applies: there exists a localn -analytic submanifold ÍxÔ of n � i passing through the origin such thatÄVÎ Ò Í Ô Nô¯ Ô ØÑ Ô � , for every point Ñ Ô �eÍ Ô . Let � denote its codimension.If � N 6 , i.e. if Í�Ô coincides with an open neighborhood of the origin inn � i , it suffices to chose for Í an open neighborhood of the origin in n � .Assuming �� = , we split the coordinates g N hgvi ' guÔ � �yn ì n � i and welet ��Ãhg Ô �·N 6 , � N = 'lklk�km' � , denote local n -analytic defining equations for

10 JOEL MERKER AND EGMONT PORTENÍ�Ô . We claim that it suffices to choose for Í the local submanifold of n �with the same equations, hence having the same codimension.

Indeed, since these equations are independent of g�i , it is first of all clearthat the vector field Q ¾ � � ¯ is tangent to Í . To conclude that every R NÊxQ ¾ � �BõRË� ¯ is tangent to Í , we thus have to prove that every õRB� õ¯ istangent to Í .

Let õR N({ � ��-� � ÊÕ��hg ' g Ô � Q ¾¼¿ � õ¯ . As a preliminary observation:

Ø�H)Q ¾ � � õR � N Ý Q ¾ � ' õR Þ N ù� ��-� � Q�ÊÕ�Qvg�i �g�i ' g Ô � QQvgu� 'and more generally, for ��d arbitrary:

7�Ã)§Q ¾ � �� õR N ù� �� Q� Ê²�Qvg � i �g�i ' g Ô � QQvgu� k

Since¯

is a Lie algebra, we have Ø�H)Q ¾ � � � õRO� ¯ . Since Ø�H)Q ¾ � � � õR doesnot involve Q ¾ � , according to its expression above, it belongs in fact to õ¯ .Also, after restriction Ø�H)Q ¾ � � � õR ÂÂ ¾ � ¹ A � ¯ Ô . By assumption,

¯ Ô is tangentto Í Ô . We deduce that, for every � �yd , the vector field

R Ô� � N Ø�H)§Q ¾ � �� õR ÂÂ ¾ � ¹ A N ù� �� Q� Ê²�Qvg � i 76 ' g Ô � QQvgu�

is tangent to Í Ô . Equivalently, qßR Ô� ��Öt#hg Ô �rN 6 for every g Ô ��Í Ô . Lettinghg�i ' guÔ � �yÍ , whence guÔ»��Í�Ô , we compute:

Ý õR �� Þ �g�i ' g Ô ��N ù� �� ÊÕ��hg�i ' g Ô � Q&��Qvgu� �g Ô �N ù� �� ù � ¹ A g� i�� Q � Ê²�Qvg � i Ø6 ' g Ô �1Q��Qvgu� hg Ô � [Taylor development]

N �ù � ¹ A g� i�� ÝºR Ô� �� Þ hg Ô �·N 6 '

so õR is tangent to Í . Finally, the property Ä ¾ �� ¾ Ò Í N ¯ �g�i ' guÔ � followsimmediately from Ä ¾ Ò Í�Ô Nø¯ ÔÏhguÔ � and the proof is complete (the Taylordevelopment argument above was crucially used, and this enlightens whythe theorem does not hold in the � � category).


1.8. Free Lie algebras and generic sets of n -analytic vector fields. Fora generic set of

µ �O� vector fields¯ A NÛ� R�°u�²i��j°³��´ , or after slightly per-

turbing any given set, one expects that¯ UMWMX 76 �§N Ä A n � . Then the Nagano

leaf Í passing through 6 is just an open neighborhood of 6 in n � . Also, oneexpects that the dimensions of the intermediate spaces

¯ Ü 76 � be maximal.To realize this intuition, one has to count the maximal number of iter-

ated Lie brackets that are linearly independent in¯ Ü

, for Ú N = ' � ' ��'lklklk ,modulo antisymmetry and Jacobi identity.

Letµ � � and let �Vi ' � � 'lklklkm' ��´ be

µlinearly independent elements

of a vector space over n . The free Lie algebra �Y µH� of rankµ

is thesmallest (non-commutative, non-associative) n -algebra ([Re1993]) hav-ing �Vi ' � � 'lklklkm' ��´ as elements, with multiplication �� ' �&Ô � å¦ ��Ô satis-fying antisymmetry 6 N ��&Ô+��Ô�� and Jacobi identity 6 N ��&Ô��Ô Ô � ��+Ô ÔØ��+Ô � ��ÔØ��+Ô Ô�� . It is unique up to isomorphism. The case

µ�N =yields only �Y = �EN n . The multiplication in �Y µH� plays the role of theLie bracket in

¯ UMWMX . Importantly, no linear relation exists between iteratedmultiplications, i.e. between iterated Lie brackets, except those generatedby antisymmetry and Jacobi identity. Thus, �Y µ²� is infinite-dimensional.Every finite-dimensional Lie n -algebra having

µgenerators embeds as a

subalgebra of �Y µ²� , see [Re1993].Since the bracket multiplication is not associative, one must carefully

write some parentheses, for instance in ��i�� , or in �»im� � ��Vi�� Ï� ,or in ��Vi�� u��»i �� . Writing all such words only with the alphabet� �Vi ' � � 'lklklkm' ��´�� , we define the length of a word � to be the number of ele-ments �� in it. For �í��d with �í� = , let � � ´ be the set of words of lengthequal to � and let �æ´ N�� â i � � ´ be the set of all words.

Define ��i� µH� to be the vector space generated by �¥i ' � � 'lklklk ' ��´ and for� �¸� , define � � µH� to be the vector space generated by words of length� � . This corresponds to¯ �

, except that in¯ �

, there might exist speciallinear relations that are absent in the abstract case. Thus, �Y µH� is a gradedLie algebra. The Jacobi identity insures (by induction) that � � � µH� � � à µ²�x�� 7á � à µ²� , a property similar to Ý ¯ Ü � 'Ï¯ ÜÏà Þ � ¯ Ü �Ïá ÜÏà . It follows that � � µ²�is generated by words of the form

�+� � �� à klklk ��"! Ò$# � ��"! Ò �Vklklk�� 'where �mÔ�� and where

= � %êi ' % � 'lklklkm' % � Ò i ' % � ÒD� µ. For instance,�»i&� � � ��u��»i �� may be written as a linear combination of such simple


words whose length is �(' . Let us denote by) �ö´ N+*� â i ) � � ´the set of these simple words, where

) � � ´ denotes the set of simplewords of length � . Although it generates �Y µH� as a vector space overn , we point out that it is not a basis of �Y µ²� : for instance, we have�Vi�� »i,� � ��ðN � � �»i��»i,� � �Ï� , because of an obvious Jacobi identity inwhich �»i-� � � �»i,� � �¬N 6 disappears. In fact, one verifies that this is theonly Jacobi relation between simple words of length . , that simple wordsof length ' have no Jacobi relation, hence a basis of �/�u7� � is�»i ' � � ' �»i-� � '�»i��»i,� � � ' � � �»i-� � � '�»i��»im��Vi-� � �Ï� ' �»i�� »i-� � ��' � � � � � � �»i �Ï� '�»i��»im��Vi��»i-� � ��' �»i��»i�� »i-� � ��Ï� ' �»i�� »i ��Ï� '� � ��Vi �»i��»i,� � ��Ï� ' � � �� »i�� »i �Ï�� ' � � � � � � �� »i �Ï��k

In general, what are the dimensions of the � � µH� ? How to find bases forthem, when considered as vector spaces ?

Definition 1.9. A Hall-Witt basis of �Y µ²� is a linearly ordered (infinite)subset 01�/´ N�� â i 02� � ´ of the set of simple words

) �@´ such that:2 if two simple words � and ��Ô satisfy 3 465�7�8:9v�� 8;3 4<5�7�8:9v=��Ô � , then�ö8>��Ô ;2 02� i´ N�� Vi ' � � 'lklklkm' ��´�� ;2 02� �´ N�� +� � �� à � = �?%¼i«8@% � � µ � ;2 02�æ´BA&C02� i´ I 01� �´ � N � �� Ô � Ô Ô � � � ' � Ô ' � Ô Ô �D02�æ´ ' � Ô 8��Ô ÔÕ�HÈ�)E��Ô»�F�ö8>��ÔG��Ô Ô�� .A Hall-Witt basis essentially consists of the choice, for every �� =

, ofsome (among many possible) finite subset 02� � ´ of

) � � ´ that generates thefinite-dimensional quotient vector space � � µ²�,H � � im µ²� . To fix ideas, anarbitrary linear ordering is added among the elements of the chosen basis02� � ´ of the vector space � � µ²�,H � � im µ²� . The last condition of the definitiontakes account of the Jacobi identity.

Theorem 1.10. ([Bo1972, Re1993]) Hall-Witt bases exist and are bases ofthe free Lie algebra �Y µH� of rank

µ, when considered as a vector space. The


dimensions 5 � µH� óI5 � i� µH� of � � µH�-H � � im µH� , or equivalently the cardinalsof 01� � ´ , satisfy the induction relation5 � µH� óJ5 � im µ²�·N =� ùKML W NÏW L XPO ��Q =� ��µR� è K 'where Q is the Mobius function.

Remind that

Q �� N STU TV= ' *XW2� N =ZY6 ' *XW2� contains square integer factors

Y¼ó = �C[H' *$W\� N �Vi �� [ is the product of ] distinct prime numbersk

Now, we come back to the system¯ A Nï� R�°±�²i��j°��´ of local n -analytic

vector fields ofa1.1, where RS° N£{ ��Á¹»i w®° � �ê�g � ½½�¾¼¿ . If the vector space¯ Ø6 � has dimension 8 µ , a slight perturbation of the coefficients w1° � ��hg � of

the R·° yields a system¯ Ô A with

¯ ÔêØ6 � of dimensionN�µ

. By an elementarycomputation with Lie brackets, one sees that a further slight perturbationyields a system

¯ Ô Ô A with¯ Ô Ô 76 � of dimension

µ � ´-^!´ i�_� N 5 � µ²� .To pursue, any simple iterated Lie bracket qÙR�° � ' qÙR·° à 'lk�klk qÙR�° ! # � ' R�° ! t k�klk tÓt

of length � is a vector field{ ��º¹»ia` � ° � � ° à �cbcbcb � ° ! # � � ° ! ½½�¾¼¿ having coefficients` � ° �ê� ° à �cbcbcb � ° ! # � � ° ! that are universal polynomials in the jetsd&� i¾ wzhg � � Nfe Q �¾ w®° � �êhg �hg �H|�} ~ � ¡ � ¡ � � ii��j°��´ � i��-� � ��n�ikj ~ml ~Bl ! # �

of order n��ó = � of the coefficients of R«i ' R � '�klklk ' R1´ . Here, ×í´ � � � � � i Nµ b ^ � á � i�_no� oR^ � i�_po denotes the number of such independent partial derivatives. A

careful inspection of the polynomials ` � ° �� ° à �cbcbcb � ° ! # � � ° ! enables to get the fol-lowing genericity statement, whose proof will appear elsewhere. It says ina precise way that

¯ UMWMX Ø6 �zN Ä A n � with the maximal freedom, for genericsets of vector fields.

Theorem 1.11. ([GV1987, Ge1988], [ P ]) If � A denotes the smallest length� such that 5 � µ²� �9b , there exists a proper n -algebraic subset q of the jetspace

d �=r iA w N n i j ~ml ~Bl ! r # � such that for every collection¯ A NË� R·°±�²i��j°³��´

ofµ

vector fields R·° N { ��º¹»i w®° � �¼�g � ½½�¾¼¿ such thatd � r iA wz76 � does not

belong to q , the following two properties hold:2 )+*-, ¯ � Ø6 �·N 5 � µH� , for every � �J� A ó = ,2 )+*-, ¯ �=r Ø6 ��N b , hence¯ UMWMX 76 ��N Ä A n � .


The number of divisors of � being an s Uctvu �Uctvu � � , one verifies that 5 � µH� ó5 � i� µH��N i� µ � �Js µ � è � Uctvu �Uctvu � � . It follows that, forµ

fixed, the integer � A of

the theorem is equivalent to Uctvu �Uctvu ´ as b ¦ � .

1.12. Local orbits of n -analytic and of (Nash) n -algebraic systems. Wenow describe a second, more concrete, simple and useful approach to thelocal Nagano Theorem 1.5. It is inspired by Sussmann’s Theorem 1.21(III)and does not involve the consideration of any Lie bracket. Theorem 1.13below will be applied in

a2.11.

As above, consider a finite set¯ A � NO� R�°±�²i��j°³��´ '¶µ ��d ';µ � = 'of nonzero vector fields defined in the cube

� �� and having n -analyticcoefficients. We shall neither consider its À � � -linear hull

¯, nor

¯ UMWMX . Wewill reconstruct the Nagano leaf passing through the origin only by meansof the flows of Rzi ' R � 'lk�klkm' R�´ .

Referring the reader toa1.3(III) for background, we denote the flow map

of a vector field R � ¯ A shortly by P8 ' g � å¦ Rxw��g �«N+y � Ç P8êR � hg � . It is n -analytic. What happens in the algebraic category ?

So, assume that the coefficients of all vector fields R � ¯ A are n -algebraic. Unfortunately, algebraicity fails to be preserved under integra-tion, so the flows are only n -analytic, in general. To get algebraicity ofNagano leaves, there is nothing else than supposing that the flows are al-gebraic, which we will do (second phrase of (5) below).

Choose now � � with 6f8 � � 8 ��i . Let Ú��Ëd with Ú�� =, let R N7R i 'lklklkm' R Ü�� / ¯ A �êÜ , let 8 N P8Öi 'lk�klk�' 8 Ü � �yn Ü with

J 8 J 8î� � , i.e. 8x� � Ü� à ,and let g¬� � �� à . We shall adopt the contracted notationRxw��g � � N R ÜwXz �� 7R iw � hg �Ï� �� for the composition of flow maps, whenever it is defined. In fact, sinceR A Ø6 �äN{y � Ç Ø6²R � 76 � N 6 , it is clear that if we bound the length ÚD�ï�ub ,then there exists � � ¢ 6 sufficiently small such that all maps P8 ' g � å¦ R|w�hg �are well-defined, with R|w��g � � � �� , at least for all 8x� � Ü� à and all g¬� � �� à .The reason why we may restrict to consider only compositions of lengthÚ��ub will appear a posteriori in the proof of the theorem below. Weshall be concerned with rank properties of P8 ' g � å¦ R}w��g � .

Let bf� =, ~ � =

, �+i ¢ 6 , �Vi ¢ 6 and let > � � �� ¦ �� , g�å¦ >1�g � ,be a n -algebraic or n -analytic map between two open cubes. Denote itsJacobian matrix by �²�<�S > ��N�e ½m�n�½�¾�¿ hg �,g i��<�³� �i��-� � . At a point g�� , the map

HOLOMORPHIC EXTENSIONS AND REMOVABLE SINGULARITIES 15> has rankµ

if and only if �²��4> has rankµ

at g . Equivalently, by linearalgebra, there is a

µÌì�µminor that does not vanish at g but all � ì � minors

withµ � = �(�r�9b do vanish at g .

For every �@��d with= ��@�¸, *�È�Fb ' ~ � , compute all the possible� ì � minors �� Z�i 'lk�klk�' ��-�Z�i ^ � _ of �²��S7> � . They are universal (homogeneous

of degree � ) polynomials in the partial derivatives of > , hence are all n -algebraic or n -analytic functions. Let ç with 6�� ç � ,�*-È¥Fb ' ~ � bethe maximal integer � with the property that there exists a minor � � �Z�� hg � ,= � Q ��×ö�� , not vanishing identically. Then the set� � � N ý g�� Z�� hg �·N 6 ' Q N = 'lklk�k�' ×æ� � �is a proper n -algebraic or analytic subset of

� �� . The principle of analyticcontinuation insures that

� �� ,� � � is open and dense.The integer ç is called the generic rank of > . For every open, connected

and nonempty subset � � � �� the restriction > J � has the same genericrank ç .Theorem 1.13. ([Me1999, Me2001a, Me2004a]) There exists an integerç with

= � ç �ïb and an ç -tuple of vector fields R�� N 7R}� i 'lklklk ' R}�� ¯ A � � such that the following six properties hold true.

(1) For every Ú N = '�klklk�' ç , the map P8³i 'lklk�k�' 8 Ü � å¦R}� Üw z �� 7R}� iw � Ø6 �� is of generic rank equal to Ú .

(2) For every arbitrary element R·Ôr� ¯ A , the map P8³i 'lklklkm' 8 � ' 8�Ô � å¦R1Ôw Ò ØR}��w$� �� 7R}� iw � Ø6 �� is of generic rank ç , hence ç is the max-imal possible generic rank.

(3) There exists an element 8-�5� � �� à arbitrarily close to the originwhich is of the special form p8-� i '�klklk ' 8h�� i ' 6 � , namely with 8,�� N 6 ,and there exists an open connected neighborhood �� of 8 � in

� �� àsuch that the map 8 å¦ R��w$� �� 7R}� iw � 76 �Ï� �� is of constant rank çin � � .

(4) Setting R �� N 7R � i 'lklk�k�' R �� , � �ð� N 7R �� i 'lklklkm' R � i � and � �� Nêó�8 �� i 'lklklkm' ó�8 � i � , we have � ��Z� R �w � 76 �·N 6 and the map � � � � ¦� �� defined by � � 8�å¦ �I�� \� R}�w 76 � is also of constant rank equalto ç in �� .

(5) The image Í � N �än�a� � is a piece of n -analytic submanifold pass-ing through the origin enjoying the most important property thatevery vector field R Ô � ¯ A is tangent to Í . If the flows of all ele-ments of

¯ A are algebraic, Í is n -algebraic.


(6) Every local n -algebraic or n -analytic submanifold Í«Ô passingtrough the origin to which all vector fields R·Ôí� ¯ A are tangentmust contain Í in a neighborhood of 6 .

In conclusion, the dimension ç of Í is characterized by the generic rankproperties (1) and (2).

Previously, Í was called Nagano leaf. Since the above statement is su-perseded by Sussmann’s Theorem 1.21(III), we prefer to call it the local¯

-orbit of 6 , introducing in advance the terminology of Part III and denot-ing it by � �! " r � �� ' 6 � . The integer ç of the theorem is �9b , just because thetarget of the maps p8³i 'lklklk ' 8 Ü � å¦ R � ÜwXz �� ØR � iw � Ø6 �� is n � . It followsthat in (4) and (5) we need � ç ó = �(�Ãbòó = compositions of flows to coverÍ .

We quickly mention an application about separate algebraicity.In [BM1949], it is shown that a local n -analytic function ¡ � � �� ¦ n isn -algebraic if and only if its restriction to every affine coordinate segmentis n -algebraic. Call the system

¯ A minimal at the origin if ��ß #" r � �� ' 6 �contains a neighborhood of the origin. Equivalently, the integer ç ofTheorem 1.13 equals b .

Theorem 1.14. ([Me2001a]) If¯ A NË� R�°u�²i��j°³��´ is minimal at 6 , a local n -

analytic function ¡ � � �� «¦ n is n -algebraic if and only it its restrictionto every integral curve of every RS° � ¯ A is n -algebraic.

Proof of Theorem 1.13. (May be skipped in a first reading.) If all vectorfields of

¯ A vanish at the origin, Í NË� 6v� . We now exclude this possibility.Choose a vector field R � i � ¯ A which does not vanish at 6 . The map8Öi�å¦ R � iw � Ø6 � is of (generic) rank one at every 8³i � � i� à . If there existsR�Ôx� ¯ A such that the map P8³i ' 8�Ô � å¦ R1Ôw Ò 7R}� iw � 76 �Ï� is of generic rank two,we choose one such R·Ô and we denote it by R��#� . Continuing in this way, weget vector fields R�� i 'lklklkm' R}�� satisfying properties (1) and (2), with ç �9b .

Since the generic rank of the map p8�i 'lk�klk�' 8 � � å¦ R ��w � �� 7R � iw � 76 �� equals ç , and since this map is n -analytic, there exists a 8 �§� � �� à arbitrarilyclose to the origin at which its rank equals ç . We claim that we can more-over choose 8 � to be of the special form p8 � i 'lklklkm' 8 �� i ' 6 � , i.e. with ¢ �� N 6 .It suffices to apply the following lemma to wzP8 � � N R �� iw � # � �� 7R}� iw � 76 �� and to R Ô � N R �� .Lemma 1.15. Let b@�Dd , b@� =

, let ç �æd ,= � ç �Ëb , let 8 � � � i� à and

let� � i� à¤£ 8·å¦ wzP8 ��N 7w·imp8 � 'lklk�k�' w � P8 �Ï� � � ��


be a n -analytic map whose generic rank equals ç ó = � . Let RSÔ be a n -analytic vector field and assume that the map � � P8 ' 8êÔ � å¦ R1Ôw Ò wzp8 �� hasgeneric rank ç . Then there exists a point p8 � ' 6 � arbitrarily close to theorigin at which the rank of � is equal to ç .Proof. Choose 8,¥·� � � i� à arbitrarily close to zero at which w has maximalrank, equal to ç ó = � . Since the rank is lower semi-continuous, there existsa connected neighborhood �¦¥ of 8,¥ in

� � i� à such that w has rank ç ó = �at every point of �a¥ . By the constant rank theorem, § � N wzn�¦¥ � is then alocal n -analytic submanifold of

� �� passing through the point wxP8 ¥ � . Tocomplete the proof, we claim that there exists 8 �¬�¨��¥ arbitrarily close to8h¥ such that the map p8 ' 8¼Ô � å¦ R�Ôw Ò wzp8 �� has rank ç at P8h� ' 6 � .

Let us reason by contradiction, supposing that at all points of the formP8h� ' 6 � , for 8,�D�©��¥ , the map � � P8 ' 8�Ô � å¦ R1Ôw Ò 7wzP8 �Ï� has rank equal to ç ó = �. Pick an arbitrary 8,�@�©��¥ . Reminding that when 8êÔ N 6 , we

have R ÔwºÒ N R Ô A N«ª ) , we observe that �äp8 ' 6 �� wzP8 � . Consequently, thepartial derivatives of � with respect to the variables 8ê� , % N = 'lklk�k�' ç ó = atan arbitrary point P8 � ' 6 � , with 8 � �¨� ¥ , coincide with the ç ó = � linearlyindependent vectors

½�¬½ w ¿ P8h� � ��n � , % N = '�klklk ' ç ó = . In fact, the tangentspace to § at the point �äp8 � ' 6 �yN wzp8 � � is generated by these ç ó = �vectors.

Reminding the fundamental property½½ wºÒ R1Ôw Ò hg � ÂÂ w Ò ¹ A N R1ÔFhg � , we deduce

[from our assumption that the map P8 ' 8 Ô � å¦ R Ôw Ò wxP8 �� has rank equal to ç ó = � ] that the vector QQ�8 Ô R Ôw Ò wzp8 �� ÂÂÂÂ w Ò ¹ A N R Ô 7wzP8 �Ï�must be linearly dependent with the ç ó = � vectors

½m¬½ w ¿ p8 � , % N = 'lklk�k�' ç ó=, for every 8ö� �a¥ . Equivalently, the vector field R·Ô is tangent to the

submanifold § . It follows that the local flow of RSÔ necessarily stabilizes§ : if g N wzP8 � ��§ , 8��F��¥ , then R1Ôw Ò �g � ��§ , for all 8�Ô§� � i� ^wX_ , where�YP8 � ¢ 6 is sufficiently small. Set ��¥ � N � p8 ' 8�Ô � � 8ð�(��¥ ' 8�ÔÌ� � � ^®wX_ � � .It is a nonempty connected open subset of

� �� à . We have thus deducedthat � �� ¥ � is contained in the ç ó = � -dimensional submanifold § . Thisconstraint entails that � is of rank � ç ó = at every point of ��¥ . However,� J �/¯ being n -analytic and of generic rank equal to ç , by assumption, itshould be of rank ç at every point of an open dense subset of ��¥ . This isthe desired contradiction which proves the lemma.

�


Hence, there exists 8,� N p8h� i 'lklklkm' 8h�� i ' 6 � �� à arbitrarily close to the

origin at which the rank of 8rå¦ R��w$� �� 7R}� iw � 76 �� is maximal (hencelocally constant) equal to ç , so we get the constant rank property (3), for asufficiently small neighborhood �a� of 8h� .

In (4), the property � �� R �° � Ø6 ��N 6 is obvious, using g R A hg ��R w � Rxw��g � :R � i w � � � �� R � w �� # � � R ��A � R �w �� # � � �� R �w � � �g �� g kSince the map gyå¦ �±�� g � is a local diffeomorphism, it is clear that themap � � 8�å¦ �I�² �� R}�w 76 � is also of constant rank ç in �a� . Thus, weobtain (4), and moreover, by the constant rank theorem, the image Í � N�än�� constitutes a local n -analytic submanifold of n � passing throughthe origin. If the flows of elements of

¯ A are all n -algebraic, clearly � andÍ are also n -algebraic.It remains to check that every vector field R Ô � ¯ A is tangent to Í . As a

preliminary, denote by R�Ôw Ò wxP8 �� , 8Ì� � �� à , 8�Ô�� i� à , the map appearing in(2), where R Ô � ¯ A is arbitrary. Reasoning as in the lemma above, we seethat R�Ô is necessarily tangent to some local submanifold § obtained as thelocal image of an open connected set where w has maximal locally constantrank. It follows that the flows and the multiple flows of elements of

¯ Astabilize this submanifold. We deduce a generalization of (2): for Ú �(�Ãb ,for R�ÔY� ¯ A � Ü , for 8�ÔY� � Ü� à , the map P8 ' 8�Ô � å ó ¦ R1Ôw Ò 7R}��w � �� ØR}� iw � 76 �Ï� �� Ï�is of generic rank ç .

In particular, for every R·Ô»� ¯ A , the map P8�Ô ' ³ ' 8 � å ó ¦ R1Ôw Ò � �±�² � R}�w 76 � isof generic rank ç . In fact, the restriction � � 8�å¦ � �² �/� R �w 76 � of this map tothe open set

� 76 ' ³ � ' 8 � � 8 �I�� is already of rank ç at every point and itsimage is the local submanifold Í , by the above construction. So the mapP8�Ô ' 8 � å ó ¦ R�Ôw Ò � �±�² � � R}�w Ø6 � must be of rank ç at every point. In particular,the vector QQ�8 Ô R ÔwºÒ � � �² �2� R �w Ø6 � ÂÂÂÂ w Ò ¹ A N R Ô e � �² �\� R �w Ø6 �,gmust necessarily be tangent to Í at the point �´�² � � R}�w 76 � ��Í . Thus, (5) isproved.

Take Í�Ô as in (6). The local flows of all vector R·Ô�� ¯ A stabilize Í�Ô .Shrinking � � if necessary, all the maps p8 ' g � å ó ¦ R|w��g � have range in Í Ô .So Í � Í�Ô , proving (6).

�

HOLOMORPHIC EXTENSIONS AND REMOVABLE SINGULARITIES 19a2. ANALYTIC CR MANIFOLDS, SEGRE CHAINS AND MINIMALITY

2.1. Local Cauchy-Riemann submanifolds of�·�

. Let �µÃi 'lklk�k�' µ � �æN � i��%�¶ji 'lk�klk�'�� %�¶ � � denote the canonical coordinates on�·�

. As before,we use the maximum norms

J ��J�N ,��u�»i�� Ü � � J � Ü J , J ¶ J»N ,��u��i�� Ü � � J ¶ Ü Jand

J µ JjN ,��u�&i�� Ü � � J µ Ü J , whereJ µ Ü J�N � �Ü �@¶v�Ü � i7è � . If � ¢ 6 , we denote

by � �� NO� µ � � � � J µ J 8?�+� the open polydisc of radius � centered at theorigin, not to be confused with

� � �� NË�� ¨%¶ð� � � � J ��Js'HJ ¶ J 8?�+� .Letd

denote the complex structure of Ä �·� , acting on real vectors asif it were multiplication by · ó = . Precisely, if � is any point, Ä�. �� isspanned by the �Ãb vectors

½½B¸ z ÂÂ . , ½½:¹ z ÂÂ . , Ú N = 'lklk�km' b , andd

acts as follows:d ½½:¸ z ÂÂ . N ½½:¹ z ÂÂ . ; d ½½:¹ z ÂÂ . N ó ½½:¸ z ÂÂ . .Choose the origin as a center point and consider a real � -codimensional

local submanifold�

of� � Å o � � passing through the origin, defined by �

Cartesian equationsµ i� ��' ¶ ��N �� NËµ K ��' ¶ �SN 6 , where the differentials� µ i 'lklklk ' � µ K are linearly independent at the origin. The functions

µ � areassumed to be of class1 �»º , where

� N �¼ ' : � , ¼Û� =, 6B� :ñ� =

,�øN � ,�;N � or

�;N;½�¾ ¡ . Accordingly,�

is said to be of class �M¿»�ÁÀ(real algebraic), � ¤ (real analytic), � � or � 3 � � .

For �� , the smallestd

-invariant subspace of the tangent space Ä&. �is given by ÄÌ". � � N Äv. �¸GÂd Äv. � and is called the complex tangent spaceto�

at � .

Definition 2.2. The submanifold�

is called:2 holomorphic if ÄÌ". � N Ä�. � at every point �� ;2 totally real if ÄÌ". � NO� 6v� at every point � � � ;2 generic if Ä�. � � d Äv. � N Ä�. �1� at every point �\� � ;2 Cauchy-Riemann (CR for short) if the dimension of Ää". � is equalto a fixed constant at every point � � � .

For fundamentals about Cauchy-Riemann (CR for short) structures,we refer the reader to [Ch1989, Ja1990, Ch1991, Bo1991, BER1999,Me2004a]. Here, we only summarize some elementary useful properties.The two

d-invariant spaces Ä+. � GÃd Ä�. � and Ä�. � � d Äv. � are of even

real dimension. We denote by ~ . the integer i� )+*�,ÅÄ�FÄv. � GÃd Äv. �B� andcall it the CR dimension of

�at � . If

�is CR, ~ . ~ is constant. Holo-

morphic, totally real and generic submanifolds are CR, with ~ N bðó i� � ,~ N 6 and ~ N böó�� respectively. If�

is totally real and generic,

1Background about Holder classes appears in Section 1(IV).

20 JOEL MERKER AND EGMONT PORTEN)+*�,ÅÄ � N b and�

is called maximally real. We denote by Æ�. the integerbDó i� )+*�,ÇÄ�FÄv. � � d Ä�. � � and call it the holomorphic codimension of�at � . It is constant if and only if

�is CR. Holomorphic, totally real,

generic and Cauchy-Riemann submanifolds are all CR and have constantholomorphic codimensions Æ N i� � , Æ N �òó@b , Æ N 6 and Æ N �íó@b��È~respectively. Submanifolds of class ��3 � � or � � will be studied in Part III.

Let�

or be a real algebraic ( �»¿»�ÁÀ ) or analytic ( � ¤ ) submanifold of��

of (real) codimension � and let � A � � . There exist complex algebraicor analytic coordinates centered at � A and ��i ¢ 6 such that

�is locally

represented as follows.

Theorem 2.3. ([Ch1989, Bo1991, BER1999, Me2004a])2 If�

is holomorphic, letting ~ N b\ó i� �\�O6 and Æ � N i� � , then~e�@Æ N b and� N ý �µ ' É i � �±� �� ì ��"� � � É i N 6 � .2 If

�is totally real, letting �+i N �Ãb;ó � � 6 andÆ N �¶ó b � 6 , then �+i@� Æ N b and

� Ný É i ' É � � � � � K �� ì ��"� � � ª , É i N 6 '-É � N 6 � .2 If�

is generic, letting ~ N �íó@b , then ~ï�È� N b and� N ý �µ ' Éä� �±� �� ì � K� � �È% o K � � ª , É�N wz�µ 'ZÊµ ':Ë�yÌÉ � � 'for some

o K-valued algebraic or analytic map w satisfying wzØ6 ��N6 whose power series converges normally in � � � � � ì � � � � � ì � K� � � .2 If

�is Cauchy-Riemann, letting ~ N�Í¦Ë )+*�, � , Æ N ��óíb��Â~;�6 , and �+i N �ÃbKó �6~¶ó¨� ��6 , then ~ï�È�+i¥�@Æ N b and� N ý �µ ' É i ' É � � �±� �� ìFe � K �� ¨% o K � grì � "� � �ª , É i N w·im�µ 'ZÊµ ':Ë�yMÉ i � '\É � N 6 � '

for someo K � -valued algebraic or analytic map w�i satisfyingw·imØ6 ��N 6 whose power series converges normally in � � � � � ì � � � � � ì� K �� .

A further linear change of coordinates may yield �jwzØ6 �eN 6 and�vw�i�76 ��N 6 .A CR algebraic or analytic manifold being generic in some local com-

plex manifold of (smaller) dimension b�ó@Æ , called its intrinsic complexi-fication, in most occasions, questions, results and articles, one deals withgeneric manifolds. In this chapter, all generic submanifolds will be of pos-itive codimension �� = and of positive CR dimension ~ � = .


2.4. Algebraic and analytic generic submanifolds and theirextrinsic complexification. Let

�be generic, represented byª , É N wz�µ 'ZÊµ ':Ë�yMÉ � . The implicit function theorem applied to the

vectorial equation Î �ÏÎ� � N w e µ 'ZÊµ ' Î á ÏÎ� g , enables to solve the variablesÊÉ � � K, or the variables

É � � K. This yields the so-called complex

defining equations for�

, most useful in applications, as stated just below.Recall that, given a power series

ª p¢ �äN {(Ð |�} ~ ª Ð ¢ Ð , ¢r� � � ,ª Ð � � ,Ñ �öd � , one defines the series

ª p¢ � � N�{ Ð |m} ~ ª Ð ¢ Ð by conjugating only

its complex coefficients. Thenª P¢ �S ª Ê¢ � , a frequently used property.

Theorem 2.5. ([BER1999, Me2004a]) A local generic real algebraic oranalytic � -codimensional generic submanifold

� G � �� may be repre-sented by

ÊÉ N�Ò Êµ ' µ ' É � , or equivalently byÉ N Ò �µ 'ÓÊµ 'ÔÊÉä� , for some

complex algebraic or analytic� K

-valued mapÒ

whose power series con-verges normally in � � � � � ì � � � � � ì � K� � � , with ��i ¢ 6 . Here,

ÒandÒ

satisfythe two (equivalent by conjugation) vectorial functional equations:Õ ÊÉ�(Ò Êµ ' µ ' Ò �µ 'ZÊµ '\ÊÉä�Ï� 'É� Ò �µ 'ZÊµ 'BÒ Êµ ' µ ' É �Ï� kConversely, if such a

� K-valued map

Òsatisfies the above, the set

� � N� �µ ' Éä� �(� �� ÊÉ<N�Ò Êµ ' µ ' Éä� � is a real local generic submanifold ofcodimension � .

The coordinates =µ ' Éä� � � � ì�� K will also be denoted by ¢«� � � . LetÖ N �× '-ØÕ� � � � ì � K be new independent complex variables. Define theextrinsic complexification Ù N � � " of

�to be the complex algebraic

or analytic � -codimensional submanifold of��ìD��

defined by the vec-torial equation

Ø ó Ò =× ' ¢ �1N 6 (the mapÒ

being analytic, we may indeedsubstitute × for

Êµ in its power series). We also write Ö N ¢ � " . Observe that�identifies with the intersection Ù G � Ö N Ê¢³� .

Lemma 2.6. ([Me2004a, Me2005]) There exists an invertible � ì � matrixÊ»p¢ ' Ö � of algebraic or analytic power series converging normally in � � � � � ì� � � � � such thatÉ ó Ò �µ ' Ö �· ÊYp¢ ' Ö � q Ø ó Ò �× ' ¢ � t .

Thus, Ù is equivalently defined byÉ ó Ò �µ ' Ö ��N 6 .

2.7. Complexified Segre varieties and complexified CR vector fields.Let Ö . ' ¢F.��F� �� be fixed and define the complexified Segre varieties Ú\ÛnÜ


and the complexified conjugate Segre varieties Ú ° Ü by:Õ ÚMÛnÜ � N ý P¢ ' Ö � �±� �� ì � �� Ö N Ö . 'xÉ�N Ò =µ ' Ö . � � �HÈ+)Ú ° Ü � NÛý P¢ ' Ö � �±� �� ì � �� ¢ N ¢F. '\ØrN�Ò =× ' ¢F. �Ý�kGeometrically, Ú1Û Ü N Ù GD� Ö N Ö .�� and Ú ° Ü N Ù GD� ¢ N ¢F.²� . We drawa diagram.

A

þ ° ¹ ° Ü ÿÞ ß®àþ Û�¹�Û Ü ÿ

° Ü °áÛnÜâ á Þäã à

Geometry of the extrinsic complexification å

generic submanifold carries a pairThe complexification of a real analytic

integral submanifolds of the complexifiedof invariant foliations which are the^Ói �hA _ and ^ A�� i�_ vector fields and whichidentify also with the complexifiedSegre varieties

We warn the reader that)+*�,EæçÙ ó@)+*�,ÂæçÚMÛnÜ§óö)�*�,EæèÚ ° Ü N �� = 'so that the ambient codimension � of the unions of Ú\Û Ü and of Ú ° Ü is invis-ible in this picture; one should imagine for instance that Ù is the three-dimensional physical space equipped with a pair of foliations by horizontalorthogonal real lines.

Next, define two collections of complex vector fields:

STTTTTU TTTTTVé Ü � N QQ&µ Ü � Kù ��¹»i Q Ò �Q&µ Ü �µ ' × ',Ø�� QQ É � ' Ú N = 'lklklk ' ~ ' �ÃÈ+)é Ü � N QQ&× Ü � Kù �Ï¹»i Q Ò �Q&× Ü =× ' µ ' É � QQ Ø � ' Ú N = 'lklklkm' ~ k

One verifies thaté Ü e É ��ó Ò �u�µ ' × '-Ø�� g 6 , which shows that the

é Ü aretangent to Ù . Similarly,

é Ü Ø �xó Ò �Ã�× ' µ '-É �� 6 , so theé Ü are also

tangent to Ù . In addition, q é Ü ' é Ü Ò t N 6 and q é Ü ' é Ü Ò t N 6 for Ú ' ÚjÔ N= 'lklklkm' ~ , so the theorem of Frobenius applies. In fact, the ~ -dimensionalintegral submanifolds of the two collections

� é Ü �²i�� Ü � � and� é Ü �²i�� Ü � �

are the ÚMÛ Ü and the Ú ° Ü . In summary, Ù carries a fundamental pair offoliations.


Observe that the vector fieldsé Ü are the complexifications of the vector

fields R Ü � N ½½Bê z � { K�Ï¹»i ½ ëÓ�½mê z �µ 'ZÊµ '\ÊÉä�Ë½½ Î � , Ú N = 'lklk�k�' ~ , that generate the

holomorphic tangent bundle Ä i � A � . A similar observation applies to thevector fields

é Ü .In general (unless

�is Levi-flat), the total collection

� é Ü ' é Ü �²i�� Ü � �does not enjoy the Frobenius property. In fact, the noncommutativity ofthis system of �6~ vector fields is at the very core of Cauchy-Riemanngeometry.

To apply Theorem 1.13, introduce the “multiple” flows of the twocollections

� é Ü �²i�� Ü � � and� é Ü �²i�� Ü � � . If � � Ù has coordinates�µÖ. ' É . ' ×ê. '-Ø . � �� ì � K� � ì � �� ì � K� � satisfying

É . N Ò �µÖ. ' ×ê. '-Ø . �andØ . NìÒ =×Ï. ' µÖ. '-É . � and if µÃi � N �µÃi � i '�klklk�' µÃi � � � � � � is a small “mul-

titime” parameter, define the “multiple” flow ofé

by:

(2.8)

é ê � =µÖ. ' É . ' ×ê. '-Ø . � � NFy � Ç =µÃi é � Á� �� NFy � Ç =µÃi � i é im �� y � Ç �µÃi � � é � º� �Ï�� N e µÖ.x�@µÃi ' Ò �µÖ.«�ÈµÃi ' ×ê. '-Ø . � ' ×ê. '-Ø . g kOf course,

é ê � Á� � �íÙ . Similarly, for ��îÙ and ×±iz� � � , defining:

(2.9)é ï � �µÖ. ' É . ' ×Ï. ',Ø . � � N �µÖ. '-É . ' ×Ï.��È×±i 'BÒ =×Ï.��@×�i ' µÖ. ' É . �Ï� '

we haveé ï � Á� � �IÙ . Clearly, Á� ' µÃi � å¦ é ê � Á� � and Á� ' ×±i � å¦ é ï � Á� � are

complex algebraic or analytic local maps.

2.10. Segre chains. Let us start from � N 6 being the origin and movevertically along the complexified conjugate Segre variety Ú A of a heightµÃi��

, namely let us consider the pointé ê � 76 � , which we shall also

denote by ð i �µÃi � . We have ð i 76 �rN 6 . Let µ � � � � . Starting from thepoint ð i =µÃi � , let us move horizontally along the complexified Segre varietyof a length µ � � � � , namely let us consider the pointð � �µÃi ' µ � � � N é ê à é ê � 76 �Ï� kNext, define ð � =µÃi ' µ � ' µR� � � N é ê�ñ é ê à é ê � 76 �Ï�� , and thenð òÃ�µÃi ' µ � ' µR� ' µ ò � � N é êó é ê�ñ é ê à é ê � 76 �Ï��Ï� 'and so on. We draw a diagram:


ô

õ

ö

÷ ñ øGù:ú ñPûýü÷ à øGù ú à û ü÷ � øGù � ü

å ÷ ó øGù ú ó ûXüþ ~�ÿ þ ~

Segre chains in åBy induction, for every Ú �öd , Ú � =

, we obtain a local complex alge-braic or analytic map ð Ü =µÃi 'lklklkm' µ Ü � , valued in Ù , defined for sufficientlysmall µÃi 'lklklkm' µ Ü � � � which satisfies ð Ü Ø6 'lklklk ' 6 �N 6 . The abbreviatednotation µ�^ Ü _ � N =µÃi 'lklklkm' µ Ü � � � � Ü will be used. The map ð Ü is called theÚ -th conjugate Segre chain ([Me2004a, Me2005]).

If we had conducted this procedure by starting withé

instead of start-ing with

é, we would have obtained maps ð·i �µÃi � � N é ê � Ø6 � , ð � �µ ^ � _ � � Né ê à é ê � 76 �Ï� , etc., and generally ð Ü �µ<^ Ü _ � . The map ð Ü is called the Ú -th

Segre chain.There is a symmetry relation between ð Ü and ð Ü . Indeed, let � be the

antiholomorphic involution of� � ì��

defined by �·P¢ ' Ö � � N ÊÖ ' Ê¢ � . Sincewe have

ÉeN Ò �µ ' × '-ØÕ� if and only ifØKN Ò �× ' µ ' É � , this involution is a

bijection of Ù . Applying � to the definitions (2.8) and (2.9) of the flowsofé

and ofé

, one may verify that �· é ê � Á� ��xN é Ïê � ��º� �Ï� . It follows thegeneral symmetry relation � e ð Ü �µ ^ Ü _ � g N ð Ü e µ ^ Ü _ g . Thus, ð Ü and ð Ü havethe same behavior.

2.11. Minimality of Ù at the origin and complexified local CR orbits.Since ð Ü 76 ��N ð Ü Ø6 ��N 6 , for every integer Ú9� =

, there exists� Ü ¢ 6

sufficiently small such that ð Ü =µ ^ Ü _ � and ð Ü �µ ^ Ü _ � are well defined and be-long to Ù , at least for all µ�^ Ü _ð� � � Ü� z . To fiw ideas, it will be conve-nient to consider that � � Ü� z is the precise domain of definition of ð Ü andof ð Ü . We aim to apply the procedure of Theorem 1.13 to the system¯ Ar� Nïý é i '�klklk�' é � ' é i 'lklk�k�' é � � .

However, there is a slight (innocuous) difference: each multitime 8 NP8Öi 'lklk�k�' 8 � � n Ü had scalar components 8ê�� n , whereas now eachµ ^ Ü _ N �µÃi 'lk�klk�' µ Ü � � � � Ü has vectorial components µ�� . It is easyto see that both ð1i and ð i are of constant rank ~ . Also, both ð � and ð �


are of constant rank �<~ , sinceé i 'lklklkm' é � ' é i 'lk�klk�' é � are linearly inde-

pendent at the origin. However, when passing to (conjugate) Segre chainsof length � � , it is necessary to speak of generic ranks and to introducesome combinatorial integers ç Ü � =

. Justifying examples may be foundin [Me1999, Me2004a].

Theorem 2.12. ([BER1996, BER1999, Me1999, Me2001a, Me2004a])There exists an integer ] A with

= � ] A � � and, for Ú N ��'lklklkm' ] A � =,

integers ç Ü with= � ç Ü � ~ such that the following nine properties hold

true.

(1) For every Ú N �v'lklklkm' ] A � = , the two maps ð Ü and ð Ü are of genericrank equal to �6~Û� ç �·� �� ç Ü . In the special case ] A N =

, theç Ü are inexistent2 and nothing is stated.

(2) For every ÚK�(] A � = , both ð Ü and ð Ü are of fixed, stabilized genericrank equal to �<~�� ç , whereç � N ç �ã� �� ç [ r �F� k

(3) Setting Q A � N �Z] A � =, there exist two points µ �^ � r _ �� r�� r andµ � ^ � r _ �¨� � � r�� r satisfying ð � r �µ �^ � r _ �ÌN 6 and ð � r �µ � ^ � r _ �N 6 which

are arbitrarily close to the origin in � � � r�� r such that ð � r and ð � rare of constant rank �<~¶� ç in neighborhoods �¦� and � � of µZ�^ � r _and of µ � ^ � r _ . The images ð � r p�� and ð � r p� � � then constitute twopieces of local n -algebraic or analytic submanifold of dimension�<~e� ç contained in Ù .

(4) Both ð � r n� � � and ð � r p� � � enjoy the most important property thatall vector fields

é i '�klklk�' é � ' é i 'lklk�k�' é � are tangent to ð � r p� � �and to ð � r n� � � .

(5) ð � r p�� and ð � r p� � � coincide together in a neighborhood of 6 inÙ .

(6) Denoting by � á � á =Ù ' 6 �this common local piece of complex analytic submanifold of Ù , itis algebraic provided that the flows of

ý é i 'lklklkm' é � ' é i '�klklk�' é � �are themselves algebraic.

2One may set �� and �� in any case.


(7) Every local complex analytic or algebraic submanifold � � Ùpassing through the origin to which

é i 'lklklkm' é � ' é i '�klklk�' é � areall tangent must contain � á � á PÙ ' 6 � in a neighborhood of the ori-gin.

(8) The integers ] A , ç � 'lklk�k�' ç � r and ç are biholomorphic invariants ofÙ .

(9) ð � r p�� and ð � r p� � � also coincide (in a neighbor-hood of the origin) with the Nagano leaf of the systemý é i 'lklklkm' é � ' é i '�klklk�' é � � , as it was constructed in Theorem 1.5.

As in [Me2004a, Me2005] (with different notations), the integer ] A willbe called the Segre type of

�.

The “orbit notation” � á � á PÙ ' 6 � anticipates the presentation and the no-tation of Section 1(III). We will abandon Lie brackets and Nagano leaves.

The complex vector fields R Ü � N ½½Bê z � { K�Ï¹»i ½ ëÓ�½mê z �µ 'ZÊµ '\ÊÉä�B½½ Î � , Ú N= 'lklklkm' ~ , are tangent to�

of equationsÉ � N Ò �u�µ 'ZÊµ '\ÊÉä� , � N = '�klklk ' � ;

their conjugates R Ü are also tangent to�

; it follows that the real and imag-inary parts

Ë�y R Ü andª ,îR Ü are also tangent to

�. We may then apply

Theorem 1.13 to the system�6Ë�y R Ü ' ª ,fR Ü �²i�� Ü � � , getting a certain real

analytic local submanifold �� ¥ �(' 6 � of�

passing through the origin.It will be called the local CR orbit of the origin in

�(terminology of

Part III).The relation between � á � á =Ù ' 6 � and � � � � �(' 6 � is as follows

([BER1996, Me1999, Me2001a, Me2004a]). Let � ° p¢ ' Ö � � N ¢ and�kÛjP¢ ' Ö � � N Ö denote the two canonical projections associated to theproduct � �� ì � �� . Let ` � N ý P¢ ' Ö � � � �� ì � �� Ö N Ê¢ � be theantiholomorphic diagonal. Observe that � ° ` G Ù �·N �

.2 The extrinsic complexification Ý �� V ��' 6 � Þ " N � á � á =Ù ' 6 � .2 The projection � ° e ` G � á � á PÙ ' 6 �hg N � � � � �(' 6 � .Concerning smoothness, a striking subtelty happens: if

�is real alge-

braic, although the local multiple flows ofé

and ofé

are complex alge-braic (thanks to their definitions (2.8) and (2.9)), the flows of

Ë�y R Ü and ofª ,9R Ü are only real analytic in general.

Example 2.13. ([Me2004a]) For the real algebraic hypersurface of� � de-

fined byª , ÉeN · = �@µ Êµ�ó = , the vector field R � N ½½Bê �>% Êµ/· = �Èµ Êµ ½½ Îgenerates Ä i � A � and the flow of � Ë�y R involves the transcendent function�� .


Theorem 2.14. ([BER1996, Me2001a]) The local CR orbit � � � � �(' 6 � isreal algebraic if

�is.

For the proof, assuming�

to be real algebraic, it is impossible, becauseof the example, to apply the second phrase of Theorem 1.13 (5) to the sys-tem

�6Ë�y R Ü ' ª ,îR Ü �²i�� Ü � � . Fortunately, this phrase applies to the complex-ified system

� é Ü ' é Ü �²i�� Ü � � , whence � á � á PÙ ' 6 � is algebraic, and thenthe local CR orbit � � � � �(' 6 ��N � ° ` G � á � á =Ù ' 6 �� is real algebraic.

Definition 2.15. The generic submanifold�

or its extrinsic complexifica-tion Ù is said to be minimal at the origin if � � � � ��' 6 � contains a neigh-borhood of 6 in

�, or equivalently if � á � á PÙ ' 6 � contains a neighborhood

of 6 in Ù .

The minimality at the origin of the algebraic or analytic complexifiedlocal generic submanifold Ù N �B� " is a biholomorphically invariantproperty; it neither depends on the choice of defining equations nor on thechoice of a conjugate pair of systems of complex vector fields

� é Ü �²i�� Ü � �and

� é Ü �²i�� Ü � � spanning the tangent space to the two foliations.Minimality at 6 reads ç N � in Theorem 2.12. For a hypersurface

�,

namely with � N =, minimality at 6 is equivalent to ] A N � .

2.16. Projections of the submersions ð � r and ð � r . Let Q A N �Z] A � = asin Theorem 2.12. If Ù is minimal at the origin, the two local holomorphicmaps ð � r �HÈ+) ð � r � � � � r�� r ó ¦ Ùsatisfy ð � r �µZ�^ � r _ �zN 6 and ð � r �µ � ^ � r _ �xN 6 and they are submersive at µ �^ � r _and at µ � ^ � r _ .

Consider the two projections � ° P¢ ' Ö � � N ¢ and �kÛ�p¢ ' Ö � � N Ö andfour compositions � ° e ð � r �µ ^ � r _ � g , � ° e ð � r �µ ^ � r _ � g and �kÛ e ð � r �µ ^ � r _ � g ,�kÛ e ð � r �µ ^ � r _ � g . Since Q A N �Z] A � =

is odd, observe that the composi-tion ð � [ r á i N é �� ends with a

éand that ð � [ r á i N é �� ends with

aé

. According to the two definitions of the flow maps, the coordinates�×Ï. ',Ø . � are untouched in (2.8) and the coordinates =µ . ' É . � are untouchedin (2.9). It follows thatÕ � ° e ð � [ r á i �µ<^ � [ r á i�_ �hgî � ° e ð � [ r �µ<^ � [ r _ �hg �HÈ+)�kÛ e ð � [ r á i��µ<^ � [ r á i�_ � g �kÛ e ð � [ r =µ<^ � [ r _ � g kCorollary 2.17. ([Me1999, BER1999, Me2004a]) If

�is minimal at the

origin, there exists a integer ] A �(�S� = (the Segre type of�

at the origin)


and there exist points µ � ^ � [ r _ � � �� [ r

and µZ�^ � [ r _ � � �� [ r

arbitrarily closeto the origin, such that the two mapsÕ � � � [ r� à! r £ µ ^ � [ r _ å ó ¦ � ° e ð � [ r �µ ^ � [ r _ � g � � � �HÈ+)� � � [ r� à! r £ µ<^ � [ r _Kåó ¦ �kÛ e ð � [ r �µ<^ � [ r _ �,g � � �are of rank b and send µ � ^ � [ r _ and µ �^ � [ r _ to the origin.a

3. FORMAL CR MAPPINGS, JETS OF SEGRE VARIETIES

AND CR REFLECTION MAPPING

3.1. Complexified CR mappings respect pairs of foliations. Let b Ô ��dwith b Ô � =

and let� Ô �¶� � Ò

be a second algebraic or analytic genericsubmanifold of codimension ��Ô�� = and of CR dimension ~KÔ N b»Ôhó��²ÔY� = .Let �+Ôr� � Ô . There exist local coordinates ¢êÔ N �µuÔ ' É Ô � � � � Ò ì � K Òcentered at � Ô in which

� Ô is represented byÊÉ Ô N�Ò Ô Êµ Ô ' ¢ Ô � , or equivalently

byÉ Ô N Ò Ô =µÃÔ ' ¢ Ô � . If ¢ Ô � " N Ö Ô N =×ÕÔ '-Ø Ô � � � � Ò ì?� K Ò

, the extrinsiccomplexification is represented by

Ø Ô NìÒ ÔF=×ÕÔ ' ¢�Ô � , or equivalently byÉ Ô NÒ Ô �µ Ô ' Ö Ô � . We shall denote by 6 Ô the origin of

� � Ò .Let ¢��

and let ��p¢ �\N �»imP¢ ��'lklklkm' � � Ò P¢ �Ï� � � q qG¢�t t � Ò be a formalpower series mapping with no constant term, i.e. ��76 �òN 6ÕÔ ; it may alsobe holomorphic namely ��p¢ � � �«� ¢³� � Ò , or even (Nash) algebraic. We have ��P¢ �Ö� " N �4Ï ¢ � " �·N �� Ö � . Define �&" p¢ ' Ö � � N ��p¢ � ' �� Ö �� .

Setµ p¢ ' Ö � � N«Ø ó Ò �× ' ¢ � , set

µ Ö ' ¢ � � N É ó Ò =µ ' Ö � , setµ ÔFp¢�Ô ' Ö Ô � � NØ Ô ó Ò Ô �× Ô ' ¢ Ô � and set

µ Ô Ö Ô ' ¢ Ô � � N©É Ô ó Ò Ô =µ Ô ' Ö Ô � . We say that the powerseries mapping � is a formal CR mapping from �(' 6 � to � Ô ' 6uÔ � if thereexists a �ÕÔ ì � matrix of formal power series "Ãp¢ ' Ê¢ � such thatµ Ô e ��P¢ ��' �� Ê¢ �,g "uP¢ ' Ê¢ ��µ P¢ ' Ê¢ �in� q q$¢ ' Ê¢�t t K Ò . By complexification, it follows that

µ Ô e ��P¢ ��' �� Ö �,g "Ãp¢ ' Ö ��µ P¢ ' Ö � in� q qG¢ ' Ö t t K Ò , namely �&"�p¢ ' Ö �DN ��P¢ ��' �� Ö �� maps =Ù ' 6 �

formally to =Ù;Ô ' 6uÔ � . By Lemma 2.6, there exist two complex analyticinvertible matrices ÊYP¢ ' Ö � and ÊÕÔÃP¢�Ô ' Ö Ô � satisfying :Õ µ P¢ ' Ö �� ÊYP¢ ' Ö � µ Ö ' ¢ ��' µ Ô P¢ Ô ' Ö Ô �� Ê Ô n¢ Ô ' Ö Ô � µ Ô Ö Ô ' ¢ Ô �V'µ Ö ' ¢ �� ÊY Ö ' ¢ ��µ p¢ ' Ö ��' µ Ô Ö Ô ' ¢ Ô �· Ê Ô Ö Ô ' ¢ Ô �zµ Ô P¢ Ô ' Ö Ô �V'in� q qG¢ ' Ö t t K and in

� q qG¢�Ô ' Ö Ôst t K Ò . So, to define a complexified formal CR map-ping � "�� PÙ ' 6 � å¦$# =Ù Ô ' 6 Ô � , we get four vectorial formal identities,


each one implying the remaining three:Õ µ Ô e ��P¢ ��' �� Ö � g "uP¢ ' Ö ��µ P¢ ' Ö � ' µ Ô e ��P¢ ��' �� Ö � g ÆÃ Ö ' ¢ � µ Ö ' ¢ ��'µ Ô e �� Ö � ' ��p¢ � g "u Ö ' ¢ � µ Ö ' ¢ � ' µ Ô e �� Ö � ' ��p¢ � g ÆÃP¢ ' Ö ��µ p¢ ' Ö ��kHere, we have set ÆHp¢ ' Ö � � N "u Ö ' ¢ � Ê»p¢ ' Ö � .

These identities are independent of the choice of local coordinates andof local complex defining equations for ��' 6 � and for � Ô ' 6uÔ � . Since �is not a true point-map, we write � � �(' 6 � ¦%# � Ô ' 6uÔ � , the index &being the initial of Formal. If � is convergent, it is a true point-map from aneighborhood of 6 in

�to a neighborhood of 6²Ô in � Ô .

ø('<ø ö ü*)*+'Zø õ üpüô Ò

õ Ò

ö Ò

å Ò

ô

õ

ö

å '-,/. ø(' )0+' ü1 � ^ ê � _ 1 à ^ ê ú à û _

1 ñ ^ ê ú ñPû _2 2 Ò

2 Ò2 þ ~�3 ÿ þ ~�3þ ~ ÿ þ ~

Complexified formal CR mappings respect pairs of foliations

If � is holomorphic in a polydisc � �� , �+i ¢ 6 , its extrinsic complex-ification �+" sends both the b -dimensional coordinate spaces

� ¢ N �-�54 k �and

� Ö N �-�54 k � to the b Ô -dimensional coordinate spaces� ¢ Ô N �-�64 k � and� Ö Ô N �-�54 k � .

Equivalently, �&" maps complexified (conjugate) Segre varieties of thesource to complexified (conjugate) Segre varieties of the target. Somestrong rigidity properties are due to the fact that � " N �� ' Ê� � must respectthe two pairs of Segre foliations.

The most important rigidity feature, called the reflection principle3, saysthat the smoothness of

�,� Ô governs the smoothness of � :2 suppose that

�and

� Ô are real analytic and that ��P¢ � � � q q ¢�t t � Ò isonly formal; statement: under suitable assumptions, ��P¢ � � �z� ¢³� � Òis in fact convergent.

3Other rigidity phenomena are: parametrization of CR automorphism groups by a jetof finite order, finiteness of their dimension, genericity of nonalgebraizable CR submani-folds, genericity of CR submanifolds having no infinitesimal CR automorphisms, etc.

30 JOEL MERKER AND EGMONT PORTEN2 suppose that�

and� Ô are real algebraic and that �4p¢ � � � q q ¢�t t � Ò is

only formal; statement: under suitable assumptions, ��p¢ � is com-plex algebraic.

After a mathematical phenomenon has been observed in a special, wellunderstood situation, the research has to focus attention on the finest, themost adequate, the necessary and sufficient conditions insuring it to holdtrue.

In this section, we aim to expose various possible assumptions forthe reflection principle to hold. Our goal is to provide a synthesisby gathering various nondegeneracy assumptions which imply reflection.For more about history, for other results, for complements and for dif-ferent points of view we refer to [Pi1975, Le1977, We1977, We1978,Pi1978, DF1978, DW1980, DF1988, BR1988, BR1990, DP1993, DP1995,DP1998, BER1999, Sh2000, BER2000, Me2001a, Me2002, Hu2001,Sh2003, DP2003, Ro2003, MMZ2003b, ER2004, Me2005].

The main theorems will be presented ina3.19 and in

a3.22 below, af-

ter a long preliminary. In these results,�

will always be assumed to beminimal at the origin. Corollary 2.17 says already how to use concretelythis assumption: to show the convergence or the algebraicity of a formalCR mapping � � �(' 6 � å¦%# � Ô ' 6 Ô � , it suffices to establish that for everyÚË�ed , the formal maps µ�^ Ü _�åó ¦$# � e � ° e ð Ü =µ<^ Ü _ � g�g are convergent oralgebraic.

Before surveying recent results about the reflection principle (withoutany indication of proof), we have to analyze thoroughly the geometry ofthe target Ù;Ô and to present the nondegeneracy conditions both on Ù Ô andon � . Of course, everything will also be meaningful for sufficiently smooth( � � or � 3 ) local CR mappings, by considering Taylor series.

These conditions are classical in local analytic geometry and they mayalready be illustrated here with a plain formal map ��P¢ � � � q qG¢�t t � Ò , notnecessarily being CR.

Definition 3.2. A formal power series mapping � � � � ' 6 � å¦$# � � Ò ' 6 Ô �with components �+� Ò p¢ � � � q qG¢�t t , %7Ô N = 'lk�klk�' bYÔ , is called

(1) invertible if b»Ô N b and ) y 4®�qÙQ�� H Q�¢�� à t#76 �� i�� #� � à � � ëN 6 ;(2) submersive if bË�<bVÔ and there exist integers

= ��%Ö = � 8 �� 8%³Fb»Ô � �9b such that ) y 4®�qÙQ�� Ò � H Q�¢ �ý^!� Ò à _ t#76 �Ï� i�� Ò � � � Ò à � � Ò ëN 6 ;(3) finite if the ideal generated by the components ��imP¢ ��'lklklkm' � � Ò P¢ � is

of finite codimension in� q q ¢�t t ; this implies b¥ÔY�9b ;


(4) dominating if bË�<b»Ô and there exist integers= � %Ö = � 8 �� 8%³Fb»Ô � �9b such that ) y 4u�qÙQ � � Ò � H Q�¢ �ý^!� Ò à _ t�p¢ �� i�� Ò � � � Ò à � � Ò ë 6 in

� q q ¢�t t ;(5) transversal if there does not exist a nonzero power series7 P¢ Ô i 'lk�klk�' ¢ Ô� Ò � � � q q$¢ Ô i 'lklk�k�' ¢ Ô� Ò t t such that

7 ��VimP¢ ��'lklklkm' � � Ò P¢ �Ï�« 6in� q q$¢�t t .

It is elementary to see that invertibility implies submersiveness whichimplies domination. Furthermore, if a formal power series is either invert-ible, submersive or dominating, then it is transversal. Philosophically, the“distance” between finite and dominating or transversal is large, whereasthe “distance” between invertible and submersive or finite is “small”.

3.3. Jets of Segre varieties and Segre mapping. The target� Ô concen-

trates all geometric conditions that are central for the reflection principle.With respect to ÙôÔ , the complexified conjugate Segre variety associatedto a fixed ¢�Ô is Ú Ô° Ò � N � =×²Ô '-Ø Ô � � �1� Ò � Ø Ô N Ò ÔF�×²Ô ' ¢�Ô � � . Here, ×ÕÔ is aparametrizing variable. For Ú�Ôx�(d , define the morphism of ÚvÔ -th jets ofcomplexified conjugate Segre varieties by:

w Ô Ü Ò =× Ô ' ¢ Ô � � N d Ü ÒÛ Ò Ú Ô° Ò � N 8 × Ô ':9 =; Ô Q=< Òï Ò Ò Ô� Ò �× Ô ' ¢ Ô �6> i��<� Ò � K Ò � < Ò |�}@? Ò � ¡ < Ò ¡ � Ü ÒBA kIt takes values in

� � Ò á i=C Ò l ? Ò l z Ò , with × K Ò � � Ò � Ü Ò4� N � Ô ^ � Ò á Ü Ò _no� Ò o Ü Ò o . If Ú Ô i � Ú Ô� , wehave of course � Ü Òà � Ü Ò � � w Ô Ü Òà N w Ô Ü Ò � .As observed in [DW1980], the properties of this morphism govern thevarious reflection principles. We shall say ([Me2004a, Me2005]) that

� Ô(or equivalently ÙôÔ ) is:

(nd1) Levi non-degenerate at the origin if wãÔ i is of rank ~KÔ��íbYÔ at �×²Ô ' ¢�Ô ��N76uÔ ' 6ÃÔ � ;(nd2) finitely nondegenerate at the origin if there exists an integer � Ô A such

that w4ÔÜ Ò is of rank b»Ô��>~ðÔ at =×ÕÔ ' ¢�Ô �ÌN 76ÃÔ ' 6ÃÔ � , for ÚjÔ N � Ô A , hencefor all Ú Ô �J� Ô A ;

(nd3) essentially finite at the origin if there exists an integer � Ô A such thatw Ô Ü Ò is a finite holomorphic map at �× Ô ' ¢ Ô ��N Ø6 Ô ' 6 Ô � , for Ú Ô N � Ô A ,hence for all ÚjÔY�?� Ô A ;

(nd4) Segre nondegenerate at the origin if there exists an integer �lÔA suchthat the restriction of w®ÔÜ Ò to the complexified Segre variety ÚxÔA (ofcomplex dimension ~ Ô ) is of generic rank ~ Ô , for Ú Ô N � Ô A , hencefor all ÚjÔY�J� Ô A ;


(nd5) holomorphically nondegenerate if there exists an integer �lÔA suchthat the map w4ÔÜ Ò is of maximal possible generic rank, equal to ~ ÔÏ�b Ô , for Ú Ô N � Ô A , hence for all Ú Ô �J� Ô A .

Theorem 3.4. ([Me2004a]) These five conditions are biholomorphicallyinvariant and: (nd1) D (nd2) D (nd3) D (nd4) D (nd5).

Being not punctual, the last condition (nd5) is the finest: as every con-dition of maximal generic rank, it propagates from any small open subet tobig connected open sets, thanks to the principle of analytic continuation.Notably, if a connected real analytic

� Ô is holomorphically nondegenerate“at” a point, it is automatically holomorphically nondegenerate “at” everypoint ([St1996, BER1999, Me2004a]).

To explain the (crucial) biholomorphic invariance of the jet map w1ÔÜ Ò ,consider a local biholomorphism ¢ Ô å¦ � Ô p¢ Ô �xN ¢ Ô Ô , where ¢ Ô ' ¢ Ô Ô � � � Ò , thatfixes the origin, � Ô� Ò p¢ Ô � � �z� ¢ Ô � , � Ô� Ò Ø6 Ô �ÌN 6 Ô , for % Ô N = 'lklk�k�' b Ô . Splittingthe coordinates ¢ Ô Ô N =µ Ô Ô ' É Ô Ô � � � � Ò ì � K Ò , the image

� Ô Ô may be similarlyrepresented by

ÊÉ Ô Ô N Ò Ô Ô Êµ Ô Ô ' ¢ Ô Ô � and there exists a � Ô ì � Ô matrix " Ô P¢ Ô ' Ö Ô �of local holomorphic functions such thatµ Ô Ô e � Ô P¢ Ô ��' � Ô Ö Ô � g " Ô P¢ Ô ' Ö Ô ��µ Ô P¢ Ô ' Ö Ô �in�z� ¢¼Ô ' Ö Ôh� K Ò , where

µ Ô�êÒ P¢�Ô ' Ö Ô � � N�Ø Ô�¼Ò ó Ò Ô�êÒ �×²Ô ' ¢�Ô � andµ Ô Ô�¼Ò p¢�Ô Ô ' Ö Ô Ô � � NìØ Ô Ô�êÒ óÒ Ô Ô�¼Ò =×ÕÔ Ô ' ¢�Ô Ô � , for �²Ô N = '�klklk�' �ÕÔ . Setting ��ÔFP¢�Ô � � N 7>YÔFp¢�Ô ��' ¡�Ô p¢�Ô �� z� ¢�Ô#� � Ò ì�«� ¢�Ô�� K Ò and replacing

Ø Ô byÒ Ô �×²Ô ' ¢�Ô � in the above equation, the right hand

side vanishes identically (sinceµ Ô P¢�Ô ' Ö Ô ��NFØ Ô�ó Ò ÔØ=×ÕÔ ' ¢�Ô � by definition) and

we obtain the following formal identity in�z� ×jÔ ' ¢�Ô�� K Ò :¡ Ô �× Ô 'BÒ Ô �× Ô ' ¢ Ô �Ï��ìÒ Ô Ô e > Ô =× Ô 'BÒ Ô �× Ô ' ¢ Ô ��' � Ô P¢ Ô � g k

Some algebraic manipulations conduct to the following.

Lemma 3.5. ([Me2004a, Me2005]) For every ��Ô N = 'lklk�k�' �²Ô and every; Ô �pd � Ò , there exists a universal rational map E Ô�¼Ò � < Ò whose expressiondepends neither on ÙôÔ , nor on �+Ô , nor on ÙôÔ Ô , such that the followingidentities in

�«� ×²Ô ' ¢�Ôh� hold true :FG ÔIHKJ ¡ < Ò ¡�L Ô Ô� ÒJNM!O Ô ÔQP < ÒSR T Ô M!O ÔVU L Ô M!O ÔWUYX Ô PYP U[ZÕÔ M X7Ô P0\:]]_^ Ô� Ò � < ÒN` R J < Ò �ï Ò L Ô� Ò� M!O ÔWUYX Ô P0\ i��<� Ò� � K Ò � ¡ < Ò � ¡ � ¡ < Ò ¡ U R J � Ò �ÛÖÒ Z Ô� Ò � M!O ÔWU L Ô M!O ÔWUYX Ô PYP0\ i�� Ò � � � Ò � ¡ � Ò � ¡ � ¡ < Ò ¡ ab�ced Ô� Ò � < Ò ` O Ô U R J < Ò �ï Ò L � Ò� M!O Ô UYX Ô P0\ i��<� Ò � � K Ò � ¡ < Ò � ¡ � ¡ < Ò ¡ a U


where the last line defines f Ô� Ò � < Ò by forgetting the jets of � Ô . Here, the E�Ô� Ò � < Òare holomorphic in a neighborhood of the constant jet

û ØQ < Ò �ï Ò Ò Ô� Ò� Ø6 ' 6 �Ï� i��<� Ò� � K Ò � ¡ < Ò � ¡ � ¡ < Ò ¡ ' 7Q � Ò �Û Ò � Ô� Ò � 76 ' 6 �� i�� Ò � � � � ¡ � Ò � ¡ � ¡ < Ò ¡ ü kSome symmetric relations hold after replacing

Ò Ô , Ò Ô Ô , ×ÕÔ , ¢�Ô , > Ô , ��Ô byÒ Ô ,Ò Ô Ô , µÃÔ , Ö Ô , >YÔ , � Ô .

The existence of fäÔ� Ò � < Ò says that the following diagram is commutative :

=Ù;Ô ' 6ÃÔ � ^hg Ò _ ,//i z Òj Þ Ò j

��

=Ù;Ô ' 6ÃÔ �i z Òj Þ Ò Òj�� á i=C Ò l ? Ò l z Ò % Ò z Ò ^$^kg Ò _ , _// � � á i/C Ò l ? Ò l z Ò '

where the biholomorphic map f ÔÜ Ò �� Ô � " � , which depends on �� Ô � " , is de-fined by its components f Ô� Ò � < Ò for �²Ô N = 'lk�klk�' �²Ô and

J ; Ô J �OÚjÔ . Thanks tothe invertibility of �+Ô , the map fäÔÜ Ò ��Ô � " � is also checked to be invertible,and then the invariance of the five nondegeneracy conditions (nd1), (nd2),(nd3), (nd4) and (nd5) is easily established ([Me2004a]).

We now present the Segre mapping of� Ô . By developing the seriesÒ Ô� Ò �× Ô ' ¢ Ô � in powers of × Ô , we may write the equations of Ù Ô under the

formØ Ô� Ò NÛ{ Ð Ò |�} ? Ò =× Ô � Ð Ò Ò Ô� Ò � Ð Ò p¢ Ô � for � Ô N = '�klklk�' � Ô . In terms of such a

development, the infinite Segre mapping of� Ô is defined to be the map-

ping l Ô � � � � Ò £ ¢ Ô å ó ¦ Ò Ô�êÒ � Ð Ò P¢ Ô �Ï� i��<� Ò � K Ò � Ð Ò |m}m? Ò � � � kLet ÚjÔ��5d . For finiteness reasons, it is convenient to truncate this infinitecollection and to define the Ú Ô -th Segre mapping of

� Ô byl Ô Ü Ò � � � Ò £ ¢ Ô å ó ¦ Ò Ô� Ò � Ð Ò P¢ Ô �� i��<� Ò � K Ò � ¡ Ð Ò ¡ � Ü ÒV� � i C Ò l ~ Ò l z Ò 'where × K Ò � � Ò � Ü Ò N �²Ô ^ � Ò á Ü Ò _no� ÒGo Ü Ò$o . If ÚjÔ� ��ÚjÔi , we have � Ü Òà � Ü Ò � Ý l Ô Ü Òà P¢�Ô � Þ N l Ô Ü Ò � P¢�Ô � .One verifies ([Me2004a]) the following characterizations.

(nd1)� Ô is Levi non-degenerate at the origin if and only if

l Ô i is of rankb»Ô at ¢�Ô N 6uÔ .(nd2)

� Ô is finitely nondegenerate at the origin if and only if there existsan integer �mÔA such that

l Ô Ü Ò is of rank b»Ô at ¢�Ô N 6ÃÔ , for all ÚvÔY�J� Ô A .(nd3)

� Ô is essentially finite at the origin if there exists an integer ��ÔA suchthat

l Ô Ü Ò is a finite holomorphic map at ¢êÔ N 6uÔ , for all ÚjÔY�J� Ô A .


(nd4)� Ô is Segre nondegenerate at the origin if there exists an integer �lÔAsuch that the restriction of

l Ô Ü Ò to the complexified Segre variety ÚxÔA Ò(of complex dimension ~ Ô ) is of generic rank ~ Ô , for all Ú Ô �J� Ô A .(nd5)

� Ô is holomorphically nondegenerate if there exists an integer �lÔAsuch that the mapl Ô Ü Ò is of maximal possible generic rank, equal

to b Ô , for all Ú Ô �J� Ô A .3.6. Essential holomorphic dimension and Levi multitype. Assumenow that

� Ô is not nececessarily local, but connected. Denote by ��ÔL Ò thesmallest integer ÚvÔ such that the generic rank of the jet mappings P¢êÔ ' Ö Ô � å¦d¥Ü ÒÛÖÒ Ú Ô° Ò does not increase after Ú Ô and denote by ~ Ô ��b ÔL Ò � ~ Ô �9b Ô the

(maximal) generic rank of p¢êÔ ' Ö Ô � å¦ d � Ò n ÒÛ Ò Ú Ô° Ò . SinceÉ Ô&å¦ Ò Ô �×²Ô ' µuÔ ' É Ô � is

of rank �²Ô according to Theorem 2.5, the (generic) rank of the zero-th orderjet map satisfieso y È ��p æ e P¢ Ô ' Ö Ô � å¦ d AÛ Ò Ú Ô° Ò N �× Ô 'BÒ Ô �× Ô ' µ Ô '-É Ô ��,g N ~ Ô �@� Ô N b Ô kThus, �ÕÔS�Ûb»ÔL Ò �ÛbYÔ . It is natural to call bVÔL Ò the essential holomorphicdimension of

� Ô because of the following.

Proposition 3.7. ([Me2001a, Me2004a]) Locally in a neighborhood of aZariski-generic point �&Ô®� � Ô , the generic submanifold

� Ô is biholomor-phically equivalent to the product

� Ô. Ò ì � � Ò � Ò n Ò , of a generic submanifold� Ô.�Ò of codimension �ÕÔ in � � Ò n Ò by a complex polydisc � � Ò � Ò n Ò .Generally speaking, we may define qYÔA�� L Ò � N o y È �6p æ �P¢�Ô ' Ö Ô � å¦ d AÛ Ò Ú Ô° Ò � ó~ Ô N � Ô and for every Ú Ô N = 'lklklk ' � Ô L Ò ,q Ô Ü Ò � L Ò � N o y È �6p æ û P¢ Ô ' Ö Ô � å¦ d Ü ÒÛ Ò Ú Ô° Ò ü ó o y È ��p æ û P¢ Ô ' Ö Ô � å¦ d Ü Ò iÛÖÒ Ú Ô° Ò ü k

One verifies ([Me2004a]) that q�Ôi � L Ò � = 'lklklk ' q+Ô� Ò n Ò � L Ò � =. With these

definitions, we have the relationso y È ��p æ û p¢ Ô ' Ö Ô � å¦ d Ü ÒÛÖÒ Ú Ô° Ò ü N ~ Ô �rq Ô A�� L Ò �rq Ô i � L Ò � �� rq Ô Ü Ò � L Ò 'for ÚjÔ N 6 ' = 'lk�klkm' �mÔL Ò ando y È �6p æ û P¢ Ô ' Ö Ô � å¦ d Ü ÒÛ Ò Ú Ô° Ò ü N ~ Ô �î� Ô �sq Ô i � L Ò � �� tq Ô� Ò n Ò � L Ò N ~ Ô �\b Ô L Ò 'for all ÚjÔ»�J� Ô L Ò . It follows that� Ô L Ò �uq Ô i � L Ò � �� rq Ô� Ò n Ò � L Ò N b Ô L Ò ó¨� Ô �?~ Ô k


Theorem 3.8. ([Me2004a]) Let� Ô be a connected real algebraic or ana-

lytic generic submanifold in�·� Ò

of codimension �ÕÔ�� =and of CR dimen-

sion ~ Ô N b Ô ó?� Ô � =. Then there exist well defined integers b Ô L Ò �©� Ô ,�mÔL Ò �p6 , q&ÔA�� L Ò � =

, q&Ô i � L Ò � =,klklk

, q&Ô� Ò n Ò � L Ò � =and a proper real alge-

braic or analytic subvariety v Ô of� Ô such that for every point � Ô � � Ô Awv Ô

and for every system of coordinates =µ²Ô ' É Ô � vanishing at �+Ô in which� Ô is

represented by defining equationsÊÉ � Ò N�Ò Ô�¼Ò ÊµÃÔ ' ¢�Ô � , �²Ô N = 'lklk�km' �²Ô , then the

following four properties hold:2 q+ÔA�� L Ò N �²Ô , �ÕÔ��b»ÔL Ò �9b»Ô and �mÔL Ò �9bYÔL Ò ó �²Ô .2 For every ÚvÔ N 6 ' = 'lklk�km' � Ô L Ò , the mapping of ÚjÔ -th order jets of theconjugate complexified Segre varieties P¢êÔ ' Ö Ô � å¦ d Ü ÒÛ Ò Ú Ô° Ò is of rankequal to ~KÔH�%q&ÔA�� L Ò � �� %q&ÔÜ Ò � L Ò at P¢�Ô. Ò ' Ê¢�Ô. Ò �·N Ø6ÃÔ ' 6ÃÔ � .2 b»ÔL Ò N �ÕÔ&�xq&Ô i � L Ò � �� xq+Ô� Ò n Ò � L Ò and for every ÚvÔr� �mÔL Ò , the

mapping of Ú Ô -th order jets of the conjugate complexified Segre va-rieties p¢�Ô ' Ö Ô � å¦ d Ü ÒÛÖÒ Ú Ô° Ò is of rank equal to bVÔL Ò at 76ÃÔ ' 6ÃÔ � .2 There exists a local complex algebraic or analytic change of co-ordinates ¢�Ô Ô N ��ÔØp¢�Ô � fixing �+Ô such that the image

� Ô Ô. Ò � N �+Ô � Ô �is locally in a neighborhood of �&Ô the product

� Ô Ô. Ò ì � � Ò � Ò n Ò ofa real algebraic or analytic generic submanifold of codimension�²Ô in

� � Ò n Ò by a complex polydisc � � Ò � Ò n Ò . Furthermore, at thecentral point � Ô � � Ô Ô. Ò � � � Ò n Ò , the generic submanifold

� Ô Ô. Ò is� Ô L Ò -finitely nondegenerate, hence in particular its essential holo-morphic dimension bVÔL Ò ÒÜ Ò coincides with b»ÔL Ò .

In particular,� Ô is holomorphically nondegenerate if and only if b Ô L Ò Nb»Ô and in this case,

� Ô is finitely nondegenerate at every point of theZariski-open subset

� Ô$Awv�Ô .3.9. CR-horizontal nondegeneracy conditions. As in

a3.1, let � N�4p¢ � � � q qG¢�t t � Ò be a formal CR mapping �(' 6 � ¦r# � Ô ' 6 Ô � . De-

compose �4p¢ ��N >1p¢ � ' ¡VP¢ �Ï� � � q q ¢�t t � ÒrìO� q q ¢�t t K Ò , as in the splitting¢ Ô N �µ Ô ' É Ô � � � � Ò ìf� K Ò. Replacing

ÉbyÒ =µ ' Ö � in the fundamental

identityµ Ô e �� Ö ��' ��P¢ �hg¬ "Ã Ö ' ¢ � µ Ö ' ¢ � , the right hand side vanishes iden-

tically (sinceµ Ö ' ¢ ��N�É ó Ò �µ ' Ö � by definition), and we get a formal

identity in� q q$µ ' Ö t t K Ò :¡ e µ ' Ò �µ ' Ö �,g� Ò Ô e >1=µ ' Ò �µ ' Ö ��' �4 Ö �hgSk


Setting Ö � N 6 , we get ¡ e µ ' Ò =µ ' 6 �hg? Ò Ô e >1�µ ' Ò =µ ' 6 �Ï� ' 6 g . In otherwords, � J Þ r maps Ú A formally to Ú ÔA Ò . The restriction � J Þ r coincides withthe formal map:� � £ µòåó ¦$# û > e µ ' Ò �µ ' 6 �,g�' Ò Ô e > e µ ' Ò =µ ' 6 �hgS' 6 g ü � �

� Ò ìy� K Ò kThe rank properties of this formal map are the same as those of its CR-horizontal part: � � £ µ¬åó ¦$# > e µ ' Ò =µ ' 6 � g � � � Ò kThe formal CR mapping � is said ([Me2004a]) to be:

(cr1) CR-invertible at the origin if ~ Ô N ~ and if its CR-horizontal partis a formal equivalence at µ N 6 ;

(cr2) CR-submersive at the origin if ~ Ô � ~ and if its CR-horizontalpart is a formal submersion at µ N 6 ;

(cr3) CR-finite at the origin if ~KÔ N ~ and if its CR-horizontalpart is a finite formal map at µ N 6 , namely the quotient ring� q qXµÃt t H e > Ü Ò �µ ' Ò =µ ' 6 �Ï� i�� Ü Ò � � Ò g is finite-dimensional (the require-ment ~ðÔ N ~ is necessary for the reflection principle below);

(cr4) CR-dominating at the origin if ~ Ô �D~ and if there exist inte-gers

= �£Ú¥ = � 8 �� 8 Ú¥P~ðÔ � � ~ such that the determinant) y 4±�qÙQKy Ü Ò � H Q&µ Ü ^ Ü Òà _ t#�µ �Ï� i�� Ü Ò � � Ü Òà � � Ò ë 6 does not vanish identicallyin� q qýµÃt t , where y Ü Ò �µ � � N > Ü Ò e µ ' Ò �µ ' 6 �,g ;

(cr5) CR-transversal at the origin if there does not exist a nonzeroformal power series

C ÔF7>Õi '�klklk�' > � Ò � � � q q�>Õi 'lklklkm' > � Ò t tsuch that

C ÔF0y¥im�µ � 'lklk�kl' y � Ò �µ �Ï� 6 in� q qXµut t , wherey Ü Ò �µ � � N > Ü Ò e µ ' Ò =µ ' 6 � g .

One verifies ([Me2004a]) biholomorphic invariance and the four implica-tions:

(cr1) D (cr2) D (cr3) D (cr4) D (cr5)'

provided that ~KÔ N ~ in the second and in the third. By far, CR-transversality is the most general nondegeneracy condition.

3.10. Nondegeneracy conditions for CR mappings. This subsection ex-plains how to synthetize the combinatorics of various formal reflectionprinciples published in the last decade.

As ina3.1, let �&" � =Ù ' 6 � ¦$# PÙ;Ô ' 6 � be a complexified formal CR

mapping between two formal, analytic or algebraic complexified genericsubmanifolds of equations 6 N�µ p¢ ' Ö � � N(Ø ó Ò �× ' ¢ � and 6 N(µ Ô p¢ Ô ' Ö Ô � � N

HOLOMORPHIC EXTENSIONS AND REMOVABLE SINGULARITIES 37Ø Ô®ó Ò Ô �×²Ô ' ¢�Ô � . By hypothesis,µ Ô ��P¢ ��' �� Ö �Ï�@ "uP¢ ' Ö ��µ p¢ ' Ö � . Denot-

ing � N > ' ¡ � � � � Ò ì/� K Ò , replacingØ

byÒ =× ' ¢ � in

µ Ô ��P¢ ��' �� Ö �Ï��"Ãp¢ ' Ö ��µ P¢ ' Ö � and developingÒ ÔF Ê> ' � � N { Ð Ò |m}m? Ò Ê> Ð Ò Ò ÔÐ Ò �� , we start

with the following fundamental power series identity in� q qX× ' ¢�t t K Ò :Ê¡»�× 'BÒ �× ' ¢ �Ï�ãìÒ Ô e Ê>1=× ':Ò =× ' ¢ ��' ��P¢ � g ùÐ Ò |m} ? Ò Ê>1�× 'BÒ �× ' ¢ �Ï� Ð Ò Ò ÔÐ Ò ��P¢ �Ï� k

Consider the ~ complex vector fieldsé i '�klklk�' é � tangent to Ù that were

defined ina2.7. For every

; N ; i '�klklk�' ; � � �Bd � , define the multiplederivation

é < N é < �i �� é < ?� . Applying them to the above ��Ô scalar equa-tions, observing that they do not differentiate the variables ¢ N =µ ' Éä� , weget, without writing the arguments:é < Ê¡R� Ò ó ùÐ Ò |�} ? Ò é < Ê> Ð Ò �kÒ Ô� Ò � Ð Ò � �· 6 'for all

; �yd � , all �ÕÔ N = '�klklk ' �²Ô and all P¢ ' Ö � �íÙ .

Lemma 3.11. ([Me2004a, Me2005]) For every % Ô N = 'lklklk ' b Ô and every; �yd � , there exists a polynomial ¨ã� Ò � < in the jetd ¡ < ¡Û Ê�4 Ö � with coefficients

being power series in p¢ ' Ö � which depend only on the defining functionsØ �zó Ò �Ã=× ' ¢ � of Ù and which can be computed by means of some combi-natorial formula, such thaté < Ê�� Ò Ö �· ¨4� Ò � < e ¢ ' Ö 'äd ¡ < ¡Û Ê�� Ö � g kConvention 3.12. Let Ú ':¾ �Od . On the complexification Ù , equippedwith either the coordinates �µ ' Ö � or �× ' ¢ � , which correspond to either re-placing

ÉbyÒ =µ ' Ö � or

ØbyÒ =× ' ¢ � , we shall identify (notationally) a

power series written under the complete formf�p¢ ' Ö 'äd Ü ��p¢ � 'äd � Ê�4 Ö ��'with a power series written under one of the following four forms:2 f e ¢ ' × ':Ò =× ' ¢ � 'äd Ü ��P¢ � '�d � Ê��× 'BÒ �× ' ¢ �Ï�,g1'2 f e ¢ ' × '�d Ü �4p¢ � 'äd � Ê�4=× 'BÒ �× ' ¢ ��hg1'2 f e µ ' Ò �µ ' Ö � ' Ö 'äd Ü ��µ ' Ò �µ ' Ö �Ï� 'äd � Ê�4 Ö � g '2 f e µ ' Ö 'äd Ü ��µ ' Ò �µ ' Ö ��'äd � Ê�� Ö � g k


Thanks to the lemma and to the convention, we may therefore write:

(3.13)é < Ý Ê¡R� Ò Ö � ó Ò Ô� Ò Ê>1 Ö � ' �4p¢ �� Þ N � f Ô� Ò � < e ¢ ' Ö 'äd ¡ < ¡Û Ê�� Ö � � ��p¢ �,g 6 '

for �²Ô N = 'lklklk ' �²Ô . Remind that ��p¢ � is not differentiated, since the deriva-tions

é < involve only½½ Û ¿ , % N = 'lk�klk�' b . This is why we write ��p¢ � after

“:”. Furthmerore, the identities “ 6 ” are understood “on Ù ”, namely

as formal power series identities in� q q$× ' ¢�t t after replacing

ØbyÒ =× ' ¢ � or

equivalently, as a formal power series identities in� q q$µ ' Ö t t after replacingÉ

byÒ =µ ' Ö � .

To understand the reflection principle, it is important to observe imme-diately that the smoothness of the power series f Ô� Ò � < is the minimum of thetwo smoothnesses of

�and of

� Ô . For instance, the power series f Ô�¼Ò � <are all complex analytic if�

is real analytic and if� Ô is real algebraic,

even if the power series CR mapping �4p¢ � was assumed to be purely formaland nonconvergent. By a careful inspection of the application of the chainrule in the development of the above equations (3.13) (cf. Lemma 3.11),we even see that each f Ô� Ò � < is relatively polynomial with respect to thederivatives of positive order 7QY�Û Ê�4 Ö �� i�� ¡ � ¡ � ¡ < ¡ .3.14. Nondegeneracy conditions for formal CR mappings. In the equa-tions (3.13), we replace ��P¢ � by a new independent variable ¢�Ô·� �� Ò , weset P¢ ' Ö �SN 76 ' 6 � , and we define the following collection of power seriesz Ô� Ò � < p¢ Ô � � N ÷ é < Ê¡ä� Ò ó ùÐ Ò |m}m? Ò é < Ê> Ð Ò �kÒ Ô� Ò � Ð Ò p¢ Ô � ú ° ¹ Û�¹ A 'for � Ô N = 'lklklk ' � Ô and

; �/d � . Here, if; N 6 , we mean that

z Ô� Ò � A P¢ Ô �ÌNó Ò Ô� Ò 76 ' ¢�Ô � . According to (3.13), an equivalent definition is:z Ô� Ò � < p¢ Ô � � N f Ô� Ò � < e 6 ' 6 'äd ¡ < ¡Û Ê��76 � � ¢ Ô g kNow, just before introducing five new nondegeneracy conditions, we makea crucial heuristic remark. When b N b Ô , ~ N ~ Ô , � N;� Ô and � Nª ) , writing Ä Ô instead of ¢ Ô the special variable above in order to avoidconfusion, we get for ��Ô N = '�klklk�' �ÕÔ and

; ÔY�yd � Ò :z Ô� Ò � < Ò Ä Ô ��N ÷ é Ô < Ò Ø Ô� Ò ó ùÐ Ò |m}m? Ò é Ô < Ò =× Ô � Ð Ò Ò Ô� Ò � Ð Ò Ä Ô � ú ° Ò ¹ Û Ò ¹ A ÒN ÷ é Ô < Ò Ò Ô� Ò �× Ô ' ¢ Ô � ó ; Ô Ò Ô� Ò � < Ò Ä Ô � ú ° Ò ¹ Û Ò ¹ A ÒN ; Ô û Ò Ô� Ò � < Ò 76 Ô � ó Ò Ô� Ò � < Ò FÄ Ô � ü k


Consequently, up to a translation by a constant, we recover withz Ô� Ò � < Ò FÄ§Ô �the components of the infinite Segre mapping

l Ô � of� Ô . Hence the next

definition generalizes the concepts introduced before.

Definition 3.15. The formal CR mapping � � �(' 6 � ¦r# � Ô ' 6ÃÔ � is called

(h1) Levi-nondegenerate at the origin if the mapping¢ Ô å¦ e f Ô�¼Ò � < 76 ' 6 'äd ¡ < ¡Û Ê�476 � � ¢ Ô � g i��<� Ò � K Ò � ¡ < ¡ �&iis of rank b»Ô at ¢�Ô N 6ÃÔ ;

(h2) finitely nondegenerate at the origin if there exists an integer �Hi suchthat the mapping¢ Ô å¦ e f Ô� Ò � < Ø6 ' 6 'äd ¡ < ¡Û Ê�4Ø6 � � ¢ Ô � g i��<� Ò � K Ò � ¡ < ¡ � Üis of rank b»Ô at ¢�Ô N 6ÃÔ , for Ú N �ui , hence for every ÚK�>�ui ;

(h3) essentially finite at the origin if there exists an integer �Hi such thatthe mapping¢ Ô å¦ e f Ô� Ò � < Ø6 ' 6 'äd ¡ < ¡Û Ê�4Ø6 � � ¢ Ô � g i��<� Ò � K Ò � ¡ < ¡ � Üis locally finite at ¢¼Ô N 6uÔ , for Ú N �ui , hence for every ÚE�J�ui ;

(h4) Segre nondegenerate at the origin if there exist an integer �²i , inte-gers �²Ô� i 'lk�klk�' �²Ô� � Ò with

= ��²Ô� � Ò ��²Ô for %7Ô N = 'lk�klk�' bYÔ and multi-indices

; i� 'lk�klkm' ; � Ò� withJ ; � Ò� J �(�±i for %7Ô N = 'lklklkm' b»Ô , such that the

determinant

) y 4|{} Q~f Ô� Ò� ¿ Ò � � < ¿ Ò ��Q�¢ Ô� Ò à 9 µ ' Ò =µ ' 6 ��' 6 ' 6 '�d ¡ < ¿ Ò �� ¡ Ê�4Ø6 � � ��µ ' Ò �µ ' 6 �� >�� i�� Ò � � � Ò à � � Òdoes not vanish identically in

� q q$µÃt t ;(h5) holomorphically nondegenerate at the origin if there exists an inte-

ger �ui , integers �ÕÔ� i 'lklklkm' �²Ô� � Ò with= �ì�²Ô� � Ò �©�²Ô for %7Ô N = 'lklk�k�' bYÔ

and multiindices; i� 'lklklkm' ; � Ò� with

J ; � Ò� J �?�±i for %7Ô N = 'lklklk ' b»Ô , suchthat the determinant

) y 4|{} Q~f Ô� Ò� ¿ Ò � � < ¿ Ò ��Q�¢ Ô� Ò à 9 6 ' 6 ' 6 ' 6 'äd ¡ < ¿ Ò �� ¡ Ê��76 � � ��P¢ � >�� i�� Ò � � � Ò à � � Òdoes not vanish identically in

� q q ¢�t t .


The nondegeneracy of the formal mapping � requires the same nonde-generacy on the target � Ô ' 6ÃÔ � .Lemma 3.16. ([Me2004a]) Let � � �(' 6 � ¦r# � Ô ' 6uÔ � be a formal CRmapping.

(1) If � is Levi-nondegenerate at 6 , then� Ô is necessarily Levi-

nondegenerate at 6 Ô .(2) If � is finitely nondegenerate at 6 , then

� Ô is necessarily finitelynondegenerate at 6HÔ .

(3) If � is essentially finite at 6 , then� Ô is necessarily essentially finite

at 6ÃÔ .(4) If � is Segre nondegenerate at 6 , then

� Ô is necessarily Segre non-degenerate at 6ÃÔ .

(5) If � is holomorphically nondegenerate at 6 , then� Ô is necessarily

holomorphically nondegenerate at 6 Ô .We now show that CR-transversality of the mapping � insures that it

enjoys exactly the same nondegeneracy condition as the target � Ô ' 6ÃÔ � .Theorem 3.17. ([Me2004a]) Assume that the formal CR mapping � � �(' 6 � ¦$# � Ô ' 6 Ô � is CR-transversal at 6 . Then the following five im-plications hold:

(1) If� Ô is Levi nondegenerate at 6HÔ , then � is finitely nondegenerate

at 6 .(2) If

� Ô is finitely nondegenerate at 6HÔ , then � is finitely nondegener-ate at 6 .

(3) If� Ô is essentially finite at 6HÔ , then � is essentially finite at 6 .

(4) If� Ô is Segre nondegenerate at 6 Ô , then � is Segre nondegenerate

at 6 .(5) If

� Ô is holomorphically nondegenerate, and if moreover � istransversal at 6 , then � is holomorphically nondegenerate at 6 .

The above five implications also hold under the assumption that � iseither CR-invertible, or CR-submersive, or CR-finite with ~ N ~EÔ or CR-dominating: this provides at least 20 more (less refined) versions of thetheorem, some of which appear in the literature.

Other relations hold true between the nondegeneracy conditions on �and on the generic submanifolds ��' 6 � and � Ô ' 6ÃÔ � . We mention some,concisely. As above, assume that � � �(' 6 � å¦r# � Ô ' 6 � is a formal CR


mapping. Since � � A FÄ"A � �r� Ä"A � Ô , a linear map � ��V� NA � Ä A �>H Ä"A � ¦Ä A � Ô H Ä"A � Ô is induced. Assume �ÕÔ N � and ~ðÔ N ~ . The next statementmay be interpreted as a kind of Hopf Lemma for CR mappings.

Theorem 3.18. ([BR1990, ER2004]) If�

is minimal at 6 and if � is CR-dominating at 6 , then � � �V� NA � Ä A �>H Ä "A � ¦ Ä A � Ô H Ä "A � Ô is an isomor-phim.

An open question is to determine whether the condition that the jacobiandeterminant ) y 4 e ½ g ¿½ ° � P¢ �,g i�� ³� � does not vanish identically in

� q qG¢�t t is suffi-

cient to insure that � � �W� NA � Ä A �>H Ä "A � ¦ Ä A � Ô H Ä "A � Ô is an isomorphism.A deeper understanding of the constraints between various nondegeneracyconditions on � , � and

� Ô would be desirable.

3.19. Classical versions of the reflection principle. Let � � �(' 6 � ¦|# � Ô ' 6 � be a formal power series CR mapping between two generic sub-manifolds. Assume that

�is minimal at 6 .

Theorem 3.20. ([BER1999, Me2004a, Me2005]) If�

and� Ô are real

analytic, if � is either Levi nondegenerate, or finitely nondegenerate, oressentially finite, or Segre nondegenerate at the origin, then ��P¢ � is conver-gent, namely �4p¢ � � �«� ¢³� � Ò . If moreover,

�and

� Ô are algebraic, then �is algebraic.

If one puts separate nondegeneracy conditions on � and on� Ô , as in

Theorem 3.17, one obtains a combinatorics of possible statements, someof which appear in the literature.

If � is finitely nondegenerate (level (2)), the (paradigmatic) proof yieldsmore information.

Theorem 3.21. ([BER1999, Me2005]) As above, let � � ��' 6 � ¦ � Ô ' 6 Ô � be a formal power series CR mapping. Assume that�

is min-imal at 6 and let ] A be the integer of Corollary 2.17. Assume also that �is �±i -finitely nondegenerate at 6 . Then there exists a

�S� Ò-valued power se-

ries mapping �öP¢ 'äd � [ r=� � � which is constructed algorithmically by meansof the defining equations of �(' 6 � and of � Ô ' 6 Ô � , such that the powerseries identity ��P¢ �· �@p¢ 'äd � [ r � � ��76 �Ï�holds in

� q qG¢�t t � Ò . If�

and� Ô are real analytic (resp. algebraic), � is holo-

morphic (resp. complex algebraic) in a neighborhood of 6 ì d � [ r=� � ��Ø6 � .


In [BER1999, GM2004], the above formula ��P¢ �§ �@p¢ 'äd � [ r=� �,��Ø6 �� isstudied horoughly in the case where

� Ô N � and � is a local holomorphicautomorphism of �(' 6 � close to the identity.

At level (5), namely with a holomorphically nondegenerate target � Ô ' 6uÔ � , the reflection principle is much more delicate. It requires the in-troduction of a new object, whose regularity properties hold in fact withoutany nondegeneracy assumption on the target � Ô ' 6 Ô � .3.22. Convergence of the reflection mapping. The reflection mappingassociated to � and to the system of coordinates =µ Ô '-É Ô � is :� Ô g Ö Ô ' ¢ � � NFØ Ô ó Ò Ô �× Ô ' �4p¢ �� q q Ö Ô ' ¢�t t K Ò7kSince � is formal, it is only a formal power series mapping. As argued inthe introduction of [Me2005], it is the most fundamental object in the ana-lytic reflection principle. In the case of CR mappings between essentiallyfinite hypersurfaces, the analytic regularity of the reflection mapping isequivalent to the extension of CR mappings as correspondences, as studiedin [DP1995, Sh2000, Sh2003, DP2003]. Without nondegeneracy assump-tion on � Ô ' 6ÃÔ � , the reflection mapping enjoys regularity properties fromwhich all analytic reflection principles may be deduced. Here is the verymain theorem of this Section 3.

Theorem 3.23. ([Me2001b, BMR2002, Me2005]) If�

is minimal at theorigin and if � is either CR-invertible, or CR-submersive, or CR-finite, orCR-dominating, or CR-transversal, then for every system of coordinates�µÃÔ '-É Ô � � � � Ò ì5� K Ò in which the extrinsic complexification Ù Ô is repre-sented by

Ø Ô N{Ò ÔF�×²Ô ' ¢�Ô � , the associated CR-reflection mapping is conver-gent, namely

� Ô g Ö Ô ' ¢ � � �z� Ö Ô ' ¢³� K Ò .If the convergence property holds in one such system of coordinates, it

holds in all systems of coordinates ([Me2005]; Proposition 3.26 below).Further, if we develope

Ò Ô =× Ô ' ¢ Ô �öN { Ð Ò |�}@? Ò =× Ô � Ð Ò Ò ÔÐ Ò p¢ Ô � , the conver-gence of

� Ô g Ö Ô ' ¢ � has a concrete signification.

Corollary 3.24. All the componentsÒ ÔÐ Ò ��p¢ �� of the reflection mapping

are convergent, namelyÒ ÔÐ Ò ��P¢ �Ï� � �z� ¢³� K Ò for every Ñ Ô �yd � Ò .

Conversely ([Me2001b, Me2005]), ifÒ ÔÐ Ò ��4p¢ �� z� ¢³� K Ò for every Ñ ÔY�d � Ò , an elementary application of the Artin approximation Theorem 3.28

(below) yields Cauchy estimates: there exist � ¢ 6 , � ¢ 6 and� ¢ 6 so

thatJÁÒ ÔÐ Ò ��P¢ �Ï�±J 8 � Ø� � Y¡ Ð Ò ¡ , for every ¢x� � � with

J ¢ J 8>� . It follows that� Ô g Ö Ô ' ¢ � � �z� Ö Ô ' ¢³� K Ò .


Taking account of the nondegeneracy conditions (ndi) and (crj), severalcorollaries may be deduced from the theorem. Most of them are already ex-pressed by Theorem 3.20, except notably the delicate case where � Ô ' 6 Ô �is holomorphically nondegenerate.

Corollary 3.25. ([Me2001b, Me2005]) If�

is minimal at the origin, if � Ô ' 6uÔ � is holomorphically nondegenerate and if � is either CR-invertibleand invertible, or CR-submersive and submersive, or CR-finite and finitewith ~ðÔ N ~ , or CR-dominating and dominating, or CR-transversal andtransversal, then ��P¢ � � �z� ¢³� � Ò is convergent.

It is known ([St1996]) that � Ô ' 6ÃÔ � is holomorphically degenerate if andonly if there exists a nonzero = ' 6 � vector field � Ô N { � Ò� Ò ¹»i Ê²Ô�sÒ P¢�Ô ��½½ ° Ò ¿ Òhaving holomorphic coefficients which is tangent to � Ô ' 6 Ô � . In thecorollary above, holomorphic nondegeneracy is optimal for the conver-gence of a formal equivalence: if

� Ô is holomorphically degenerate, if��Ô ' ¢�Ô � å ó ¦ y � Ç ��Ôh�yÔ � P¢�Ô � denotes the local flow of ��Ô , where �lÔ�� ,¢ Ô � � � Ò , there indeed exist ([BER1999, Me2005]) nonconvergent power

series �ðÔhP¢�Ô � � � q q ¢�Ôst t such that ¢�Ô&å¦$# y � Ç ��ÔFP¢�Ô � �yÔ � p¢�Ô � is a nonconvergentformal equivalence of

� Ô .The invariance of the reflection mapping is crucial.

Proposition 3.26. ([Me2002, Me2004a, Me2005]) The convergence ofthe reflection mapping is a biholomorphically invariant property. Moreprecisely, if ¢¼Ô Ô N y&ÔFp¢�Ô � is a local biholomorphism fixing 6²Ô and trans-forming � Ô ' 6ÃÔ � into a generic submanifold � Ô Ô ' 6ÃÔ � of equations

ÊÉ Ô Ô� Ò NÒ Ô Ô� Ò Êµ Ô Ô ' ¢ Ô Ô � , � Ô N = 'lklk�km' � Ô , the composed reflection mapping of y Ô � � � �(' 6 � ¦$# � Ô Ô ' 6ÃÔ � defined by� Ô Ô� ÒQ� g Ö Ô Ô ' ¢ � � NJØ Ô Ô ó Ò Ô Ô e × Ô Ô ' y Ô ��4p¢ �� gN@Ø Ô Ô ó ùÐ Ò |�}@? Ò =× Ô Ô � Ð Ò Ò Ô ÔÐ Ò e y Ô ��4p¢ �� ghas components

Ò Ô ÔÐ Ò e y�Ô ��P¢ �Ï� g given by formulasÒ Ô ÔÐ Ò e y Ô ��P¢ �Ï� g �� ÔÐ Ò û e Ò ÔÐ Ò� ��P¢ �� g Ð Ò� |�} ? Ò ü 'where the local holomorphic functions

� ÔÐ Ò depend only on the biholomor-phism ¢¼Ô Ô N y�ÔFp¢�Ô � (they have an infinite number of variables, but the neces-sary Cauchy estimates insuring convergence are automatically satisfied).


A few words about the proof of the main Theorem 3.23. Althoughthe classical reflection principle deals only with the “reflection identi-ties” (3.13), to get the most adequate version of the reflection principle,it is unavoidable to understand the symmetry between the variables ¢ andthe variables Ö N Ê¢ � " .

The assumption that ��" maps formally =Ù ' 6 � to PÙôÔ ' 6ÃÔ � is equivalentto each one of the following two formal identities:

STTTU TTTV¡» Ö �5N ùÐ Ò |m}m? Ò >1 Ö � Ð Ò Ò ÔÐ Ò ��P¢ ��'¡VP¢ ��N ùÐ Ò |m}m? Ò >1p¢ � Ð Ò Ò ÔÐ Ò e �4 Ö � g '

on Ù , namely after replacing eitherÉ

byÒ =µ ' Ö � or

ØbyÒ =× ' ¢ � . The

symmetry may be pursued by considering the two families of derivations:Õ é < � N é i � < � é i � < à �� é � � < ? �HÈ+)é < � N é i � < � é i � < à �� é � � < ? 'where

; N ; i ' ; � 'lklk�k�' ; � � �0d � . Applying them to the two formalidentities above, if we respect the completeness of the combinatorics, wewill get four families of reflection identities. The first pair is obtained byapplying

é < to the two formal identities above:

STTTU TTTVé < ¡» Ö � N ùÐ Ò |�}@? Ò é < Ý >ã Ö � Ð Ò Þ Ò ÔÐ Ò ��4p¢ ��'6 N ùÐ Ò |�}@? Ò >1P¢ � Ð Ò é < Ý Ò ÔÐ Ò e �� Ö �hg Þ k

The second pair is obtained by applyingé < , permuting the two lines:

STTTU TTTVé < ¡VP¢ ��N ùÐ Ò |m}m? Ò é < Ý >1P¢ � Ð Ò Þ Ò ÔÐ Ò e �� Ö �,g�'6 N ùÐ Ò |m}m? Ò >ã Ö � Ð Ò é < Ý Ò ÔÐ Ò ��P¢ �Ï� Þ k

We immediately see that these two pairs are conjugate line by line. Ineach pair, we notice a crucial difference between the first and the second

line: whereas it is ¡ and the power > Ð Ò (or ¡ and > Ð Ò ) that are differentiatedin each first line, in each second line, only the components

Ò ÔÐ Ò � � (orÒ ÔÐ Ò � � ) of the reflection mapping, which are the right invariant functions,


are differentiated. In a certain sense, it is forbidden to differentiate ¡ and> Ð Ò (or ¡ and > Ð Ò ), because the components > ' ¡ � of � need not enjoy areflection principle. In fact, in the proof of the main Theorem 3.23, onehas to play constantly with the four reflection identities above.

Since we cannot summarize here the long and refined proof, we onlyformulate the main technical proposition. Denote by

d �° � the � -th jet of apower series � P¢ � � � q qG¢�t t K Ò , for instance

d �° Ò ÔÐ Ò � � for some Ñ Ô»��d � Ò . Re-

mind that ð Ü and ð Ü are (conjugate) Segre chains. Let × K Ò � � � � � N �²Ô ^ � á � _no� o � o .Proposition 3.27. ([Me2005]) For every Ú��0d and every �?�0d , thefollowing two properties hold:2 if Ú is odd, for every Ñ ÔY�yd � Ò :Ý d&�° Ò ÔÐ Ò � � Þ e ð Ü e µ<^ Ü _ g�g � �z� µ<^ Ü _��6i=C Ò l ~Bl ! Y2 if Ú is even, for every Ñ Ô»��d � Ò :÷=d �Û Ò ÔÐ Ò � � ú e ð Ü e µ ^ Ü _ gÝg � �z� µ ^ Ü _ � i/C Ò l ~ml ! k

With � N 6 and Ú N �Z] A , thanks to Corollary 2.17, we deduce from thismain proposition that

Ò ÔÐ Ò ��P¢ �Ï� � �z� ¢³� Ð Ò for every Ñ Ô �5d � Ò . This yieldsTheorem 3.23.

The main tool in the proof of this proposition is an approximation theo-rem saying that a formal power series mapping that is a solution of someanalytic equations may be corrected so as to become convergent and still asolution.

Theorem 3.28. (ARTIN [Ar1968, JoPf2000]) Let n N�oor�

, let b/�Ddwith b?� =

, let g N hg�i 'lklklkm' g � � �?n � , let ~ �?d , with ~ � =, let � N��²i 'lk�klk�' � � � �în � , let �æ�îd with �æ� =

and let frim�g ' � � 'lk�klkm' f K �g ' � �be an arbitrary collection of formal power series in n � g ' �� that vanish atthe origin, namely f �Ã76 ' 6 �xN 6 , � N = '�klklk ' � . Assume that there exists aformal mapping ��hg �«N �Vi�hg ��'lklklk ' � � �g �� ænrq qsgut t � with �4Ø6 �zN 6 suchthat f¦��hg ' ��g �� 6?*-È nrq qsgut t ' W�� N = '�klklk ' � kLet �ðhg � � N g�i¼nrq qsgut tê� �� òg � nrq qsgut t be the maximal ideal of nrq qsgut t . For everyinteger × � =

, there exists a convergent power series mapping � i hg � �n � gj� � such thatf¦� e g ' �kiÌhg �,g 6?*�È(nrq q gut t ' W�� N = 'lk�klk�' � '


that approximates ��hg � to order ×<ó = :� i hg �� hg � ,:�v) e �ð �»� i g kAs an application of the main Theorem 3.23, an approximation property

for formal CR mappings holds.

Theorem 3.29. ([Me2005]) Under the assumptions of Theorem 3.23, forevery integer × � =

, there exists a convergent power series mapping0 i p¢ � � �z� ¢³� � Ò with 0 i P¢ � ��P¢ � ,��j)�I�Kp¢ �� i (whence �@76 � N 6 ),that induces a local holomorphic map from �(' 6 � to � Ô ' 6 Ô � .Corollary 3.30. ([Me2001b, Me2005]) Assume that b�Ô N b , that �ÕÔ N � ,that

�is minimal at the origin, and that � � �(' 6 � ¦r# � Ô ' 6ÃÔ � is a

formal (invertible) equivalence. Then�

and� Ô are biholomorphically

equivalent.

It is known ([St1996, BER1999, GM2004]) that a minimal holomor-phically nondegenerate real analytic generic submanifold of

� �has finite-

dimensional local holomorphic automorphism group. Unique determina-tion by a jet of finite order follows from a representation formula, as inTheorem 3.21. More generally:

Corollary 3.31. ([Me2001b, BMR2002, Me2005]) Assume that ~\Ô N ~and � Ô N � , that �(' 6 � is minimal at the origin and that � Ô ' 6 � is holo-morphically nondegenerate. There exists an integer ¼ N ¼�=~ ' � � such that,if two local biholomorphisms � i ' � � � �(' 6 � ¦ � Ô ' 6 � have the same ¼ -th jet at the origin, then � i N �+� .

From an inspection of the proof, Theorem 3.29 holds without the as-sumption that �(' 6 � is minimal, but with the assumption that its CR orbitshave constant dimension in a neighborhood of 6 . However, the case whereCR orbits have arbitrary dimension is delicate.

Open question 3.32. Does formal equivalence coincide with biholomor-phic equivalence in the category of real analytic generic local submani-folds of

��whose CR orbits have non-constant dimension ?

3.33. Algebraicity of the reflection mapping. We will assume that both�and

� Ô are algebraic. Remind that Theorem 3.20 shows the algebraic-ity of � under some hypotheses. A much finer result is as follows. Itsynthetizes all existing results ([We1977, SS1996, CMS1999, BER1999,Za1999]) about algebraicity of local holomorphic mappings.


Theorem 3.34. ([Me2001a]) If � is a local holomorphic map �(' 6 � ¦ � Ô ' 6uÔ � , if�

and� Ô are algebraic, if

�is minimal at the origin and if� Ô is the smallest (for inclusion) local real algebraic manifold containing�4 �B� , then the reflection mapping

� Ô g Ö Ô ' ¢ � is algebraic.

Trivial examples ([Me2001a]) show that the algebraicity of� Ô g need not

hold if� Ô is not the smallest one.

In fact, Theorem 3.34 also holds (with the same proof) if one assumesonly that the source

�is minimal at a Zariski-generic point: it suffices

to shrink�

and the domain of definition of � around such points, gettinglocal algebraicity of

� Ô g there, and since algebraicity is a global property,� Ô g is algebraic everywhere.An equivalent formulation of Theorem 3.34 uses the concept of tran-

scendence degree, studied in [Pu1990, CMS1999, Me2001a]. With b�ÔL Òbeing the essential holomorphic dimension of � Ô ' 6 Ô � defined in

a3.6, set¼ Ô L Ò � N b Ô ó5b ÔL Ò . Observe that � Ô ' 6 Ô � is holomorphically nondegenerate

precisely when ¼+ÔL Ò N 6 . Denote by� q ¢�t the ring of complex polynomials

of the variable ¢z� �� and by� P¢ � its quotient field. Let ¢êÔ N ��p¢ � be a local

holomorphic mapping as in Theorem 3.34. and let� p¢ � ��iHP¢ ��'lklklkm' � � Ò P¢ �Ï�

be the field generated by the components of � .Theorem 3.35. ([Me2001a]) With the same assumptions as in The-orem 3.34, the transcendence degree of the field extension

� P¢ � ¦� p¢ � ��p¢ �� is less than or equal to ¼&ÔL Ò .Corollary 3.36. ([CMS1999, Za1999, Me2001a]) If

�is minimal at a

Zariski-generic point and if the real algebraic target� Ô does not contain

any complex algebraic curve, then the local holomorphic mapping � isalgebraic.

However, in case � is only a formal CR mapping, it is impossible toshift the central point to a nearby minimal point. Putting the simplest rankassumption (invertibility) on � , we may thus formulate delicate problemsfor the future.

Open question 3.37. Let � be a formal equivalence between two real an-alytic generic submanifolds of

�·�which are minimal at a Zariski-generic

point.2 Is the reflection mapping convergent ?2 Is � uniquely determined by a jet of finite order when the target isholomorphically nondegenerate ?

48 JOEL MERKER AND EGMONT PORTEN2 Is � convergent under the assumption that the real analytic target� Ô does not contain any complex analytic curve ?

For� Ô algebraic containing no complex algebraic curve and

�min-

imal at 6 , the third question has been settled in [MMZ2003b]. How-ever, the assumption of algebraicity of

� Ô is strongly used there, be-cause these authors deal with the transcendence degree of the field ex-tension

� p¢ � ¦ � p¢ � ��p¢ �Ï� , a concept which is meaningless if� Ô is

real analytic. For further (secondary) results and open questions, werefer to [BMR2002, Ro2003]. This closes up our survey of the for-mal/algebraic/analytic reflection principle.

A generic submanifold� �¶�·�

is called locally algebraizable at oneof its points � if there exist local holomorphic coordinates centered at � inwhich it is Nash algebraic. Unlike partial results, the following questionremains up to now unsolved.

Open problem 3.38. ([Hu2001, HJY2001, Ji2002, GM2004, Fo2004])Formulate a necessary and sufficient condition for the local algebraizabil-ity of a real analytic hypersurface

� � �·�in terms of a basis of the

(differential) algebra of its Cartan-Hachtroudi-Chern invariants.

To conclude, we would like to mention that the complete theory of CRmappings may be transferred to systems of partial differential equationshaving finite-dimensional Lie symmetry group. This aspect will be treatedin subsequent publications ([Me2006a, Me2006b]).


III: Systems of vector fields and CR functions

Table of contents

1. Sussman’s orbit theorem and structural properties of orbits . . . . . . . . . . . . . . . 50.2. Finite type systems and their genericity (openess and density) . . . . . . . . . . . . . .63.3. Locally integrable CR structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.4. Smooth generic submanifolds and their CR orbits . . . . . . . . . . . . . . . . . . . . . . . . . 87.5. Approximation and uniqueness principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.

[7 diagrams]

Beyond the theorems of Frobenius and of Nagano, Sussmann’s theorem pro-vides a means, valid in the smooth category, to construct all the integral manifoldsof an arbitrary system of vector fields, as orbits of the pseudo-group actions ofglobal flows. The fundamental properties of such orbits: lower semi-continuity ofdimension, local flow box structure, propagation of embeddedness, intersectionwith a transversal curve in the one-codimensional case, are essentially analogous,but different from the ones known in foliation theory. Orbits possess wide appli-cations in Control Theory, in sub-Riemannian Geometry, in the Analysis of LinearPartial Differential Equations and in Cauchy-Riemann geometry.

Let � � T br� � , � b F U��U6� , be a linear PDE system with unknown T , where� is smooth and where �� Üe� i�� Ü ��´ is an involutive (in the sense of Frobenius)system of smooth vector fields on � � having complex-valued coefficients. SinceLewy’s celebrated discovery of an example of a single equation � T b�� in � �without any solution, a major problem in the Analysis of PDE’s is to find ade-quate criterions for the existence of local solutions. Condition (P) of Nirenberg-Treves has appeared to be necessary and sufficient to insure local integrability ofa single equation of principal type having simple characteristics. The problem ofcharacterizing systems of several linear first order PDE’s having maximal spaceof solutions is not yet solved in full generality; several fine questions remain open.

Following Treves, to abstract the notion of systems involving several equa-tions, an involutive structure on a smooth � -dimensional real manifold _ isa � -dimensional complex subbundle � of `��u�x_ satisfying � U �[¡£¢ � .The automatic integrability of smooth almost complex structures (those with�¥¤ � b `$�$�x_ ) and the classical (non)integrability theorems for smoothabstract CR structures (those with �§¦ � b ��¨ � ) are inserted in this generalframework.

Beyond such problematics, it is of interest to study the analysis and the geome-try of subbundles � whose space of solutions is maximal, viz the preceding ques-tion is assumed to be solved, optimally: in a neighborhood of every point of _ ,there exist M ��© � P local complex valued functions ª i�U��U ª � ¬« having linearly


independent differentials which are solutions of �ª Ü b ¨ . Such involutive struc-tures are called locally integrable. Some representative examples are provided bythe bundle of anti-holomorphic vector fields tangent to various embedded genericsubmanifolds of ` � . According to a theorem due to Baouendi-Treves, every lo-cal solution of � T b ¨ may be approximated sharply by polynomials in a set offundamental solutions ª i U��U ª � ¬« , in the topology of functional spaces as ® 3 � � ,��¯�ß #" , or ° Ô .

In a locally integrable structure, the Sussmann orbits of the vector fields±³² � Ü UY´¶µ � Ü are then of central importance in analytic and in geometrical ques-tions. They show up propagational aspects, as for instance: the support of a func-tion or distribution solution T of � T b ¨ is a union of orbits. The approximationtheorem also yields an elegant proof of uniqueness in the Cauchy problem. Fur-ther propagational aspects will be studied in the next chapters, using the methodof analytic discs. Sections 3, 4 and 5 of this chapter and the remainder of thememoir are focused on embedded generic submanifolds.a

1. SUSSMANN’S THEOREM AND STRUCTURAL PROPERTIES

OF ORBITS

1.1. Integral manifolds of a system of vector fields. Ordinary differentialequations in the modern sense emerged in the seventieth century, concomi-tantly with the infinitesimal calculus. Nowadays, in contemporary mathe-matics, the abstract study of vector fields is inserted in several broad areasof research, among which we perceive the following.2 Control Theory: controllability of vector fields on � � and real

analytic manifolds; nonholonomic systems; sub-Riemannian ge-ometry ([GV1987, Bel1996]).2 Dynamical systems: singularities of real or complex vectorfields and foliations; normal forms and classification; phase di-agrams; Lyapunov theory; Poincare-Bendixson theory; theory oflimit cycles of polynomial and analytic vector fields; small divi-sors ([Ar1978, Ar1988]).2 Lie-Cartan theory: infinitesimal symmetries of differential equa-tions; classification of local Lie group actions; Lie algebras of vec-tor fields; representations of Lie algebras; exterior differential sys-tems; Cartan-Vessiot-Kahler theorem; Janet-Riquier theory; Car-tan’s method of equivalence ([Ol1995, Stk2000]).2 Numerical analysis: systems of (non)linear ordinary differentialequations; methods of: Euler, Newton-Cotes, Newton-Raphson,Runge-Kutta, Adams-Bashforth, Adams-Moulton ([De1996]).

HOLOMORPHIC EXTENSIONS AND REMOVABLE SINGULARITIES 512 PDE theory: Local solvability of linear partial differential equa-tions; uniqueness in the Cauchy problem; propagation of singu-larities; FBI transform and control of wave front set ([ES1993,Trv1992]).

To motivate the present Part III, let us expose informally two dual questionsabout systems of vector fields. Consider a set

¯of local vector fields de-

fined on a domain ofoS�

. Frobenius’ theorem provides local foliations bysubmanifolds to which every element of

¯is tangent, provided

¯is closed

under Lie brackets. However, for a generic set¯

, the condition q ¯�'Ï¯ t �?¯fails and in addition, the tangent spaces spanned by elements of

¯are of

varying dimension. To surmount these imperfections, two inverse optionspresent themselves:

Sub: find the subsystems¯ Ô � ¯

which satisfy Frobenius’ conditionq ¯ Ô 'Ï¯ Ôht �?¯ Ô and which are maximal in an appropriate sense;

Sup: find the supsystems¯ Ô Ð ¯

which have integral manifolds andwhich are minimal in an appropriate sense.

CHEVALLEY

LOBRY

SUSSMANN

NAGANO

CHOW¯ W�·CN�¯vector fieldsystem

Initial

� ��¯ i¯ �¯ �sub-systemsintegrable Unique

sup-systemintegrable

KÆHLER

CARTAN

VESSIOT

Several

Searching for integrable sub- or supsystems

The first problem Sub is answered by the Cartan-Vessiot-Kahler the-orem, thanks to an algorithm which provides all the minimal Frobenius-integrable subsystems

¯ Ô of¯

(we recommend [Stk2000] for a presen-tation). Generically, there are infinitely many solutions and their cardi-nality is described by means of a sequence of integers together with theso-called Cartan character of

¯. In the course of the proof, the Cauchy-

Kowalevskaya integrability theorem, valid only in the analytic category, isheavily used. It was not a serious restriction at the time of E. Cartan, but,in the second half of the twentieth century, the progress of the Analysis ofPDE showed deep new phenomena in the differentiable category. Hence,one may raise the:

Open problem 1.2. Find versions of the Cartan-Vessiot-Kahler theoremfor systems of vector fields having smooth non-analytic coefficients.


The Cauchy characteristic subsystem of¯

([Stk2000]) is always invo-lutive, hence the smooth Frobenius theorem applies to it4. However, forintermediate systems, the question is wide open. Possibly, this question isrelated to some theorems about local solvability of smooth partial differen-tial equations (cf. Section 3) that were established to understand the HansLewy counterexample (

a3.1).

The second problem Sup is already answered by Nagano’s theorem(Part II), though only in the analytic category, with a unique integrableminimal supsystem

¯ UMWMX Ð ¯ . In the general smooth category, the strongerChevalley-Lobry-Stefan-Sussmann theorem, dealing with flows of vectorfields instead of Lie brackets, shows again that there is a unique integrablesup-system of

¯which has integral manifolds. As this theorem will be

central in this memoir, it will be exposed thoroughly in the present Sec-tion 1.

1.3. Flows of vector fields and their regularity. Let n N0oor�

. Let¸be a open connected subset of n � . Let g N hg�i 'lklk�k�' g � � � ¸

. LetR N { ��º¹»i ÊÕ��hg �D½½�¾�¿ be a vector field defined over¸

. Throughout thissection, we shall assume that its coefficients Êj� are either n -analytic (ofclass �»¤ ), of class � � , or of class ��3 � � , where ¼D� =

and 6 ��:9� =(see

Section 1(IV) for background about Holder classes).By the classical Cauchy-Lipschitz theorem, through each point g A � ¸ ,

there passes a unique local integral curve of the vector field R , namely a lo-cal solution g+P8 �·N hgji P8 � '�klklk�' g � p8 �� of the system of ordinary differentialequations: �Hg�i�P8 �,H �Z8 N Ê�i��g+P8 ��'lklklk klklkm' �Ãg � P8 �-H �Ó8 N Ê � �g+P8 �Ï� 'which satisfies the initial condition g+Ø6 �EN g A . This solution is definedat least for small 8ð�Bn and is classically denoted by 8�å¦ y � Ç P8êR � �g A � ,because it has the local pseudogroup propertyy � Ç P8 Ô R � e y � Ç P8êR � �g A � g NFy � Ç e p8�� 8 Ô � R g hg A � 'whenever the composition is defined. Denote by � ¾ r the largest connectedopen set containing the origin in n in which

y � Ç P8êR � �g A � is defined. Oneshows that the union of various � ¾ r , for g A running in

¸, is an open con-

nected set � � of n ì n � which contains� 6j� ì¹¸

. Some regularity withrespect to both variables 8 and g A is got automatically.

Theorem 1.4. ([La1983], [ P ]) The global flow � � £ p8 ' g A � å¦y � Ç P8�R � hg A � � ¸of a vector field R N { ��º¹»i Ê²�¼�g � Q ¾¼¿ defined in the

4We are grateful to Stormark for pointing out this observation


domain¸

has exactly the same smoothness as R , namely it is � ¤ , � � or� 3 � � .As a classical corollary, a local straightening property holds : in a neigh-

borhood of a point at which R does not vanish, there exists a � ¤ , � � or � 3 � �change of coordinates gHÔ N guÔ hg � in which the transformed vector field isthe unit positive vector field directed by the gHÔ i lines, viz R1Ô N Q H QvguÔ i .

Up to the end of this Section 1, we will work with n N�o.

1.5. Searching integral manifolds of a system of vector fields. Let�

be a smooth paracompact real manifold, which is ��¤ , � � or ��3 á i � � , where¼ � =, 6 �¸:Û� =

. Let¯ � N � R·°±�±° |@º be a collection of vector fields

defined on open subsets¸ ° of

�and having �¥¤ , � � or ��3 � � coefficients,

where ` is an arbitrary set. It is no restriction to assume thatI ° |-º ¸ ° N�

, since otherwise, it suffices to shrink�

. Call¯

a system of vectorfields on

�.

Problem 1.6. Find submanifolds × of�

such that each element of¯

istangent to × .

To analyze this (still imprecise) problem, let » L denote the collection ofall � ¤ , � � or � 3 � � functions defined on open subsets of

�, and call the

system¯

of vector fields » L -linear if every combined vector field >Ì�@�Ç¡vRbelongs to

¯, whenever > ' ¡��t» L and � ' Rf� ¯ . Here, >Ì�æ� ¡vR is defined

in the intersection of the domains of definition of > , ¡ , � and R . To studythe problem, it is obviously no restriction to assume that

¯is » L -linear.

For �\� � arbitrary, define¯ º� � � NO� Rº� � � R?� ¯ � kSince

¯is » L -linear, this is a linear subspace of Ä&. � . So Problem 1.6 is to

find submanifolds × satisfying Ä+.±×ôÐ ¯ Á� � , for every �\��× . Notice thatan appropriate answer should enable one to find all such submanifolds.Also, suppose that × i and × � are two solutions with × � � ×�i . Then theproblem with the pair �(' × � is exactly the same as the problem with thepair 7×�i ' × � � . Hence a better formulation.

Problem 1.6’. Find all the submanifolds × �0�of smallest dimension

that satisfy Ä�.±×ôÐ ¯ º� � , for every � ��× .

The classical Frobenius theorem ([Fr1877, Sp1970, BER1999, Bo1991,Ch1991, Stk2000, Trv1992]) provides an answer in the (for us simplest)case where

¯is closed under Lie brackets and is of constant dimension:

every point �\� � admits an open neighborhood foliated by submanifolds

54 JOEL MERKER AND EGMONT PORTEN× satisfying ÄVÎ�× N ¯ ØÑ � , for every Ñ(�c× . The global properties ofthese submanifolds were not much studied until C. Ehresmann and G. Reebendeavoured to understand them (birth of foliation theory). A line withirrational slope in the � -torus oaH½¼§� � shows that it is necessary to admitsubmanifolds × of

�which are not closed. Let

½�Ldenote the manifold

structure of�

.

Definition 1.7. An immersed submanifold of ��' ½�L¬� is a subset of × of�equipped with its own smooth manifold structure

½ i , such that the in-clusion map % � Ø× ' ½ i � ¦ �(' ½�Lò� is smooth, immersive and injective.

Thus, to keep maximally open Problem 1.6’, one should seek immersedsubmanifolds and make no assumption about closedness under Lie brack-ets. For later use, recall that an immersed submanifold × of

�is em-

bedded if its own manifold structure coincides with the manifold inheritedfrom the inclusion × �ô�

. It is well known ([CLN1985]) that an im-mersed submanifold × is embedded if and only if for every point �5�@× ,there exists a neighborhood ¾¥. of � in

�such that the pair ¶¾¥. ' × G ¾». �

is diffeomorphic to o L W�¿ L '�o L W�¿ i � .1.8. Maximal strong integral manifolds property. In order to under-stand Problem 1.6’, for heuristic reasons, it will be clever to discuss thedifferences between the two possibilities

¯ º� � N Ä&.�× and¯ º� �ÁÀ Äv.±× .

Consider an arbitrary » L -linear system of vector fields Â¯ containing¯

,for instance

¯itself. Let �î� � and define the linear subspace Â¯ Á� � � N� ÂR Á� � � ÂRf� Â¯ � .

Definition 1.9. An immersed submanifold × of�

is said to be:2 a strong Â¯ -integral manifold if ÄVÎÖ× N Â¯ ØÑ � , at every point Ñ��× ;2 a weak¯

-integral manifold if Ä¥Î�×øÐ Â¯ ØÑ � , at every point Ñ¬� × .

In advance, the answer (Theorem 1.21 below) to Problem 1.6’ states thatit is possible to construct a unique system of vector fields Â¯ containing¯

, whose strong integral manifolds coincide with the smallest weak¯

-integral manifolds × . Further definitions are needed.

A system of vector fields Â¯ is said to have the strong integral manifoldsproperty if for every point ��

, there exists a strong Â¯ -integral sub-manifold × passing through � . A maximal strong Â¯ -integral manifold ×is an immersed Â¯ -integral manifold with the property that every connectedstrong Â¯ -integral manifold which intersects × is an open submanifold of

HOLOMORPHIC EXTENSIONS AND REMOVABLE SINGULARITIES 55× . Thus, through a point � � � , there passes at most one maximal strongÂ¯ -integral submanifold. Finally, the system Â¯ has the maximal strong in-tegral manifolds property if, through every point �Ë� �

, there passes amaximal strong Â¯ -integral manifold. The » L -linear systems Â¯ containing¯

are ordered by inclusion. We then admit that Problem 1.6’ is essentiallyreduced to:

Problem 1.6”. How to construct the (a posteriori unique) smallest (forinclusion) » L -linear system of vector fields Â¯ containing

¯which has the

maximal strong integral manifolds property ?

1.10. Taking account of the Lie brackets. Here is a basic geometric ob-servation inspired by Frobenius’ and Nagano’s theorems.

Lemma 1.11. Assume the » L -linear system Â¯ has the strong integral man-ifolds property. Then for every two vector fields ÂR ' ÂR Ô �ÃÂ¯ and for every �in the intersection of their domains, the Lie bracket Ý ÂR ' ÂR1Ô Þ º� � belongs toÂ¯ Á� � .Proof. Indeed, let × be a strong Â¯ -integral manifold, namely satisfyingÄ × N Â¯ ÂÂ i . If ÂR ' ÂR�ÔY� Â¯ , the two restrictions ÂR ÂÂ i and ÂR1Ô ÂÂ i are tangent to× . Hence the restriction to × of the Lie bracket Ý ÂR ' ÂR Ô Þ is also tangent to× . In conclusion, at every ��y× , we have Ý ÂR ' ÂR1Ô Þ Á� � ��Äv.±× N Â¯ Á� � . �

So it is a temptation to believe that the smallest system¯ UMWMX of vector

fields containing¯

which is closed under Lie brackets does enjoy themaximal integral manifolds property. However, just after the statementof Nagano’s theorem (Part II), we have already learnt by means of Exam-ple 1.6(II) that in the � � and � 3 � � categories, the consideration of

¯ UMWMX isinappropriate.

1.12. Transport of a vector field by the flow of another vector field.To understand why

¯ UMWMX is insufficient, it will be clever to recall one ofthe classical definitions of the Lie bracket between two vector fields. Let�y� � and let � be a vector field defined in a neighborhood of � . Denoteby �/FÑ � the value of � at a point Ñ (this is a vector in Ä¥Î � ), by ¡ � =� �the push-forward of � by a local diffeomorphism ¡ , and by Ñ å¦ � ² FÑ �[instead of Ñðå¦ y � Ç ³ � � ØÑ � ] the local diffeomorphism at time

³induced

by the flow of � . If R is a second vector field defined in a neighborhood


of � , the Lie bracket between � and R at � is defined by:

(1.13) qÁ� ' R1t�Á� � � NÅÄ *-,²VÆ A 9 Rº� � ó�=� ² � � ØR�� ² Á� ��Ï�³ > kObserve that for every fixed

³ ëN 6 , the two vectors RÌÁ� � and�� ² � � 7R�� ² Á� ��Ï� belong Ä�. � .

Ç �mor-

diffeo-local

phism integralcurveof È

� : integral curve of É � : integral curve of ÉÉ=Ê

Infinitesimal version:the Lie bracket

ÈwËÌÉÎÍ Ê Ë�Ï5Ð(ÐÑÓÒÔ Ê ú Ü ûÏ Õ à5Ö × :

É=ÊBË Õ àØÖ × ÐÈwË�Ï5Ð

ËÌÉ Ê ÐVÙ5ËÌÈwË�Ï5Ð(ÐÈwË�Ï5Ðú Ô Ê û Ù úÛÚRú Ô Í Ê ú Ü û�û�û Ç #�Ü ^ .m_Ï Õ Í Ê Ö ×É=Ê(Ë Õ Í Ê Ö × Ð

Definition of the system Ý½Þ ß�à and of the Lie bracket

We explain how to read the right hand side of the diagram. In it:the integral curve of � passing through � is denoted by Q ; the inte-gral curve of R passing through the point � ² Á� � for

³very small is

denoted by Ñ ² � � ; its image by the local diffeomorphism � ² is denotedby � ² Ñ ² � � � ; the vector RÌ=� ² Á� �� is tangent to Ñ ² � � at the point� ² º� � ; the vector �� ² � � ØR�� ² Á� ��Ï� , transported by the differerentialof � ² , is in general distinct from the vector RÌÁ� � ; in fact, the differenceRÌÁ� � ó�� ² � � 7R=� ² º� �Ï�� divided by

³, tends to q� ' R1t#º� � as

³ ¦ 6 .Essentially,

¯ UMWMX collects all vector fields obtained by taking infini-tesimal differences (1.13) between vectors RÁ� � and transported vectors�� ² � � 7R�� ² Á� ��Ï� , and then iterating this processus to absorb all multipleLie brackets.

As suggested in the left hand side of the diagram, instead of taking theinfinitesimal differences, it is more general to collect all the vectors of theform =� ² � � ØRÌÁ� �� . This is the clue of Sussmann’s theorem. In fact, thesystem Â¯ which is sought for in Problem 1.6” should not only contain

¯ UMWMX ,but should also collect all the vector fields of the form �� ² � � ØR � , where

³is not an infinitesimal.

Lemma 1.14. Let Â¯ be a » L -linear system of vector fields containing¯

which has the strong integral manifolds property. Let �� , let � ' R?� ¯be two arbitrary vector fields defined in a neighborhood of � and let Ñ N� ² º� � be a point in the integral curve of � issued from � , with

³ � o small.


Then the linear subspace Â¯ ØÑ � necessarily contains the transported vector�� ² � � 7Rº� �Ï� .Proof. Let × be a strong Â¯ -integral manifold passing through � . AsÂ¯ µ²�EN ÄY´³× at every point

µ �Û× , and as¯

is contained in Â¯ , it fol-lows that the restricted vector field � J i is tangent to × . Consequently,the integral curve of � issued from � is locally contained in × , hence thepoint Ñ N � ² Á� � belongs to × .

Moreover, as¯

is contained in Â¯ , the vector Rº� � is tangent to × at � .The differential =� ² � � being a linear isomorphism between Ä+.±× and ÄVÎ�× ,it follows that the vector =� ² � � ØRº� �Ï� belongs to the tangent space Ä¥Î�× ,which coincides with Â¯ ØÑ � by assumption.

�1.15. The smallest

¯-invariant system of vector fields

¯ W�·hN . Based onthis crucial observation, we may introduce the smallest » L -linear systemof vector fields

¯ W�·CN (“inv” abbreviates “invariant”) containing¯

whichcontains all vectors of the form =� ² � � 7R � , whenever � ' R�� ¯ and

³ � o .It follows that �� ² � � e ¯ W�·CN º� � g N<¯ W�·hN =� ² º� �Ï� : the distribution of linearsubspaces �ðå¦ ¯ W�·CN º� �«� Äv. � is invariant under the local flow maps.

In [Su1973], it is shown that¯ W�·hN is concretely and finitely generated

as stated in Lemma 1.16 below. At first, some more notation is needed todenote the composition of several local diffeomorphisms of the form � ² .Let á denote the system of all tangent vector fields to

�, defined on open

subsets of�

. Let Úf��d with Úf� =and let � N �� i 'lklk�k�' � Ü � �Åá Ü

be a Ú -tuple of vector fields defined in their domains of definition. If³ N ³ i 'lklklkm' ³ Ü � � o Ü

is a Ú -tuple of “time” parameters, we will denote by� ² º� � the point� i² � e �� Ü² z º� �� g � NFy � Ç e ³ i,� i e �� y � Ç ³ Ü � Ü º� �Ï�� g6g 'whenever the composition is defined. The Ú -tuple ³ i 'lk�klk�' ³ Ü � will alsobe called a multitime parameter. For

³fixed, the map �<å¦ � ² º� � is

a local diffeomorphism between two open subsets of�

. Its local in-verse is the map �ïå¦ õ� �â ² Á� � , where õ� � N �� Ü 'lklklkm' � i � � ¯ Ü

andõ ³ � N ³ Ü 'lk�klk�' ³ i � . Moreover, if we define ³ ' ³ Ô � � N ³ i 'lklklkm' ³ Ü ' ³ Ô i 'lklklkm' ³ Ô Ü Ò �for general

³�N ³ i 'lk�klk�' ³ Ü � � oSÜ and³ Ô N ³ Ô i 'lklklkm' ³ Ô Ü Ò � � o�Ü Ò , we have��Ô² Ò � � ² N =��Ô ' � � ^ ² Ò � ² _ .

After shrinking the domains of definition, the composition of local dif-feomorphisms � ² is clearly associative, where it is defined. It follows thatthe set of local diffeomorphisms � ² constitutes a pseudogroup of local


diffeomorphisms. Here, the term “pseudo” stems from the fact that the do-mains of definitions have to be adjusted; not all compositions are allowed.

Lemma 1.16. ([Su1973]) The system¯ W�·CN is generated by the » L -linear

combinations of all vector fields of the form =� ² � � ØR � , for all RO� ¯ , allÚ -tuples � N =� i '�klklk�' � Ü � � ¯ Üof elements of

¯and all multitime

parameters³SN ³ i 'lklklkm' ³ Ü � � o Ü .

The definitions and the above reasonings show that the » L -linear system¯ UMWMX is a subsystem of the » L -linear system¯ W�·CN (of course, every system

is contained in á ): ¯��?¯ UMWMX �?¯ W�·hN � á kIn general, at a fixed point ��

, the inclusions¯ Á� �æ� ¯ UMWMX º� �\�¯ W�·CN Á� �z� á Á� �«N Äv. � may be all strict.

Example 1.17. Ono ò

, consider the system¯

generated by the three vectorfields QQvg�i ' g�i QQvg � ' ç i7è ¾ à � QQvg<� kThen it may be checked that

STTTTU TTTTV¯ 76 �§N�o Q ¾ � '¯ UMWMX Ø6 ��N�o Q ¾ �äã o Q ¾ à '¯ W�·hN Ø6 ��N�o Q ¾ � ã o Q ¾ à ã o Q ¾vñ 'á Ø6 �«N�o Q ¾ �äã o Q ¾ à ã o Q ¾ ñ ã o Q ¾Có k

Theorem 1.18. ([Na1966, Su1973]) In the ��¤ , � � and ��3 � � categories, thesystem

¯ W�·CN is the smallest one containing¯

that has the maximal strongintegral manifolds property. In the � ¤ category,

¯ W�·hN N9¯ UMWMX .Further structural properties remain to be explained.

1.19.¯

-orbits. The maximal strong integral manifolds of¯ W�·hN may be de-

fined directly by means of¯

, without refering to¯ W�·hN , as follows. Two

points � ' Ñ�� are said to be¯

-equivalent if there exists a local diffeo-morphism of the form � ² , � N =� i '�klklk�' � Ü � , ³òN ³ i 'lklklk ' ³ Ü � , Ú/�9d ,with � ² Á� � N Ñ . This clearly defines an equivalence relation on

�. The

equivalence classes are called the¯

-orbits of�

and will be denoted eitherby � º� � or shortly by � , when the reference to one point of the orbit issuperfluous.

Concretely, two points � ' Ñ¬� � belong to the same¯

-orbit if and only ifthere exist a continuous curve Ñ � qß6 ' = t ¦ �

with Ñ 76 �SN � and Ñ = �SN Ñ


together with a partition of the interval qß6 ' = t by numbers 6 N«³ A 8 ³ i¬8³ � 8 �� 8 ³ Ü N =and vector fields � i 'lklklkm' � Ü � ¯ such that for each% N = 'lklklkm' Ú , the restriction of Ñ to the subinterval q ³ � i ' ³ ��t is an integral

curve of � � . Such a curve will be called a piecewise integral curve of¯

.Let �B� �

. Then its¯

-orbit � º� � may be equipped with the finesttopology which makes all the maps

³ å¦ � ² Á� � continuous, for all Ú�� =,

all � N �� i '�klklk�' � Ü � � ¯ Ü and all multitime parameters³�N ³ i 'lk�klk�' ³ Ü � .

This topology is independent of the choice of a central point � inside agiven orbit ([Su1973]). Since the maps

o Ü £ ³ å¦ � ² º� � � �are al-

ready continuous, the topology of � Á� � is always finer than the topologyinduced by the inclusion � º� � ��

. It follows that the inclusion mapfrom � Á� � into

�is continuous. In particular, � º� � is Hausdorff.

1.20. Precise statement of the orbit theorem. We now state in lengththe fundamental theorem of Sussmann, based on preliminary versions dueto Hermann ([He1963]), to Nagano ([Na1966]) and to Lobry ([Lo1970]).It describes

¯-orbits as immersed submanifolds (1), (2) enjoying the ev-

erywhere accessibility conditions (3), (4), together with a local flow-boxproperty (5), useful in applications.

Theorem 1.21. (SUSSMANN [Su1973, Trv1992, BM1997, BER1999,BCH2005], [ P ]) The following five properties hold true.

(1) Every¯

-orbit � , equipped with the finest topology which makesall the maps

³ å¦ � ² º� � continuous, admits a unique differentiablestructure with the property that � is an immersed submanifold of�

, of class � ¤ , � � or � 3 � � .(2) With this topology, each

¯-orbit � is simultaneously a (con-

nected) maximal weak integral manifold of¯

and a (connected)maximal strong integral manifold of the

¯-invariant » L -linear sys-

tem¯ W�·CN ; thus, for every point �\� � , it holds Ä+.H� Á� ��N�¯ W�·CN º� � ,

whence in particular )�*�, ¯ W�·CN FÑ �N )+*-,�� º� � is constant for allÑ belonging to a given¯

-orbit � Á� � .(3) For every �<� �

, Úc� =, � � ¯ Ü

,³ � o Ü

such that � ² º� �is defined, the differential map =� ² � � makes a linear isomorphismfrom Ä�.H� Á� ��N�¯ W�·CN º� � onto Ä Ç Ü ^ .m_ � º� ��N�¯ W�·hN �� ² Á� �� .

(4) For every � ' Ñ¸� �belonging to the same

¯-orbit, there ex-

ists an integer Ú � =, there exists a Ú -tuple of vector fields� N �� i 'lk�klk�' � Ü�� ¯®Ü

and there exists a multitime³ � N

60 JOEL MERKER AND EGMONT PORTEN ³ � i 'lk�klk�' ³ �Ü � � o Ü such that � N � ² � ØÑ � and such that the differ-ential at

³ � of the map

(1.22)o Ü £ ³ å¦ � ² ØÑ � �D� Á� �

is of rank equal to )�*�,9� º� � .(5) For every �� , there exists an open connected neighborhood å�.

of � in�

and there exists a �¥¤ , � � or ��3 � � diffeomorphism

(1.23)

� � ì � � � £ ³ '-æØ� å ó ¦ wz ³ '-æØ� �çåÕ. 'where ç N )+*-,�� Á� � , where

� NO� g¬� o � J g J 8 = � , such that:2 wzØ6 ' 6 ��N � ;2 the plaque wE � � ì5� 6v� � is an open piece of the¯

-orbit of � ;2 each plaque wE � � ìæ�Óæ � � is contained in a single¯

-orbit;and:2 the set of

æ � � � � such that w e � � ìæ�Óæ � g is contained in thesame

¯-orbit � º� � is either finite or countable.

In general, foræ ëN 6 , the ç -dimensional plaques wz � � ìË�Óæ � � have

positive codimension in the nearby orbits. We draw a diagram, in whichç N )+*-,�� º� �N =, with the nearby

¯-orbits � � , �òÔ� , �� and �rÔ� having

dimensions � , � , � and�.

.è ÜL é Òà

é ñé àé Òñ Î+¹ âÇ #=êÜ Ù ^ .m_

é�ë ^ .m_

Local orbit flow box theorem

Property (4) is crucial: the maps (1.22) of rank )+*�,9� º� � are used todefine the differentiable structure on � º� � ; they are also used to obtainthe local orbit flow box property (5), as follows.

Let �ö� � and choose ÑK�9� º� � with Ñ5ëN � , to fit with the diagrams( Ñ N � would also do). Assuming that (4) holds, set ç � N )+*�,9� º� � ,introduce an open subset Ä � in some ç -dimensional affine subspace passingthrough

³ � inoSÜ

so that the restriction of the map (1.23) to Ä � still has rank

HOLOMORPHIC EXTENSIONS AND REMOVABLE SINGULARITIES 61ç at³�N ³ � . Introduce also an Øbyó ç � -dimensional local submanifold Í�.

passing through � with Ä+.�Í®. ã Ä�.Ã� º� �«N Ä�. � and set ÍxÎ � N õ� â ² ØÍã. � .Notice that ÄVÎÏÍxÎ ã Ä»Î � º� �rN ÄVÎ � , since the multiple flow map � ² � �stabilizes � Á� � . Then, as one of the possible maps w whose existence isclaimed in (5), we may choose a suitable restriction of:Ä � ì Í�Î £ ³ '-æØ� å ó ¦ � ² æØ� � �(k

ìí � í ~ # �

î Üî Ü ï ë ø�ð ü î Ñ

î Ñð Î+¹ Ç Ü ^ .m_Local foliation by the multitime flow map

1.24. Characterization of embedded¯

-orbits. A smooth manifold ×together with an immersion % � × ¦ �

is called weakly embedded if forevery manifold ¨ , every smooth map � � ¨ ¦ �

with � 7¨ � � × , then� � ¨ ¦ × is in fact smooth ([Sp1970]; the diagram is also borrowed).

An immersion of the real line in ñ � that is not weakly embedded

Proposition 1.25. ([Bel1996, BM1997, BCH2005]) Each¯

-orbit is count-able at infinity (second countable) and weakly embedded in

�.

As the multiple flows are diffeomorphisms, embeddability propagates.

Proposition 1.26. ([Bel1996, BM1997, BCH2005]) Let � be an¯

-orbitin�

and let ç � N )+*-,�� . The following three conditions are equivalent:2 � is an embedded submanifold of�

;2 for every point ��ô� , there exists a straightening map w asin (1.23) with � G wE � � ì � � � �·N wE � � ìæ� 6j� � ;2 there exists at least one point at which the preceding propertyholds.


Conversely, � is not embedded in�

if and only if for every � � � and for every local straightening map w centered at � as in (1.23), the setof

æ � � � � such that w e � � � ì(�Óæ � g is contained in � N � º� � isinfinite (nonetheless countable).

1.27. Local¯

-orbits and their smoothness. For ¾ running in the collec-tion of all nonempty open connected subsets of

�containing � , consider

the localized¯J�ò

-orbit of � in ¾ , denoted by � Y¾ ' � � . If � �u¾ � � ¾·i ,then � Y¾ � ' � �x� � Y¾·i ' � �+G ¾ � , so the dimension of � Y¾ ' � � decreasesas ¾ shrinks. Consequently, the localized

¯-orbit � Y¾ ' � � stabilizes and

defines a unique piece of local5 ¯ -integral submanifold through � A , calledthe local

¯orbit of � A and denoted by ��ß #" Á� � . In the CR context, this con-

cept will be of interest in Parts V and VI. Sometimes,¯

-orbits (in�

) arecalled global, to distinguish them and to emphasize their nonlocal, non-pointwise nature.

From the flow regularity Theorem 1.4 and from Theorem 1.21, it fol-lows:

Lemma 1.28. Global and local¯

-orbits are as smooth as¯

, i.e. � ¤ , � �or ��3 � � . Furthermore,Äv.Ã� �ß #" Á� �z� Ä�.H� �(' � �·N�¯ W�·hN º� ��'for every �� . This inclusion may be strict in the smooth categories � �and ��3 � � , whereas, in the �V¤ category, local and global CR orbits have thesame dimension.

In the � 3 � � category, the maximal integral curve of an arbitrary ele-ment of

¯is � 3 á i � � , trivially because the right hand sides of the equations�Hg Ü P8 �,H �Z8 N Ê Ü hg+p8 �� , Ú N = 'lklk�km' b , are � 3 � � . May it be induced that

general¯

-orbits are � 3 á i � � ? Trivially yes if )+*-, ¯ N =at every point.

Another instance is as follows. Letµ �ôd with

= � µ � b?ó =and let

¯ A N � R�°��²i��j°³��´ be a system of � 3 � � vector fields defined ina neighborhood of the origin in

o��that are linearly independent there.

Consider the system¯

generated by linear combinations of elements of¯ A . Achieving Gaussian elimination and a linear change of coordinates,we may assume that

µgenerators of

¯, still denoted by R§i 'lklklkm' R1´ , take

the form R�� N ½½�¾¼¿ � { � ´��¹»i Ê²�Á�uhg ' � �@½½�óP� , % N = 'lklklk 'Öµ, with hg ' � � N

5In certain references, local ô -orbits are considered as germs. Knowing by experiencethat the language of germs becomes misleading when several quantifiers are involved incomplex statements, we will always prefer to speak of local submanifolds of a certainsmall size.

HOLOMORPHIC EXTENSIONS AND REMOVABLE SINGULARITIES 63hg�i 'lk�klk�' gu´ ' �²i '�klklk�' � � ´ � and with ÊÕ�Á�uhg ' � � of class � 3 � � in a neighborhoodof the origin.

We claim that if � 76 � has (minimal possible) dimensionµ, then it is� 3 á i � � . This happens in particular if

¯is Frobenius-integrable.

Indeed, the local graphed equations of � 76 � must then be of the form�m� N �Ó�H�g � , � N = 'lklk�k�' bó µ , with the � � of class at least � 3 � � , thanks to thelemma above. Observe that the R·� are tangent to this submanifold if andonly if the � � satisfy the complete system of partial differential equations½ g �½�¾¼¿ hg ��N Ê²�Á�Ãhg ' �4�g �� , for % N = 'lklklkm'Öµ

, � N = 'lklklkm' b�ó µ , implying directlythat the � � are ��3 á i � � . In general, this argument shows that if )+*�,�� º� �coincides with )+*�, ¯ º� � , the orbit is � 3 á i � � at � .

Example 1.29. However, this improvement of smoothness is untrue when)+*�, ¯ º� � � = �9)+*-,�� º� � �9bEó = .Indeed, pick the function õ 3 � � N õ 3 � � �ö � of ö@� o

equal to zero forö �(6 and, for ö ��6 , defined by:õ 3 � � �ö �·N Õ ö 3 á+� ' *XW\6 8?:ö� = 'ö 3 H½Ä � o ö ' *$W\: N 6 kThis function is � 3 � � on

o, but for 0q ' ; � ¢ ¼ ' : � , it is not � « � < in

any neighborhood of the origin. Then ino ò

equipped with coordinateshg ' � ' ö ' 8 � , consider the hypersurface q of equation:6 N 8®ó£õ 3 á i � � �� õ 3 � � �ö � 'Then q is � 3 � � , not better. The two vector fields R«i � N ½½�¾ and R � � N½½�ó �öq gäõ 3 � � ¼ó÷� � t ½½�ø � Ý õ Ô 3 á i � � �� õ 3 � � �ö � Þ ½½ w have ��3 � � coefficients and aretangent to q . We claim that q is the local

� R«i ' R � � -orbit of the origin.Otherwise, there would exist a local two-dimensional submanifoldý ö N ¡Vhg ' � � ' 8 N ��hg ' � � � with Rzi and R � tangent to it. ThenqßRxi ' R � t N õ 3 � � êóS� �¬½½�ø should also be tangent. However, at points76 ' � ' ¡VØ6 ' � � ' ��Ø6 ' � �� , with � negative and arbitrarily small, R«i , R � and R|�

are equal to the three linearly independent vectors½½�¾ , ½½�ó and õ 3 � � ¼ó÷� �ò½½�ø .�

a2. FINITE TYPE SYSTEM AND THEIR GENERICITY (OPENESS AND

DENSITY)

2.1. Systems of vector fields satisfying¯ UMWMX N�¯ . Let

�be a � 3 (

= �(¼\�� ) connected manifold of dimension bf� =. By á , denote the system of


all vector fields defined on open subsets of�

(it is a sheaf). Let¯ A N�� R�°±�²i��j°��´ ' µ � = 'be a finite collection of � 3 i vector fields defined on

�, namely RS°ö�á �B� . Unlike in the � ¤ category, in the � 3 category, áä �B� is always

nonempty and quite large, thanks to partitions of unity. For this reason,we shall not work in the real analytic category, except in some specificlocal situations. The set of linear combinations of elements of

¯ A withcoefficients in � 3 i �('�ox� will be denoted by

¯(or

¯ i ) and called the� 3 i �B� -linear hull of¯ A .

Definition 2.2. A ��3 i �B� -linear system¯9� á is said to be of finite type

at a point �� if¯ UMWMX º� ��N Ä�. � .

If¯ UMWMX is of finite type at every point, then

¯ UMWMX NË¯ W�·CN N á and there isjust one maximal

¯-integral manifold in the sense of Sussmann:

�itself.

In 1939, Chow had already shown that the equality¯ UMWMX N á implies the

everywhere accessibility condition: every two points of�

may be joinedby integral curves of

¯. In 1967, Hormander established that every second

order partial differential operator ¨ � N RS� i � �� BR��´ �ùfíi��ùf A on adomain � � o �

whose top order part is a sum of squares of � � vectorfields R·° , = � Êe� µ

, such that¯ UMWMX N á is � � -hypoelliptic, namely¨ò><�c� � implies ><�¶� � . Vector field systems satisfying

¯ UMWMX N áhave been further studied by workers in hypoelliptic partial differentialequations and in nilpotent Lie algebras: Metivier, Stein, Mitchell, Stefan,Lobry and others.

In the next Parts V and VI, we will focus on propagational aspects thatare enjoyed by the (more general) smooth systems

¯that satisfy

¯ W�·hN N á ,but possibly

¯ UMWMX Á� � ëN áäº� � at every �0� �. Nevertheless, for com-

pleteness, we shall survey in the present section some classical geometricproperties of finite type systems.

2.3. Lie bracket flags, weights, privilegied coordinates and distanceestimate. Define

¯ i � N�¯and by induction, for �K�?d with �� K� ¼ ,

define¯ � to be the � 3 � -linear hull of

¯ � i �îq ¯ i 'Ï¯ � i t . Concretely,¯ � is

generated over � 3 � by iterated Lie brackets of length �(� of the form:R � N qßR � � ' qßR � à 'lk�klk�' qÙR � ! # � ' R � ! t k�klk tÓt ' = �J��(� kJacobi’s identity insures that q ¯ � � 'Ï¯ � à t �î¯ � �Ïá � à .

Denote¯ � º� � � Nùú}y ��4,Ä � RÌÁ� � � R�� ¯ � � . Clearly,

¯is of finite type at�� if and only it there exists an integer �Yº� � �(¼ with¯ K ^M.m_ Á� ��N Äv. � .

The smallest �Yº� � is sometimes called the degree of non-holonomy of¯

at

HOLOMORPHIC EXTENSIONS AND REMOVABLE SINGULARITIES 65� . Other authors call it the type of¯

at � , which we will do. The function�9å¦ �»Á� � �cq = ' ¼�t I � �(� is upper-semi-continuous: �YØÑ � � �»Á� � for Ñnear � .

Combinatorially, at a finite type point, it is of interest to introduce theLie bracket flag:� 6v� �?¯ i Á� �z�?¯ � Á� �x� �� ?¯ � º� �z� �� ?¯ K ^ .m_ º� �·N Äv. �(kThen a finite type point � is called regular if the integers b � ØÑ � � N)+*�, ¯ � ØÑ � remain constant in some neighborhood of � . It is elementary toverify ([Bel1996]) that, at such a regular point, the dimensions are strictlyincreasing: 6 8fb4i�Á� � 8îb � Á� � 8 �� 8fb K ^ .�_ º� �SN b k

Fix � , not necessarily regular. A local coordinate system � i 'Ï� � 'lklklkm'�� centered at � is linearly adapted at � if:

STTTTTTTTTTTU TTTTTTTTTTTV

¯ i º� ��Nuú}y ��4#. 9 QQ � i 'lklklk ' QQ � � � ^ .�_ > '¯ � º� ��Nuú}y ��4#. 9 QQ � i 'lklklk ' QQ � � � ^ .�_ 'lk�klk�' QQ � � à ^ .m_ > 'klk�k�klklkmklklkmklklk�k�klk�klk�k�klklkmklklklk�klk�klk�k�klklkmklklkmklklk�k�klk�klk�k�klklkmklklk¯ K ^ .m_ º� ��Nuú}y ��4#. 8 QQ � i 'lklklkm' QQ � � � ^ .�_ 'lklk�k�' QQ � � à ^ .m_ '�klklk QQ � � C ú Ü û ^ .m_ A k

Let us assign weightsÉ � to such linearly adapted coordinates

� � as fol-lows: the first group � i 'lklklk '�� ^ .m_ � being linked to

¯ i º� � , their weightswill all be equal to one:

É i N �� N É � � ^ .m_ N =. The second group � � � ^ .m_ á i '�klklk�'Ï� � à ^ .m_ � , linked to the quotient

¯ �vÁ� �-H±¯ i Á� � , will be assigneduniform weight two:

É � � ^ .m_ á i N �� N É � à ^ .m_ N � , and so on, untilÉ � C ú Ü û # � ^ .�_ á i N �� N É � C ú Ü û ^ .m_ N �»Á� � .Provided

¯is of finite type at every point, we claim that the original

finite collection¯ A produces what is called a sub-Riemannian metric; then

by means of weights, the topology associated to this metric may be com-pared to the manifold topology of

�in a highly precise way.

Indeed, let us define the (infinitesimal) sub-Riemannian length of a vec-tor ûm.ò� ¯ i Á� � by:J J û . J J r � N *�È/W ý �ü � i � �� $ü �� i7è � � ûm. N ü¥ivRxi�º� � � �� %ü&´�R1´±Á� � � k


For û .æë� ¯ i º� � , we setJ J ûm. J J r N � . The length of a piecewise � i curveÑ p¢ � , ¢x�/qÙ6 ' = t , will be the integral:Äýy È o 4�� r Ñ � � Nùý iA J J � Ñ p¢ �-H �Z¢ J J r �Z¢ k

Finally, the distance associated to the finite collection¯ A is:� r º� ' Ñ � � N *-È WÐmþ . Æ Î ÄXy È o 4�� r Ñ � k

Assume for instance �Yº� � N � , so that b � Á� � N b . If the coordinatesare linearly adapted, the tangent space Ä&. � then splits in the “horizontal”space, the � i 'lk�klkm'Ï� � � ^ .�_ � -plane, together with a (not unique) “vertical”space generated e.g. by the remaining coordinates. It is then classical thatthe distance from � to a point of coordinates � i 'lklk�k�'�� close to � enjoysthe estimate:� r e � ' � i 'lklklkm'�� hg ÿ¶J � i J � �� J � � � ^ .m_ J � J � � � ^ .�_ á i J i7è � � �� J � � J i7è � kHere, the abbreviation

ª ÿ zmeans that there exists

� ¢ =with� i z 8 ª 8 � z

. Notice that the successive exponents coincide withthe weights

É i '�klklk�'-É � � ^ .m_ ' É � � ^ .m_ á i 'lk�klk�' É � . In particular, to reach a pointof coordinates Ø6 'lklklk ' 6 ' � '�klklk ' � � , it is necessary to flow along

¯ A duringa time �{�-�54 k � i7è � . Observe that

J � i J � �� J � � J is equivalent to the dis-tance from � to

�induced by any Riemannian metric. Thus, the modified

distance � r is just obtained by replacing eachJ � � J by

J � � J i7è Î ¿ , up to a mul-tiplicative constant.

To generalize such a quantitative comparison between the � r -distanceand the underlying topology of

�, linearly adapted coordinates appear

to be insufficient. For; N ; i '�klklk�' ; ´ � �Od ´ , denote by R < the

J ; J-th

order derivation R < �i R < à� �� R < j´ . Beyond linearly adapted coordinates, onemust introduce privileged coordinates, whose existence is assured by thefollowing.

Theorem 2.4. ([Bel1996]) There exist local coordinates � i 'lklklkm'�� cen-tered at � that are privileged in the sense that each

� � is of order exactlyequal to

É � with respect to¯ A -derivations, namely, for % N = '�klklk ' b :R Ð � � J . N 6 ' for all Ñ with

J Ñ J � É ��ó = 'R < �¿ � � J .�ëN 6 ' for some; �� with

J ; �� JÃN É � kOnly if �Yº� � N � , linearly adapted coordinates are automatically privi-

leged ([Bel1996]). As soon as �Yº� � � � , privileged systems are unavoid-able.


Theorem 2.5. ([Bel1996]) For�

in a neighborhood of � , the estimate:� r Á� ' � i 'lklklkm'�� ÿ¶J � i J i7è Î � � �� J � � J i7è Î ~holds if and only if the coordinates are privileged.

For � ¢ 6 small, define the anisotropic ball� r Á� ' � � � N �� r º� 'Ï�»� 8 � � .

Corollary 2.6. ([Bel1996]) There exist��¢ =

such that=� �� º¹»i qÁó � Î ¿ ' � Î ¿ t �� r º� ' � �x� � �� º¹»i qÁó � Î ¿ ' � Î ¿ t k2.7. Local basis. At a non-regular point, the integers b Ü º� � ,Ú N = 'lklk�k�' �Yº� � are not necessarily strictly increasing. Thus, it isnecessary to express the combinatorics of the Lie bracket flag with moreprecision, in terms of what is sometimes called Hormander numbers ~ � ,�m� . From now on, we shall assume that the

µvector fields R�° , = �OÊy� µ

,are linearly independent at � and have � � or �V¤ coefficients. In bothcases, the formal Taylor series of every coefficient exists.

In the flag� 6v� �?¯ i Á� �z�?¯ � Á� �x� �� ?¯ �Ãº� �z� �� ?¯ K ^ .m_ º� �·N Äv. �(klet ~�i denote the smallest ÚÛ� � such that the dimension of

¯ Ü º� � islarger than the dimension of

¯ i º� � (at a regular point, ~yi N � ) and set�ui � N )+*�, ¯ � �HÁ� � ó µ � = . Similarly, let ~ � denote the smallest ÚE� = � ~yisuch that the dimension of

¯1Ü Á� � is larger than the dimension of¯ � � º� �

(at a regular point, ~ � ND� ) and set � � � N )+*�, ¯ � à Á� � óB)+*-, ¯ � �Hº� � .By induction, let ~�� á i denote the smallest Úc� = �+~�� such that thedimension of

¯ Ü Á� � is larger than the dimension of¯ � � º� � and set � � á i � N)+*�, ¯ � �� ÃÁ� � ó@)+*�, ¯ � � º� � .

Since � is a point of finite type, the process terminates until ~ g N �Yº� �reaches the degree of non-holonomy at � , for a certain integer �D� =

. Wethus have extracted the interesting information, namely the strict flag oflinear spaces:¯ i º� �x�î¯ � � º� �z�?¯ � à º� �z� �� ?¯ �� º� �·N Äv. �('with Lie bracket orders

= 8 ~yiE8�~ � 8 �� 8�~�g , whose successivedimensions may be listed parallelly:µ 8 µ � �±iz8 µ � �±i�� 8 �� 8 µ � ùi��<��¬g � � k


Next, let�;N � i 'lklk�km'�� be linearly adapted coordinates, vanish-

ing at � . We shall denote them by =¶ ' �²i ' � � 'lklklkm' �@g � , where ¶ � o ´ ,�HiK� o � � , � � � o � à , . . . , �@g@� o � � . As in the preceding paragraph, weassign weight

=to the ¶ -coordinates, weight ~ i to the �Hi -coordinates,

weight ~ � to the � � -coordinates, . . . , weight ~ g to the �mg -coordinates.The weight of a monomial

� � N ¶ < � Ð �i � Ð à� �� Ð �g is obviously defined asJ ; J � ~�i J Ñ i J �´~ � J Ñ � J � �� ´~�g J Ñ g J . We say that a formal power seriesÊ» �»�äN Ê»P¶ ' �Hi 'lklklkm' �@g � is an s�¼ � if all its monomials have weight �©¼ .Also, ÊY �»� is called weighted homogeneous of degree ¼ ifÊän¢C¶ ' ¢ � � �Ãi ' ¢ � à � � 'lk�klk�' ¢ � � �@g �·N ¢ 3 ÊY=¶ ' �Ãi ' � � '�klklk�' �mg � 'for all ¢ � o . As in the case of

o q q$µ²i '�klklk�' µ � t t with all weights equal to=,

every formal series ÊY=¶ ' �Hi ' � � 'lk�klk�' �@g � may be decomposed as a countablesum of weighted homogeneous polynomials of increasing degree.

Dually, we also assign weights to all the basic vector fields:½½:¹� will have

weight ó = , whereas for � N = '�klklk ' ~�g , the½½ � � � , ¾«N = 'lklklkm' �-� , will have

weight ó�~�� . The weight of a monomial vector field� � ½½:¸�¿ is defined to be

the sum the weights of� � with the weight of

½½:¸�¿ . Every vector field havingformal power series coefficients may be decomposed as a countable sumof weighted homogeneous vector fields having polynomial coefficients.

Theorem 2.8. ([Bel1996, BER1999]) Assume the local vector fields Rx° ,Ê N = 'lklk�k�'Öµ, have � � or �V¤ coefficients and are linearly independend at� . If the � � or � ¤ coordinates

��N P¶ ' �Hi ' � � 'lk�klk�' �@g � centered at � arepriveleged, then each R·° may be developed as:R�° N ÂR�°·�Js�76 � 'where each vector field:ÂR�° � N QQk¶H° � ùi��<��&i ùi�� &° � � � � =¶ ' �Ãi '�klklk�' �B� i � QQ �:� � � 'is homogeneous of degree ó = and has as its coefficients some polynomials�&° � � � � N �&° � � � � =¶ ' �Ãi 'lklklk ' �:� i � that are independent of �m� and are homoge-neous of degree ~��xó = .

A crucial algebraic information is missing in this statement: what are thenondegeneracy conditions on the �Y° � � � � that insure that the system is indeedof finite type at � with the combinatorial invariants ~ � and �-� ? The realproblem is to classify vector field systems that are of finite type, up to localchanges of coordinates. At least, the following may be verified.


Theorem 2.9. ([Bel1996, BER1999]) The vector fields ÂR�° , Ê N = 'lk�klk�'Öµ,

form a finite type system Â¯ A at � having the same combinatorial invariants~�� and � � and satisfying the same distance estimate as � r in Theorem 2.5.Moreover, the linear hull of Â¯ A generates a Lie algebra Â¯ UMWMX with the nilpo-tency property that all Lie brackets of length �?~ gx� = all vanish.

2.10. Finite-typisation of smooth systems of vector fields. As previ-ously, let

¯ A N � R�°u�²i��j°³��´ be a finite collection of ��3 i vector fieldsglobally defined on a connected manifold

�of class � 3 (

= � ¼B� � )and of dimension bÛ� =

. Let¯

be its ��3 i �B� -linear hull. Ifµ/N =

,then

¯ UMWMX N;¯, hence

¯cannot be of finite type, unless b N =

. So weassume be�;� and

µ �ô� . We want to perturb¯

slightly to õ¯ so as toget finite-typeness at every point: õ¯ UMWMX Á� �äN Ä�. � at every �/� � . Sincethe composition of Lie brackets of length � requires coefficients of vectorfields to be at least � � , if ¼\89� , then necessarily õ¯ UMWMX N õ¯ 3 stops at length¼ .

At a central point, say the origin in n � , and for n -analytic vector fieldsystems, the already presented Theorem 1.11(II) yields small perturbationsthat are of finite type at 6 . Of course, the same local result holds true forcollections of vector fields that are � � , or even � 3 i with ¼ large enough.Now, we want a global theorem.

What does it mean for õ¯ to be close to¯

? A vector field RÛ� á �B�may be interpreted as a section of the tangent bundle, in particular a � 3 imap

� ¦ Ä � . The most useful topology on the set � « ��' × � of all� « maps from a manifold�

to another manifold × (e.g. × N Ä � withq N ¼Kó = ) is the strong Whitney topology; it controls better than the so-called weak topology the behaviour of maps at infinity in the noncompactcase. Essentially, > ' ¡E�� « �(' × � are (strongly) close to each other if alltheir partial derivatives of order � q , computed in a countable collectionof charts w [ � ¾ [ ¦ o �

and � [ � å [ ¦ o �covering

�and × , ]E�yd , are� [ -close, the smallness of � [ ¢ 6 depending on the pair of charts 7w [ ' � [ � .

Precise definitions may be found in the monograph [Hi1976]. We thentopologize this way the finite product á �B� ´ .

Already two vector fields may well be of finite type on a manifold ofarbitrary dimension, e.g.

½½B¸ � and{ ��º¹ � � � ii ½½:¸�¿ on

o·�.

Theorem 2.11. ([Lo1970]) If the connected manifold�

of dimension b �� is � � á � à , then the set of pairs of vector fields¯ A � N �� ' R � �táä � � � on�

whose � � à á � i -linear hull¯

satisfies¯®� à á �òN�¯ , is open and dense in

the strong Whitney topology.


According to [Su1976], the smoothness� � � � á � à in [Lo1970] was

improved to� � �� in Lobry’s thesis (unpublished). We will summarize

the demonstration in the case� �� . However, since neither �4� � nor� � á � à are optimal, we will improve this result afterwards (Theorem 2.16

below).

Proof. Openness is no mystery. For denseness, we need some preliminary.If�

and × are two � « manifolds, we denote byd « �(' × � the bundle ofq -th jets of � « maps from

�to × . We recall that, to a � « map > � � ¦× is associated the q -th jet map � « > � � ¦ d « �(' × � , a continuous

map that may be considered as a kind of intrinsic collection of all partialderivatives of > up to order q . Let � � d « �(' × � ¦ �

be the canonicalprojection, sending a jet to its base point. For � � � , the fiber � i º� � maybe identified with

o i ? l ~Bl � , where × � � � � « � N ~ ^ � á « _no� o « o counts the number ofpartial derivatives of order �uq of maps

o � ¦ o �.

We will state a lemma which constitutes a special case of the jet transver-sality theorem. This particular statement (Lemma 2.12 below) generalizesthe intuitively obvious statement that any �4A curve graphed over

oBì5� 6v�u�ino � may always be slightly perturbed to avoid a given fixed � i curve q .Call a subset q � d « �(' × � algebraic in the jet variables if in every

pair of local charts, it possesses defining equations that are polynomials inthe jet variables >�� , = �@�� ~ , :@��d � ,

J : J ��q , whose coefficients areindependent of the coordinates

� � � . Of course, after a local diffeomor-phism

� å¦ Ê� �»� of�

, a general polynomial in the jet variables which isindependent of

�almost never remains independent of

Ê�in the new coor-

dinates. Nevertheless, in the sequel, we shall only encounter special setsq � d « ��' × � which, in any coordinate system, may be defined as zerosets of such special polynomials.

For instance, taking � Ü ' ¶R� ' ¶R� � Ü � as coordinates ond i �(' × � , where= �OÚ��b N )+*�, � and

= �(�\� ~ N )+*�,f× , a change of coordinates� å¦ Ê� �»� induces � Ü ' ¶ä� ' ¶R� � Ü � å ó ¦ Ê� Ü 'ZÊ¶R� 'ÓÊ¶ä� � Ü � , whereÊ¶R� N ¶R� is un-

changed but the new jet variables ¶Ý� � Ü NB{ �� ¹»i Ê¶R� � � ½ Ï¸ �½:¸ z involve the variables�(orÊ�). Nevertheless, the equations

� ¶Ý� � Ü N 6 ' = �J�\��~ ' = ��Ú�� b®�saying that the first (pure) jet vanishes are equivalent to

� Ê¶�� Ü N 6 ' = �¨��~ ' = ��ÚE�9b4� , since the invertible Jacobian matrixe ½ Ï¸ �½:¸ z g may be erased:

vanishing properties in a jet bundle are intrinsic !A theorem due to Whitney states that real algebraic sets are stratified,

i.e. are finite unions of geometrically smooth real algebraic manifolds.The codimension of q is thus well-defined.


Lemma 2.12. ([Hi1976]) Assume q � d « �(' × � is algebraic in the jetvariables and of codimension � = �()+*-, � . Then the set of maps >�� « �(' × � whose q -th prolongation � « > � � ¦ d « �(' × � does not meetq at any point is open and dense in the strong Whitney topology.

Although � « > is only continuous, the fact that the bad set q is algebraicenables to apply the appropriate version of Sard’s theorem that is used inthe jet transversality theorem.

We shall apply the lemma by defining a certain bad set q which, ifavoided, means that a pair of vector fields on

�is of finite type at ev-

ery point.Assume

� �B� � � and let =� ' R � �xá �B� � . Both vector fields have� � � i coefficients. With q � N �Ãb ó = , denote byd � � i �áä �B� � � the fiber

bundle of the �ub ó = � -th jets of these pairs. In some coordinates providedby a local chart ¾ £ Ñ�å¦ � i ØÑ � '�klklk�'Ï�&� FÑ �Ï� � o·� , with ¾ ��

open, wemay write � Ne{ i�� »� ½½:¸ ¿ and R Np{ i�� R1�¼ �»� ½½B¸ ¿ . In such achart, the �ub�ó = � -th jet map �j� � i �� ' R � � ¾ ó ¦ d � � i !áx�Õ �B��J�ò¥� isconcretely given by:¾ £ � å ó ¦ e Q �¸ ��¼ �»��' Q �¸ R1�ê �»�,g �H|m} ~ � ¡ � ¡ � � � i � i�� kWe denote by � � � � and R1� � � the corresponding jet variables. A �� localdiffeomorphism

� å¦ Ê�KN«Ê� �»� induces a triangular transformation involv-ing the chain rule between these jets variables, with coefficients dependingon the �ub -th jet of

Ê� �»� , some of which are only �4A , which might be un-pleasant. Fortunately, our bad set q will be shown to be algebraic withrespect to the jet variables �� and R1� � � in any system of coordinates.

Let �� ' R � � áä � � � . To write shortly iterated Lie brackets, we de-note �H)�� R � N qÁ� ' R1t , so that �H)�� R N q� ' q� ' R1tÓt , �H)�=� � � R Nq� ' qÁ� ' qÁ� ' R1t-tÓt and so on. Also, we set �H) A �� R � N R . Define a subsetq ��d � � i �á��j �B�Ï� as a union q N qxÔ I q�Ô Ô I q�Ô Ô Ô , where:2 firstly q Ô is defined by the �Ãb equations �� A N R�� A N 6 ;2 secondly, q Ô Ô is defined by requiring that all the b ì b minors of the

following b ì �Ãb � matrixe �H) A �� R¸�H) i �� R �� H) � � i �� R g·'vanish;2 thirdly, qxÔ Ô Ô is defined similarly, after exchanging � with R .


Lemma 2.13. In the vector space of real b ì �Ãb � matrices, isomorphic too � � à , the subset of matrices of rank �OFb ó = � is a real algebraic subset ofcodimension equal to Fb � = � .

Without obtaining a complete explicit expression, it is easily verifiedthat �Ã) � =� � ØR � , 6 � �Ë� �Ãbæó =

, is a universal polynomial in the jetvariables � � � � and R1� � � . Under a local change of coordinates

� å¦ Ê� �»� ,if the two vector fields � and R transform to � and to R (push-forward),all the multiple Lie brackets �Ã) � =� � R then transform to �H) � � � R , thanksto the invariance of Lie brackets. Geometrically, the vanishing of eachof the b ì b minors defining qxÔ Ô and q�Ô Ô Ô means the linear dependenceof a system of b vectors, thus it is an intrinsic condition. Consequently,although the jet variables �� and R�� are transformed in an unpleasantway through diffeomorphisms, the sets q Ô , q Ô Ô and q Ô Ô Ô may be defined byuniversal polynomials in the jet variables �� and R1� � � , that are the samein any system of local coordinates.

The lemma above and an inspection of a part of the complete expressionof the �H) � �� 7R � , 6ö�{� ��ub�ó =

provides the following information.Details will be skipped.

Lemma 2.14. The two subsets q Ô Ô and q Ô Ô Ô ofd � � i �áä �B� � � are both al-

gebraic in the jet variables and of codimension Øb � = � outside q«Ô .To conclude the proof of the theorem, we have to show that arbitrar-

ily close to �� ' R � , there are pairs of finite type. Since qzÔ has codimen-sion �Ãb ¢ )+*-, � , a first application of the avoidance Lemma 2.12 yieldsa perturbed pair, still denoted by �� ' R � , with the property that at everypoint �c� �

, either �/º� � ëN 6 or RÁ� � ëN 6 . Since qxÔ Ô and qxÔ Ô Ô bothhave codimension b�� = ¢ )+*�, � , a second application of the avoidanceLemma 2.12 yields a perturbed pair such that the two collections of �Ãbvector fields �H) � �� R , 6 �(��p�Ãb\ó = , and �H) � ØR � � , 6E�(��p�Ãb�ó = ,generate Ä � at every point �� . The proof is complete.

�To improve this theorem, let

µ �e� and consider the set á �B� ´ of col-lections of

µvector fields globally defined on

�that are � 3 i for some¼ � � to be chosen later. If

¯ A N � Rzi ' R � 'lklklkm' R1´�� , is such a col-lection, its elements may be expressed in a local chart � i 'lklklkm'�� asR·° N�{ ��Á¹»i w®° � �¼ �»� ½½B¸�¿ , for Ê N = '�klklk�'�µ

. Since the coefficients are ��3 i ,it is possible to speak of

¯ « only for q � ¼ . We want to determine thesmallest regularity ¼ such that the set of

µ-tuples

¯ A��¥á �B� ´ that are offinite type at every point of

�is open and dense in á �B� ´ for the strong

Whitney topology.


As ina1.8(II), let 5 3 µ²� denote the dimension of the subspace � 3 µ²�

of the free Lie algebra �Y µH� that is generated as a real vector space bysimple words (abstract Lie brackets) of length � ¼ . Then 5 3 µ²� is themaximal possible dimension of

¯ 3 Á� � at a point �� . We know that¯ 3

is generated by simple iterated Lie brackets of the formÝ R�° � ' Ý R�° à 'lklklk ' Ý R�°�� # � ' R�° � Þ k�klk Þ Þ 'for all q��ì¼ and for certain (not all) Ê�� with

= �BÊ�i ' Ê � 'lklk�k�' Ê «l i ' Ê « � µthat depend on the choice of a Hall-Witt basis (Definition 1.9(II)) ofC 3 µ²� .

We choose ¼ minimal so that 5 3 µ²� � ��)�*�, � N �Ãb . This fixesthe smoothness of

�. For " N µ � = 'lklklk ' 5 3 µH� , we order linearly asR�� N { ��º¹»i �� »� ½½:¸�¿ the chosen collection of iterated Lie brackets that

generate¯ 3 . If q N q4*" � denotes the length of R�� , namely R�� ¯ « ^��P_ of

the form R�� N Ý R·° � 'lklk�km' Ý R�° � ú�� û # � ' R�° � ú�� û Þ k�klk Þ , there are universal differ-ential polynomials ` � ° �¼�cbcbcb � ° � ú�� û in the 0q4*" � ó = � -th jet of the coefficients wã° � �such that �� ê �»��N ` � ° �¼�cbcbcb � ° � ú�� û e-d « ^��P_ i¸ wz �»�,g . Also, in a fixed local systemof coordinates, we form the b ì 7�Ãb � matrixe w�i � � klklk w�´ � �Å�4´ á i � � klklk � � � � � g i�� kSimilarly as in the proof of the previous theorem, we define a “bad” subsetq ofd 3 i �áä �B� ´ � by requiring that the dimension of

¯ 3 Á� � is ��Fb ó = �at every point �(� �

. This geometric condition is intrinsic and neitherdepends on the choice of local coordinates nor on the choice of a Hall-Wittbasis. Concretely, in a local system of coordinates, q is described as thezero-set of all b ì b minors of the above matrix. Thanks to Lemma 2.13 andto an inspection of a portion of the explicit expressions of the jet polyno-mials ` � ° ��cbcbcb � ° � ú�� û e-d « ^��P_ i¸ wz �»�,g , we may establish the following assertion.

Lemma 2.15. The so defined subset q N ý )+*-, ¯ 3 º� � �ñb5ó = '�� ofd 3 i �áä �B� ´ � is algebraic in the jet variables and of codimensionØb�� = � .

Then an application of the avoidance Lemma 2.12 yields that, after anarbitrarily small perturbation of

¯ A , still denoted by¯ A , we have

¯ 3VÁ� �zNÄ�. � for every � � � . Equivalently, the type �Yº� � of � is finite at everypoint and satisfies �Yº� � �(¼ .Theorem 2.16. Let

µ � � be an integer and assume that the connectedb -dimensional abstract manifold�

is � 3 , where ¼ is minimal with theproperty that the dimension 5 3 µH� of the vector subspace � 3 µ²� , of the free


Lie algebra �Y µH� havingµ

generators, that is generated by all brackets oflength � ¼ , satisfies 5 3 µ²� ��1)+*-, � N �1b kThen the set of collections of

µvector fields

¯ A �¥áä � � ´ that are of type� ¼ at every point is open and dense in á �B� ´ for the strong Whitneytopology.

A more general problem about finite-typisation of vector field structuresis concerned with general substructures of a given finite type structure.

Open question 2.17. Given a finite type collection näA N�� ²i�� , ��, of � 3 i vector fields on

�of class � 3 with the property that n 3 Á� �äNÄ�. � at every point and given a � 3 i subsystem

¯ A NË� R�°u�²i��j°³��´ , �¬� µ ��ó = , of the form R·° Nc{ i�� ®° � � �� , is it always possible to perturbslightly the functions �1° � � � � ¦ o

so as to render¯ A of finite type at

every point ? If so, what is the smallest regularity ¼ , in terms ofµ, � and

the highest type of n A at points of�

?

Finally, we mention a result similar to Theorem 2.16 that is valid inthe �� category and does not use any Lie bracket. It is based on Suss-mann’s orbit Theorem 1.21. The reference [Su1976] deals with severalother genericity properties, motivated by Control Theory.

Theorem 2.18. ([Su1976]) Assumeµ �� and ¼\�� . The set of collections¯ A Nc� R�°u�²i��j°³��´ of

µvector fields on a connected � 3 manifold

�so that�

consists of a single¯

-orbit, is open and dense in á �B� ´ equipped withthe strong Whitney � 3 i topology.

2.19. Transition. The next Section 3 exposes the point of view of Analy-sis, where vector field systems are considered as partial differential opera-tors, until we come back to the applications of the notion of orbits to CRgeometry in Section 4.a

3. LOCALLY INTEGRABLE CR STRUCTURES

3.1. Local insolvability of partial differential equations. Until the1950’s, among analysts, it was believed and expected that all linear par-tial differential equations having smooth coefficients had local solutions([Trv2000]). In fact, elliptic, parabolic, hyperbolic and constant coeffi-cient equations were known to be locally solvable. Although his thesissubject was to confirm this expectation in full generality, in 1957, Hans


Lewy ([Lew1957]) exhibited a striking and now classical counterexam-ple of a � � function ¡ in a neighborhood of the origin of

o � , such thatRx> N ¡ has no local solution at all. Here, R N ½½ Ïê � µ ½½�� is the gen-erator of the Cauchy-Riemann anti-holomorphic bundle tangential to theHeisenberg sphere of equation û N µ Êµ in

� � , equipped with coordinates�µ ' Éä�·N � �È%�¶ ' üò�È%*û � .From the side of Analysis, almost absent in the two grounding

works [Po1907] and [Ca1932] of Henri Poincare and of Elie Cartan,Lewy’s discovery constituted the birth of smooth linear PDE theoryand of smoooth Cauchy-Riemann geometry. Later, in 1971, the simplertwo-variables Mizohata equation

½m�½:¸ ó¨% � Ü ½m�½:¹ N ¡ was shown by Grushinto be non-solvable, if Ú is odd, for certain ¡ . One may verify that the set ofsmooth functions ¡ for which Lewy’s or Grushin’s equation is insolvable,even in the distributional sense, is generic in the sense of Baire. For Ú N =

,the Mizohata vector field

½½B¸ ó©% ��½½:¹ intermixes the holomorphic andantiholomorphic structures, depending on the sign of

�.

In 1973 answering a question of Lewy, Nirenberg ([Ni1973]) exhibiteda perturbation

½½:¸ ó±% � = �æwz ��' ¶ ��½½:¹ of the Mizohata vector field, wherew is � � and null for� �e6 , such that the only local solutions of RS> N 6

are the constants. A year later, in [Ni1974], he exhibited a perturbationof the Lewy vector field having the same property. A refined version is asfollows.

Let � be a domain ino � , exhausted by a countable family of compact

sets �Å� , � N = ' � 'lklklk with �Å� � ª ÈÓ4Ì�Ç� á i . If >0�¶� � =� '³�S� , definethe Frechet semi-norms �6�H7> � � N ,��u� ¸ | Ç � � ¡ � ¡ �<� J Q&�¸ >1 �»�±J and topologize� � �� 'Ö�x� by means of the metric �Y > ' ¡ � � Nø{ ��¹»i � � ^ � À _i á � � ^ � À _ . Considerthe set Â� � N! R N �ù �Ï¹»i Ê��H �»��QQ � � � Ê6�Ì�\� � =� '³�S�#"z'equipped with this topology for each coefficient Ê�� .Theorem 3.2. ([JT1982, Ja1990]) The set of Rî� Â� for which the solutionsüy�\� i �� '³�� of Räü N 6 are the constants only, is dense in Â� .

These phenomena and others were not suspected at the time of Lie, ofPoincare, of E. Cartan, of Vessiot and of Janet, when PDE theory wasfocused on the algebraic complexity of systems of differential equationshaving analytic coefficients. In 1959, Hormander explained the behavior ofthe Lewy counter-example, as follows. The references [Trv1970, Trv1986,


ES1993, Trv2000] provide further survey informations about operators ofprincipal type, operators with multiple characteristics, pseudodifferentialoperators, hypoelliptic operators, microlocal analysis, etc.

Let ¨ N ¨� ��' Q ¸ �@N { �H|�} ~ � ¡ � ¡ � � Ê � �»� Q&�¸ be a linear partial dif-ferential operator of degree ~ having � � complex-valued coefficientsÊ � � � ¦ �

defined in a domain � � oS�. Its symbol ¨� ��'-Ø�� N{ ¡ � ¡ � � Ê � �»� =% Ø�� is a function from the cotangent Ä � � � ì?o �

to�

. Its principal symbol is the homogeneous degree ~ part ¨ � ��'-Ø�� N{ ¡ � ¡ ¹ � Ê � �»� P% ØÕ� � . The cone of points ��'-Ø�� í� ì o � A � 6v� � such that¨ � ��'-ØÕ�SN 6 is the characteristic set of ¨ , the locus of the obstructions toexistence as well as to regularity of solutions > of ¨� ��' Q ¸ � > N ¡ .

The real characteristics of ¨ are called simple if, at every character-istic point � A '-Ø A � with

Ø A ëN 6 , the differential �%$Ö¨ � N { � Ü ¹»i ½'& ?½ $ z � Ø Üwith respect to

Øis nonzero. It follows from homogeneity and from Eu-

ler’s identity that the zeros of ¨ are simple, so the characteristic set is aregular hypersurface of � ì o � A � 6j� � . One can show that this assump-tion entails that the behaviour of ¨ is the same as that of ¨ � : in a certainrigorous sense, lower order terms may be neglected. In his thesis (1955),Hormander called such operators of principal type, a label that has stuck([Trv1970]).

Call ¨ solvable at a point� A �I� if there exists a neighborhood ¾ of

� Asuch that for every ¡��\� � ¶¾ � , there exists a distribution > supported in ¾that satisfies ¨r> N ¡ in ¾ . In 1955, Hormander had shown that a principaltype partial differential operator ¨ is locally solvable if all the coefficientsÊ � �»� , J : J�N ~ , of its principal part ¨ � are real-valued. On the contrary,if they are complex-valued, in 1959, he showed:

Theorem 3.3. ([Ho1963]) If the quantity�ù Ü ¹»i Q�¨� ��'-ØÕ�Q Ø Ü Q�¨ � ��'-ØÕ�Q � Ü

is nonzero at a characteristic point � A '-Ø A � �DÄ�� , for someØ A ëN 6 , then¨ is insolvable at

� A .With ¨ � ��' Q ¸ � � N { ¡ � ¡ ¹ � Ê � �»� Q&�¸ , denote by

� � � i� ��'-Ø�� the prin-

cipal symbol of the commutator ÝÁ¨ � ��' Q ¸ � ' ¨ � ��' Q ¸ � Þ , obviously zeroif ¨ � has real coefficients. The above necessary condition for local solv-ability may be rephrased as: if ¨ is locally solvable at

� A , then for allØ � oS� A � 6v� : ¨ � ��',Ø��N 6 N D � � � i� ��'-Ø��N 6 k


This condition explained the non-solvability of the Lewy operator appro-priately.

3.4. Condition (P) of Nirenberg-Treves and local solvability. The geo-metric content of the above necessary condition was explored and general-ized by Nirenberg-Treves ([NT1963, NT1970, Trv1970]). Recall that theHamiltonian vector field associated to a function > N >ã ��'-ØÕ� �\� i �� ì�o��&�is � � � Nø{ �Ü ¹»i û ½m�½ $ z ½½B¸ z ó ½m�½B¸ z ½½ $ z ü . A bicharacteristic of the real part` ��',Ø�� of ¨ � ��',Ø�� is an integral curve of � º , namely:� ��Ó¢ N o � �H) $ ` ��'-ØÕ� ' � Ø�Z¢ N ó o � �H) ¸ ` ��'-ØÕ� kIt follows at once that the function ` ��'-Ø�� must be constant along itsbicharacteristics. When the constant is zero, a bicharacteristic is calleda null bicharacteristic. In particular, null bicharacteristics are contained inthe characteristic set, which explains the terminology.

Then Hormander’s necessary condition may be interpreted as follows.Let

� ��'-ØÕ� be the imaginary part of ¨ � ��',Ø�� . An immediate computationshows that the principal symbol of q ` ��' Q ¸ ��'�� ' Q ¸ � t is given by:� i ��'-ØÕ��N �ù Ü ¹»i

( Q `Q Ø Ü ��',Ø�� Q �Q � Ü ��'-Ø�� ó Q �Q Ø Ü ��',Ø�� Q `Q � Ü ��'-Ø��#) kEquivalently, � im ��'-Ø��·N �� EH �Z¢ � ��'-Ø��kTheorem 3.3 says that the nonvanishing of

� i at a characteristic point en-tails insolvability. In fact, Nirenberg-Treves observed that if the order ofvanishing of

�along the null characteristic of ` is odd then insolvability

holds. Beyond finite order of vanishing, what appeared to matter is onlythe change of sign. Since the equation ¨r> N ¡ has the same solvabilityproperties as µ·¨ò> N ¡ , for all µ�� A � 6v� , this led to the following:

Definition 3.5. ([NT1963, NT1970]) A differential operator ¨ of prin-cipal type is said to satisfy condition (P) if, for every µ�� A � 6v� , thefunction

ª ,B�µS¨ � � does not change sign along the null bicharacteristic ofË�y =µ�¨ � � .The next theorem has been shown for ¨ having � ¤ coefficients and in

certain cases for ¨ having � � coefficients by Nirenberg-Treves, and fi-nally, in the general � � category by Beals-Fefferman (sufficiency) and byMoyer (necessity).


Theorem 3.6. ([NT1963, Trv1970, NT1970, BeFe1973, Trv1986]) Con-dition (P) is necessary and sufficient for the local solvability in R�� of aprincipal type linear partial differential equation ¨r> N ¡ .

Except for complex and strongly pseudoconvex structures, little isknown about solvability of � � systems of PDE’s, especially overdeter-mined ones ([Trv2000]). In the sequel, only vector field systems (order~ N =

), studied for themselves, will be considered.

3.7. Involutive and CR structures. Following [Trv1981, Trv1992,BCH2005], let

�be a � ¤ , � � or � 3 � � ¼Ë�ô� , 6î8¸:08 =

) paracom-pact Hausdorff second countable abstract real manifold of dimensionQ � =

and leté

be a � ¤ , � � or � 3 i � � complex vector subbundle of� Ä � � N��+* Ä � of rank q , with= � qæ� Q . Denote by

é . its fiber ata point �D� � . Denote by , the orthogonal of

éfor the duality between

differential forms and vector fields. It is a vector subbundle of� Ä � � ,

whose fiber at a point �� isé.-. N ý �¸� � Ä��. � � � N 6s�²È é . � .

The characteristic set � � N , G Ä�� (real Ä�� ) is in general not a vectorbundle: the dimension of �4A. may vary with � , as shown for instance by thebundle generated over

o � by the Mizohata operator Q ¸ ó % � Q ¹ .From now on, we shall assume that the bundle

éis formally integrable,

i.e. that q é ' é t � é . Thené

defines:2 an elliptic structure if �². N 6 for all �� ;2 a complex structure ofé . ã é . N�� Ä+. � for all � � � ;2 a Cauchy-Riemann (CR for short) structure if

é . G é . N0� 6v� forall �\� � ;2 an essentially real structure if

é . N é . , for all �\� � .

In general,é

will be called an involutive structure if q é ' é t N é .Let us summarize basic linear algebra properties ([Trv1981, Trv1992,BCH2005]). Every essentially real structure is locally generated by realvector fields. Every complex structure is elliptic. If

éis a CR structure

(often called abstract), the characteristic set � is in fact a vector subbundleof Ä�� of rank Q ó�� q ; this integer is the codimension of the CR structure.A CR structure is of hypersurface type if its codimension equals

=.

3.8. Local integrability and generic submanifolds of�·�

. The bundleéis locally integrable if every �B� �

has a neighborhood ¾�. in whichthere exist Ö � N Q ó q functions µHi 'lklk�k�' µäÛ � ¾». ¦ �

of class � ¤ , � �or ��3 � � whose differentials � µHi 'lklklkm' � µäÛ are linearly independent and span

HOLOMORPHIC EXTENSIONS AND REMOVABLE SINGULARITIES 79, J�ò Ü (or equivalently, are annihilated by sections ofé

). In other words, thehomogeneous PDE system

é > N 6 has the best possible space of solutions.Here is a canonical example of locally integrable structure. Consider a

generic submanifold�

of��

of class � ¤ , � � or � 3 � � , ¼�� =, 6��(:f� =

,as defined in

a2.1(II) and in

a4.1 below. Let �ð��6 be its codimension and

let ~ N b óF�f�<6 be its CR dimension. Let ÄÌ" � N Ä � G¨d Ä � (areal vector bundle) and let

� Ä � N¸�/* Ä � . Define the two complexsubbundles Ä i � A � and ÄÌA�� i � N Ä i � A � of

� Ä � whose fibers at a point�� are:Õ Ä i � A. � N�� í.z�¨% d �r. � �r.í�yÄ ". � � NO�10 .í� � Äv. � � d20 . N ó�% 0 .²� 'Ä A�� i. � N�� í.Ìó % d �í. � �í.í��Ä ". � � NË�10 .í� � Ä+. � � d30 . N % 0 .²� kGeometrically, Ä i � A � and ÄA�� i � are just the traces on

�of the holomor-

phic and anti-holomorphic bundles Ä i � A � � and Ä A�� i � � , whose fibers at apoint � are

{ � Ü ¹»i Ê Ü ½½Bê z ÂÂ . and{ � Ü ¹»i " Ü ½½ Ïê z ÂÂ . . They satisfy the Frobenius

involutivity conditions qMÄ i � A �(' Ä i � A � t � Ä i � A � and q ÄÌA�� i �(' ÄA�� i � t �ÄÌA�� i � . More detailed background information may be found in [Ch1991,Bo1991, Trv1992, BER1999].

On such an embedded generic submanifold�

, choose as structure bun-dleé

just ÄÌA�� i � �<� Ä � . Then clearly, the b holomorphic coordinatefunctions µHi 'lklklkm' µ � are annihilated by the anti-holomorphic local sections{ � Ü ¹»i " Ü ½½ Ïê z of ÄA�� i � and they have linearly independent differential, atevery point of

�. A generic submanifold embedded in

�S�carries a locally

integrable involutive structure. Conversely:

Lemma 3.9. Every locally integrable CR structure is locally realizable asthe anti-holomorphic structure induced on a generic submanifold embed-ded

�1�.

Proof. Indeed, if a real Q -dimensional ��¤ , � � or ��3 � � manifold�

bearsa locally integrable CR structure, the map

0 N =µÕi '�klklk�' µäÛ � produces anembedding of the open set ¾¥. as a local generic submanifold

� � N40 Y¾¥. �of� Û , with

0 � é ��N ÄA�� i � .

�A locally integrable CR structure is sometimes called locally realizable

or locally embeddable.

3.10. Levi form. Leté

be an involutive structure, not necessarily locallyintegrable and let 5Ö.��5�H. � Ä��. � be a nonzero characteristic covector at� .


Definition 3.11. The Levi form at � in the characteristic codirection 5�.��².<A � 6v� � Ä��. � A � 6v� is the Hermitian form acting on two vectors �ò. '�6 .r�é º� � as: 7 . � 8 Ü e �í. ' 6 . g � N =�<% 5�. e Ý � ' 6 Þ gS'where � ,

6are any two sections of

édefined in a neighborhood of � and

satisfying � º� �ÌN �r. , 6 º� �äN96 . . The resulting number is independentof the choice of such extensions � ,

6.

For the study of realizability of CR structures of codimension one, non-degeneracy of the Levi form, especially positivity or negativity, is of cru-cial importance. An abstract CR structure of hypersurface type whose Leviform has a definite signe is said to be strongly pseudoconvex, since, aftera possible rescaling of sign of a nonzero characteristic covector, all theeigenvalues of its Levi form are positive.

3.12. Nonembeddable CR structures. After Lewy’s discovery, the firstexample of a smooth strictly pseudoconvex CR structure in real dimension�

which is not locally embeddable was produced by Nirenberg in 1973([Ni1973]), cf. Theorem 3.3 above. For CR structures of hypersurfacetype, Nirenberg’s work has been generalized in higher dimension under theassumption that the Levi form is neither positive nor negative, in any char-acteristic codirection. Let b5�� and let � i N =

, � Ü N óä� , Ú N � 'lklklk ' b .

Theorem 3.13. ([JT1982, BCH2005]) There exists a � � complex-valuedfunction ¡ N ¡V ��' ¶ ' � � defined in a neighborhood of the origin in

� � ì�oand vanishing to infinite order along

�� i N ¶ji N 6v� such that the vectorfields: ÂR1� � N QQ Êµ:� ó % � ��µ:�� = � ¡V ��' ¶ ' � ��KQQ�� 'are pairwise commuting and such that every � i solution � of ÂR1�� N 6 , � N= ' � 'lk�klk�' b defined in a neighborhood of the origin must satisfy

½ g½ � 76 ��N 6 .This entails that the involutive structure spanned by the ÂR1� is not lo-

cally integrable at 6 . One may establish that the set of such ¡ is generic.Crucially, the Levi-form is of signature ØbKó = � .Open problem 3.14. Find versions of generic non-embeddability for CRstructures of codimension

=having degenerate Levi-form. Find higher

codimensional versions of generic non-embeddability.


3.15. Integrability of complex structures and embeddability ofstrongly pseudoconvex CR structures. Let us now expose positiveresults. Every formally integrable essentially real structure

é N Ë�y éis

locally integrable, thanks to Frobenius’ theorem; however the conditionq é ' é t � é entails the involutivity ofË�y é

only in this special case. Also,every analytic formally integrable CR structure is locally integrable: itsuffices to complexify the coefficients of a generating set of vector fieldsand to apply the holomorphic version of Frobenius’ theorem. For complexstructure, the proof is due to Libermann (1950) and to Eckman-Frolicher(1951), see [AH1972a, Trv1981, Trv1992].

Theorem 3.16. Smooth complex structures are locally integrable.

This deeper fact has a long history, which we shall review concisely. Onreal analytic surfaces, isothermal coordinates where discovered by Gaussin 1825–26, before he published his Disquisitiones generales circa superfi-cies curvas. In the 1910’s, by a nontrivial advance, Korn and Lichtensteintransferred this theorem to Holder continuous metrics.

Theorem 3.17. Let � � � N v � ü � � � C � ü �ÓûY� 7 �Óû � be a � A�� 76�89:ö8 = �Gaussian metric defined in some neighborhood of 6 in

o � . Then thereexists a � i � � change of coordinates �ü ' û � å¦ ;:ü ' :û � fixing 6 and a �4A�� function :q N :q®;:ü ' :û � such that::q e �<:ü � �@�=:û � g N v �Óü � �f� C �Óü � û � 7 � û � k

A modern proof of this theorem based on the complex notation andon the Q formalism was provided by Bers ([Be1957]) and by Chern([Ch1955]). In the monograph [Ve1962], deeper weakenings of smooth-ness assumptions are provided.

As a consequence of this theorem, complex structures of class � A�� onsurfaces may be shown to be locally integrable. Let us explain in lengththis corollary.

At first, remind that an almost complex structure on �Ãb -dimensionalmanifold

�is a smoothly varying field

dfN d . � . | L of endomorphismsof Äv. � satisfying

d . � d . N ó ª ) . Thanks tod

, as in the standard complexcase, one may define Ä A�� i. � � NÛ� �í.z�¨% d .��r. � �r.í�yÄv. � � and then thebundle

é � N ÄA�� i � is a complex structure in the PDE sense ofa3.7 above.

Conversely, given a complex structureé �� Ä � , then locally in some

neighborhood ¾V. of an arbitrary point �\� � , there exist local coordinates � i 'lklk�k�'�� ' ¶ji 'lklklkm' ¶ � � vanishing at � so that b complex vector fields of


the form: 0 � � N �ù Ü ¹»i Ê Ü � �»Q ¸ z �È%�ù Ü ¹»i " Ü � �»Q ¹ z '

with Ê Ü � �u76 �äN � Ü � � N " Ü � �uØ6 � , spané J�ò Ü . The associated almost complex

structure is obtained by declaring that, at a point of coordinates ��' ¶ � , onehas:d 8 �ù Ü ¹»i Ê Ü � �»Q ¸ z A N �ù Ü ¹»i " Ü � �VQ ¹ z@�HÈ+) d 8 �ù Ü ¹»i " Ü � �»Q ¹ z A N ó �ù Ü ¹»i Ê Ü � �»Q ¸ z kLemma 3.18. The bundle

é �ï� Ä � satisfies q é ' é t � é if and only if,for every two vector fields � and

6on�

, the Nijenhuis expression:×æ�� '>6�� N q d � 'äd?6 t&ó d q�� '�d@6 t&ó d q d � '>6 t�ó�q � '�6 tvanishes identically.

The proof is abstract nonsense. Also, one verifies that ×ö7>�� ' ¡ 6�� N>�¡x×ö!� '�6¬� for every two smooth local function > and ¡ : the expression× is of tensorial character. In symplectic and in almost complex geometry([MS1995]), the following is settled.

Definition 3.19. The almost complex structure is called integrable if, insome neighborhood ¾V. of every point � � � there exist b complex-valuedfunctions µHi 'lklk�k�' µ � � ¾». ¦ �

of class at least � i and having linearlyindependent differentials such that � µ Ü � d�N % � � µ Ü , for Ú N = 'lklklkm' b .

One verifies that it is equivalent to requireé µ Ü N 6 , Ú N = 'lklk�km' b : inte-

grability of an almost complex structure coincides with local integrabilityofé N ÄA�� i � .

Now, we may come back to the integrability Theorem 3.16. To an arbi-trary Gaussian metric ¡ N � �� as in Theorem 3.17, with v ¢ 6 , 7�¢ 6 andv 7 ó C � ¢ 6 , are associated both a volume form and an almost complexstructure:�BA � Ä À � N · v 7 ó C � �ÓüDC � û �ÃÈ+) d À � N =· v 7 ó C � 9 ó C ó 7v C >ökConversely, given a volume form and an almost complex structure

don a

surface, an associated Riemannian metric is provided by:¡¥ � ' � � � N �BAw� Ä � 'äd � � kAccording to Korn’s and Lichtenstein’s theorem, there exist coordinatesin which the metric is conformally flat, equal to q¬��Óü � �@�Óû � � . In these


coordinates, the associated complex structure is obviously the standardone:

d Q%E N Q � andd Q � N óÌQBE . In fact, any local change of coordi-

nates �ü ' û � å¦ ;:ü ' :û � which respects orthogonality of the curvilinear co-ordinates, i.e. transforms the Gaussian isothermal metric to a similar one:q¬��2:ü��®�@�=:û�� , commutes with

d, so that the map ü �ì%Óû?å¦ :ü �ì%F:û is

holomorphic. In conclusion:

Theorem 3.20. ��A�� öØ6 8 : 8 = �complex structures are locally inte-

grable.

The generalization to several variables of the theorem of Korn and Licht-enstein is due to Newlander-Nirenberg, who solved a question raised byChern. The proof was modified and the smoothness assumption was per-fected by several mathematicians: Nijenhuis-Woolf ([NW1963]), Mal-grange, Kohn, Hormander, Nirenberg, Treves ([Trv1992]) and finally Web-ster ([We1989c]) who used the Leray-Koppelman Q homotopy formula to-gether with the Nash-Moser rapidly convergent iteration scheme for solv-ing nonlinear functional equations.

Theorem 3.21. ( �� , 6<8 : 8 =: [NN1957]; � i � � , 6<8 : 8 =

:[NW1963, We1989c]; � � : [Trv1992]) Suppose that on the real manifold�

of dimension �ÃbË� . , the formally integrable complex structureé

is� � or ��3 i � � , ¼@�ï� , 6y8O: 8 =. Then there exist local complex-valued

coordinates �µÃi 'lklklkm' µ � � annihilated byé

which are � � or � 3 � � .Finally, an elementary linear algebra argument ([Trv1981, Trv1992,

BCH2005]) enables to deduce local integrability of � � or � 3 i � � ellip-tic structures from the above theorem. In fact, elliptic structures are shownto be locally isomorphic to

� Û ì o «� Û , equipped with½½ Ïê ¿ , ½½ ° � .

Problem 3.22. Is a formally integrable involutive structure havingpositive-dimensional characteristic set locally integrable ?

Again the history is rich. Integrability results are known only forstrongly pseudoconvex CR structures of hypersurface type. Solving a ques-tion raised by Kohn in 1965, Kuranishi ([Ku1982]) showed in 1982 that � �strongly pseudoconvex abstract CR structures of dimension �/G are locallyrealizable. His delicate proof involved a study of the Neumann operator inR·� spaces, for solving the tangential Cauchy-Riemann equations, togetherwith the Nash-Moser argument. In 1987, Akahori ([Ak1987]) modified thetechnique of Kuranishi and included the case of dimension H .


In 1989, to solve an associated linearized problem, instead of the Neu-mann operator, Webster used the totally explicit integral operators ofHenkin.

Theorem 3.23. ([We1989a, We1989b]) Let�

be a strongly pseudoconvex �Ãbzó = � -dimensional CR manifold of class � � . Then�

admit, locally neareach point, a holomorphic embedding of class � 3 , providedb �?. ' ¼ �� = ' Q ��I�¼r�@'1bKó ��k

The main new ingredient in his proof was Henkin’ local homotopy op-erator Q L on a hypersurface

� ��:> N Q L ¨� > � �%E� Q L > � ' >��Ø6 ' = � ó W�� , '

known to hold for bï��. . For this reason, Webster suspected the exis-tence of refinements based on an insider knowledge of Q techniques. In1994, using a modified homotopy formula yielding better � 3 -estimates,Ma-Michel [MM1994] improved smoothness:¼\� =KJ ' Q �(¼¬� = ��kUp to now, the five dimensional remains open. In fact, the solvability ofQ L > N ¡ for a 76 ' = � -form on a hypersurface of

� � requires a specialtrick which does not lead to a homotopy formula. Nagel-Rosay [NR1989]showed the nonexistence of a homotopy formula in the ' -dimensional case,emphasizing an obstacle.

Open problem 3.24. Find generalizations of the Kuranishi-Akahori-Webster-Ma-Michel theorem to higher codimension, using the integral for-mulas for solving the Q L due to Ayrapetian-Henkin. Replace the assump-tion of strong pseudoconvexity by finer nondegeneracy conditions, e.g.weak pseudoconvexity and finite type in the sense of Kohn.

3.25. Local generators of locally integrable structures. Abandoningthese deep problems of local solvability and of local realizability, let ussurvey basic properties of locally integrable structures. Thus, let

ébe a� � or ��3 i � � locally integrable structure of rank q on a �43 � � or � � mani-

fold�

of dimension Q . Denote by Ö N Q ó%q the dimension of , N éL-.

Let �\� � and let� . denote the dimension of �Õ. N , G Ä �. � . Notice that Ö ó � . � �( Ö ó � . � � � .x� *q�ó Ö � � . �SN Ö �rq N Q just below.

Theorem 3.26. ([Trv1981, Trv1992, BCH2005]) There exist real coordi-nates: e7� i 'lk�klk�'�� Û � Ü ' ¶ji 'lk�klk�' ¶6Û � Ü ' ü¥i 'lk�klk�' ü � Ü ' �Ãi 'lk�klk�' � «l Û á � Ü g·'


defined in a neighborhood ¾¥. of � and vanishing at � , and there exist � �or � 3 � � functions w»� N wÌ�u ��' ¶ ' ü ' � � with wÌ�HØ6 ��N 6 , �jwÌ�u76 � N 6 , � N= 'lklklkm' � . , such that the differentials of the Ö functions:( µ Ü � N�� Ü �¨%Ó¶ Ü ' Ú N = 'lk�klkm' Ö ó � . 'É � � N ü �ã�È%jwÌ�u ��' ¶ ' ü ' � ��' � N = 'lklk�km' � .span , J�ò Ü .

Since �jwÌ�u76 �¬N 6 , there exist unique coefficients " �Á� � N " �º� �u ��' ¶ ' ü ' � �such that the vector fields:

�Å� � N � Üù � ¹»i " �º� � QQ~ü � ' Ú N = '�klklk�' � . 'satisfy �Å� � É � à � N � � �#� � à , for �Ãi ' � � N = 'lklklkm' � . . Define then the q vectorfields:

STTTTTU TTTTTVR Ü � N QQ Êµ Ü ó %

� Üù � ¹»i QYw �Q Êµ Ü � � ' Ú N = 'lk�klk�' Ö ó � . 'R Ô� � N QQ �:� ó %

� Üù � ¹»i QYw �Q �:� � � ' � N = 'lklk�k�' q�ó Ö � � . kClearly, 6 N R Ü � �µ ÜÏà �·N R Ü É � �·N R1Ô� =µ Ü �·N R1Ô� � É � à � , hence the structurebundle

é J�ò Ü is spanned by the R Ü ' R1Ô� . One may verify the commutationrelations ([Trv1981, Trv1992, BCH2005]):6 N Ý R Ü � ' R ÜÏà Þ N Ý R Ü ' R Ô� Þ N Ý R Ô� � ' R Ô� à Þ '6 N Ý R Ü ' �Å� Þ N ÝºR Ô� � ' �Ç� à Þ N qÁ�Å� � ' �Å� à t k

Remind that if an involutive structureé

is CR, then� . is independent of� and equal to the codimension Q ó��wq N � � . It follows that Ö ó � N Ö ó Q ��wq N q , or q�ó Ö � � N 6 : this means that the variables ��²i 'lklklkm' � «l Û á � Ü �disappear.

Corollary 3.27. In the case of a CR structure of codimension�, the local

integrals are: ( µ Ü � N�� Ü �È%Ó¶ Ü ' Ú N = 'lklklk ' q 'É � � N ü �1�¨%jwÌ�u ��' ¶ ' ü � ' � N = 'lklklk ' � '


and a local basis for the structure bundleé JÌò Ü is:

R Ü � N QQ Êµ Ü ó % �ù � ¹»i QYw �Q Êµ Ü � � ' Ú N = 'lk�klkm' q kWe recover a generic submanifold embedded in

� Û which is graphed bythe equations ûä� N wÌ�u ��' ¶ ' ü � , as in Theorem 2.3(II), or as in Theorem 4.2below.

3.28. Local embedding into a CR structure. But in general, the coor-dinates �Ãi 'lklk�k�' � «l Û á � Ü � are present. A trick ([Ma1992]) is to introduceextra coordinates P¢³i 'lklk�km' ¢ «l Û á � Ü � and to define a new structure on theproduct ¾V. ì\o «� Û á � Ü generated by the following local integrals:

STU TV:µ Ü � N µ Ü ' Ú N = '�klklk�' Ö ó � . ':µ Ü � N � Ü Û á � Ü �È%Z¢ Ü Û á � Ü ' Ú N Ö ó � .z� = 'lklk�km' q ':É � � N É � ' � N = '�klklk�' � . k

The associated structure bundle õé is spanned by:

STU TVõR Ü � N R Ü ' Ú N = '�klklk�' Ö ó � . 'õR Ô� � N =� R Ô� � %� QQ�¢=� ' � N = '�klklk�' q ó Ö � � . k

It is a CR structure of codimension� . on ¾V. ì/o «l Û á � Ü . All analytico-

geometric objects defined in ¾¥. can be lifted to ¾V. ìfo «l Û á � Ü , just bydeclaring them to be independent of the extra variables p¢mi 'lklklkm' ¢ «l Û á � Ü � .Such an embedding enables one to transfer elementarily several theo-rems valid in embedded Cauchy-Riemann Geometry, to the more generalsetting of locally integrable structures. For instance, this is true of mostof the theorems about holomorphic or CR extension of CR functions pre-sented Part V. In addition, most of the results stated in

a3,a4 and

a5 below

hold in locally integrable structures.

3.29. Transition. However, for reasons of space and because the possi-ble generalizations which we could state by applying this embedding trickwould require dry technical details, we will content ourselves to just men-tion these virtual generalizations, as was done in [MP1999]. For fur-ther study of locally integrable structures, we refer mainly to [Trv1992,BCH2005] and to the references therein.

In summary, in this Section 3, we wanted to show how our approach isinserted into a broad architecture of questions about solvability of partial


differential equations, about the problem of realizability of abstract CRstructures, as well as into hypo-analytic structures. Thus, even if some ofthe subsequent surveyed results (exempli gratia: the celebrated Baouendi-Treves approximation Theorem 5.2) were originally stated in the contextof locally integrable structures, even though we could as well state them inthis context or at least in the context of locally embeddable CR structures(as was done in [MP1999]), we shall content ourselves to state them in thecontext of embedded Cauchy-Riemann geometry, just because the verycore of the present memoir is concerned by Several Complex Variablestopics: analytic discs, envelopes of holomorphy, removable singularities,etc. a

4. SMOOTH GENERIC SUBMANIFOLDS AND THEIR CR ORBITS

4.1. Definitions of CR submanifolds and local graphing equations. Webegin by some coordinate-invariant geometric definitions. Some implicitlemmas are involved (the reader is referred to [Ch1989, Ch1991, Bo1991,BER1999]). Let

ddenote the complex structure of Ä �S� (see

a2.1(II)). A

real connected submanifold� ��·�

of class at least � i is called:2 Totally real if Ä�. � G¨d Äv. � Nø� 6j� for every �(� � . Then�

has codimension ��b and is called maximally real if � N b . Thecomplex vector subspace �í. � N Äv. � � d Ä�. � of Äv. �1� has com-plex dimension �Ãb§ó�� . If M æON�PRQ Ü � � denotes any

�-linear projection

of Äv. �� onto ��. and if S4. is a small neighborhood of � in�·�

, thenM æON�P Q Ü � G S4. � is maximally real in ��. .2 Generic if Ä�. � � d Äv. � N Äv. �1� for every �D� � . Then�

hascodimension �9�¸b and is maximally real only if � N b . ThenÄv. � G¨d Äv. � is the maximal

�-linear subspace of Ä+. � and has

complex dimension equal to the integer ~ � N b�óI� , called the CRdimension of

�. It is obviously constant, as � runs in

�.2 Cauchy-Riemann (CR for short) if the maximal

�-linear subspaceÄv. � GÇd Äv. � of Ä�. � has constant dimension ~ (necessarily �9b )

for � running in�

. If�

has codimension � , the integer Æ � N�·ó�b§�è~ is called the holomorphic codimension of�

. Then for �running in

�, the complex vector subspaces �í. � N Äv. � � d Äv. �

of Äv. �1� have constant complex codimension Æ , which justifies theterminology. If M æON�P Q Ü � � denotes any

�-linear projection of Ä+. � �

onto ��. , and if S4. is a small open neighborhood of � in�·�

, thenT� . � N M æUN�P Q Ü � G S�. �


is a generic submanifold of�·� " .

Ina2.1(II), we have graphed totally real, generic and, generally, Cauchy-

Riemann local submanifolds� ��·�

, but only in the algebraic and in theanalytic category. In the smooth category, the intrinsic complexification��É � N 6j� disappears, but ��. N Ä�. � � d Ä�. � still exists, so that furthergraphing functions are needed.

Theorem 4.2. ([Ch1989, Ch1991, Bo1991, BER1999, Me2004a]) Let� �c�1�be a real submanifold of codimension � and let �f� � . There

exist complex algebraic or analytic coordinates centered at � and �Yi ¢ 6such that

�, supposed to be � º , where

� N � or where� N ¼ ' : � ,¼\� = , 6��:ö� = , is locally represented as follows:2 If

�is totally real, letting �+i N �Ãb\óJ��p6 and Æ N ��ó bf�p6 ,

then ��i��ÈÆ N b and� N ý É i ' É � � � e � K �� ì % o K � gòìy� " �ª , É i N w·i� Ë�y»É i � '-É � N � � Ë�yMÉ i � � 'for some

o K � -valued �Mº map w·i and some� " -valued �»º map � �

satisfying w·i�76 ��N 6 and � � 76 ��N 6 .2 If�

is generic, letting ~ N �íó@b , then ~ï�È� N b and� N ý �µ '-É � �±� �� ì e � K� � ì % o K g � ª , É(N wx Ë�y µ ' ª ,>µ ':Ë�yMÉä� � 'for some

o K-valued �»º map w satisfying wz76 ��N 6 .2 If

�is Cauchy-Riemann, letting ~ N�Í¦Ë )+*�, � , Æ N ��óíb��Â~;�6 , and �+i N �ÃbKó �6~¶ó¨� ��6 , then ~ï�È�+i¥�@Æ N b and� N�ý =µ ' É i ' É � � �í� �� ì e � K �� ì % o K � g ì�� " �ª , É i N w·i� Ë�y µ ' ª ,?µ ' Ë�yMÉ i ��'xÉ � N � � Ë�y µ ' ª ,>µ ' Ë�yMÉ i �R�&'

for someo K � -valued �»º map w·i with w·im76 �¶N 6 and some� " -valued � º map � � with � � 76 �ÛN 6 which is CR (defini-

tion ina4.25 below) on the local generic submanifold

� i � Ný =µ ' É i � �±� �� ìFe � K �� ì % o K � g � ª , É i N w�im Ë�y µ ' ª ,Jµ ':Ë�y2É i � � .An adapted linear change of coordinates insures that the differentials at

the origin of the graphing maps all vanish.

A CR manifold�

being always locally graphed above a generic sub-manifold of

�� " , the remainder of this memoir will mostly be devotedto the study of � � or � 3 � � generic submanifolds of

��. The above local

representation of a generic�

will be constantly used.


4.3. CR vector fields. Let�

be generic, of class at least � i , representedby û N wz ��' ¶ ' ü � as above, in coordinates =µ ' Éä�yN � �ì%¶ ' ü�� %*û � .Sometimes, we shall also write û N wx�µ ' ü � , being it clear that w is notholomorphic with respect to µ . Here, we provide a description in coordi-nates of three useful families of vector fields.

There exist ~ anti-holomorphic vector fields defined in� �� ì e � K� � ì % o K g of the form:

R Ô Ü N QQ Êµ Ü � Kù ��¹»i Ê Ô� � Ü ��' ¶ ' ü � QQ ÊÉ � 'whose restrictions to

�span ÄÌA�� i � . To compute the coefficients ÊÕÔ� � Ü , the

conditions 6 R Ô Ü w��Ã ��' ¶ ' ü � ó¥û�� yield:

�·wÌ� � Ïê z N Kù � ¹»i =% � � � � ó wÌ� � E � � Ê Ô �Á� Ü kIn matrix notation, the solution is: Ê�Ô N � =%%V K � K ó w<E � i � w Ïê , withÊÕÔ N ØÊ²Ô� � Ü � i�� Ü �

�i��<�� K and w Ïê N wÌ� � Ïê z � i�� Ü ��i��<�³� K both of size � ì ~ . SinceqÙÄ A�� i �(' Ä A�� i � t � Ä A�� i � , the R Ô Ü commute. They are extrinsic.

Also, there exist ~ intrinsic sections of� Ä � of the form:

R Ü � N QQ Êµ Ü � Kù ��¹»i Ê�� Ü ��' ¶ ' ü � QQ/ü � 'written in the coordinates ��' ¶ ' ü � of

�, which span the structure bundleÄ A�� i � � � Ä � . Since ��' ¶ ' ü � are coordinates on

�, restricting theR Ô Ü JML to

�amounts to just drop the terms �� ½½��P� in each

½½ ÏÎ � appearing inR Ô Ü . Hence:

Lemma 4.4. One has Ê6� � Ü N i� Ê²Ô� � Ü .Another argument is to first introduce the � vector fields:

�Å� N Kù � ¹»i " �Á� �H ��' ¶ ' ü � QQ/ü � 'that are uniquely determined by the conditions� � � � � à N �Ç� � É � à J L��·N �Å� � �ü � à �È%#wÌ� à ��' ¶ ' ü ��&k


Equivalently, the coefficients " �Á� � satisfy:� � �� à N Kù � ¹»i � � à � � �¨%jwÌ� à � E � � " �º� � � 'whence, in matrix notation: " N WV K � K �¨%jw<E � i . Here, " N *" �º� � � i��<�� Ki�� Kand w<E N wÌ� � E � � i�� Ki��<�³� K both are � ì � matrices.

Similarly as in Corollary 3.27, the R Ü defined above span the structurebundle

é N Ä A�� i � having the local integrals µHi 'lklk�k�' µ � ' É i J L�'lklk�k�' É K J L ,if and only if they satisfy 6 N R Ü É � JML��§N R Ü q�ü �ã�È%jwÌ�u ��' ¶ ' ü � t . Seek-ing the R Ü under the form R Ü N ½½ Ïê z � { K� ¹»i Æ �Á� Ü ��' ¶ ' ü � � � , it followsfrom

� � �#� � à N �Ç� � !ü/� à �È%jwÌ� à ��' ¶ ' ü �� that Æ,� � Ü N ó�%jwÌ� � Ïê z . Reexpressingexplicitly the � � in terms of the

½½ E � as achieved above, we finally get in

matrix notation Ê N P%%V K � K ó w<E � i � w Ïê . This yields a second, more intrin-sic computation of the coefficients Ê<� � Ü and a second proof of Ê<� � Ü N i� Ê²Ô� � Ü .

If õ is a � � or ��3 � � function on�

, its differential may be computed as

�Óõ N �ù Ü ¹»i R Ü �õ � � µ Ü ��ù Ü ¹»i R Ü �õ � � Êµ Ü � Kù ��¹»i �Ç�u�õ � � É � ÂÂ L k

Lemma 4.5. ([Trv1981, BR1987, Trv1992, BCH2005]) The following re-lations hold:( R Ü � =µ Ü�à �SN � Ü �� Ü�à ' R Ü É � ��N 6 ' �Ç�H=µ Ü ��N 6 ' �Å� � É � à JML��·N � � �#� � à 'qÙR Ü � ' R Ü à t N qßR Ü ' �Å�Öt N qÁ�Å� � ' �Å� à t N 6 k4.6. Vector-valued Levi form. Let �\� � and denote by ��. the projection� Ä�. � ó ¦ � Äv. �>H e Ä i � A. � ã ÄA�� i. � g

.

Definition 4.7. The Levi map at � is the Hermitian� K

-valued form actingon two vectors �r. '�6 .í��Ä i � A. �

as:XY 7 . � Ä i � A. � ì Ä i � A. � ó ¦ � Äv. �>H�e Ä i � A. � ã Ä A�� i. ��g7 .j��í. '�6 . � � N =�<% ��. e Ý � ' 6 Þ Á� �,g·'where � '�6 are any two sections of Ä i � A � defined in a neighborhood of �satisfying � º� �ÌN �r. , 6 º� �äN96 . . The resulting number is independentof the choice of such extensions � '�6 .

As � varies, this yields a smooth bundle map. The Levi map

7 . is non-degenerate at � if its kernel is null:

7 .j��í. '�6 . ��N 6 for every6 . implies

HOLOMORPHIC EXTENSIONS AND REMOVABLE SINGULARITIES 91�r. N 6 . On the opposite,�

is Levi-flat if the kernel of

7 . equals Ä i � A. �at every � . If

�is a hypersurface =� N = �

, one calls

7 . the Levi form at� . Then�

is strongly pseudoconvex at � if the Levi form

7 . is definite,positive or negative. These definitions agree with the ones formulated ina3.10 for abstract CR structures.

Theorem 4.8. ([Fr1977, Ch1991]) In a neighborhood ¾�. of a point �\� �in which the kernel of the Levi map is of constant rank and defines a � ¤ ,� � or �¥3 i � � ¼\�� ' 6��:ö� = � distribution of ~yi -dimensional complexplanes ¨®Î � Ä»Î � ,

� Ñ�� ¾». , the distribution is Frobenius-integrable,hence

�is � ¤ , � � or � 3 i � � foliated by complex manifolds of dimension~�i .

In particular, a Levi-flat generic submanifold of CR dimension ~ is foli-ated by ~ -dimensional complex manifolds. These observations go back toSommer (1959). In [Ch1991], one founds a systematic study of foliationsby complex and by CR manifolds.

4.9. CR orbits. Let� �ø��

be generic and consider the system¯

ofsections of Ä " � . To apply the orbit Theorem 1.21, we need

�to be at

least � � , in order that the flows are at least � i . By definition, a weak Ä " � -integral submanifold

��O�satisfies Ä+. � ÐBÄ". � , at every point �5� �

.Equivalently,

�has the same CR dimension as

�at every point.

In the theory of holomorphic extension exposed in Part V, local andglobal CR orbits will appear to be adequate objects of study. They consti-tute one of the main topics of this memoir.

Proposition 4.10. ([Trp1990, Tu1990, Tu1994a, Me1994, Jo1996,MP1999, MP2002]) Let

� �Ë�·�be generic of class � ¤ , � � or � 3 � � with¼\�� and 6��:ö� = .

(a) The (global) ÄÌ" � -orbits are called CR orbits. They are denotedby � $&% or by � $&% �(' � � , if the reference to one of point � �D� $&%is needed.

(b) The local CR orbit of a point �Ë� �is denoted by � �! #"$&% �(' � � .

It is a local submanifold embedded in�

, closed in a sufficientlysmall neighborhood of � in

�.

(c) Local and global CR orbits are � ¤ , � � or � 3 � < , for every;

with6�8 ; 8?: .

(d)�

is partitioned in global CR orbits. Each global CR orbit is injec-tively immersed and weakly embedded in

�, is a CR submanifold

of��

contained in�

and has the same CR dimension as�

.


(e) Every (immersed) CR submanifold�<�;�

having the same CRdimension as

�contains the local CR orbit of each of its points

(f) CR orbits of the smallest possible real dimension �<~ N� Í¦Ë )�*�, � satisfy Ä�.H� $&% N Ä". � $&% at every point, hence arecomplex ~ -dimensional submanifolds.

According to Example 1.29, CR orbits should be �43 i � � , not smoother.But in generic submanifolds, they also can be described as boundariesof small attached analytic discs ([Tu1990, Tu1994a, Me1994]) and the� 3 � � A N[Z <1\ � � 3 � < smoothness of the solution in Theorem 3.7(IV) ex-plains (c).

Let us summarize some structural properties of CR orbits, useful in ap-plications. A specialization of Theorem 1.21(4) yields the following.

Proposition 4.11. For every �\� � , there exist ÚK�yd , sections R i '�klklk�' R Üof ÄÌ" � and parameters

³ � i 'lk�klk�' ³ �Ü � o such that R Ü² �z �� ØR i² � � º� �� SN �and the map ³ i 'lklklkm' ³ Ü � å ó ¦ R Ü² z �� ØR i² � Á� �� is of rank)+*�,�� $&% �(' � � at ³ � i 'lk�klkm'-³ �Ü � .

The dimension of any � $&% is equal to �<~<� ç , for some ç � d with6 � ç �F� . Denote:2 � � � á�� the union of CR orbits of dimensionN �6~ï� ç ;2 � â� � á�� the union of CR orbits of dimension ��<~e� ç ;2 � �� á�� the union of CR orbits of dimension ��<~e� ç .

The function � å¦ )+*-,�� $&% �(' � � is lower semicontinuous. It followsthat � â� � á�� is open in

�and that � �� á�� is closed in

�.

Let �� , let �. be a small piece of � $&% �(' � � passing through � ,of dimension �6~0� ç . , for some integer ç . with 65� ç .\�«� , and let � .be a local � � or � 3 � � submanifold of

�passing through � and satisfyingÄ�. � . ã Ä�.H�§. N Äv. � . Call � . a local orbit-transversal. Implicitly, �í. N^]

if ç . N � . Then, in a sufficiently small neighborhood of � :2 lower semi-continuity: �í. G � â� � á�� Ü á i N^];2 equivalently: ��. G � â� � á�� Ü N � . G � � � á�� Ü ;2 � . G � �� á�� Ü is closed.

Proposition 4.12. If�

is � ¤ , then at every point � � � , for every orbit-transversal ��. passing through � , the closed set �í. G � �� á�� Ü is a localreal analytic subset of ��. containing � .


A CR-invariant subset of�

is a union of CR orbits. A closed (for thetopology of

�) CR-invariant subset is minimal if it does not contain any

proper subset which is also a closed CR-invariant subset.

Problem 4.13. Describe the possible structure of the decomposition of�

in CR orbits.

There are differences between embedded and locally embeddablegeneric submanifolds, which we shall not discuss, assuming that

�is embedded in

��or in ¨ � �� . Also, the � ¤ category enjoys special

features.Indeed, if

�is a connected real analytic hypersurface, Proposition 4.12

entails that each minimal closed invariant subset of�

is either an embed-ded complex hypersurface or an open orbit; also if

�contains at least one

CR orbit of maximal dimension �Ãb ó = � (hence an open subset of�

),all its CR orbits of codimension one are complex Fb ó = � -dimensional em-bedded submanifold of

�(a real analytic subset of codimension one in

oconsists of isolated points). In the smooth category things are different.

So, let�

be a connected � � or ��3 � � ( ¼(�¸� ' 6@��:ï� =) hypersur-

face of��

. Its CR-orbits are either �ÃbDó = �-dimensional, i.e. open in�

, or 7�Ãb�ó�� -dimensional and Ä " � -integral, hence complex Øb�ó = � -dimensional hypersurfaces immersed in

�.

Proposition 4.14. ([Jo1999a]) In the smooth category, the structure of ev-ery minimal closed CR-invariant subset

�of�

has one of the followingtypes:

(i)� NB�

consists of a single embedded open CR orbit;(ii)

� N � ° |@º � $&% � ° NO�is a union of complex hypersurfaces, each

being dense in�

, withÍ � � ) ` N�Í � � ) o ;

(iii)� N�� ° |-º � $&% � ° has empty interior in

�and is a union union

of complex hypersurfaces, each being dense in�

, withÍ � � ) ` NÍ � � ) o ;

(iv)�

consists of a single complex hypersurface embedded in�

.

Furthermore, the closure, with respect to the topology of�

, of everyCR orbit of dimension 7�Ãb¬óy� � is a minimal closed CR-invariant subset of�

.

These four options are known in foliation theory ([HH1983, CLN1985]).One has to remind that each CR orbit contained in

�is dense in

�. In the

first two cases, the trace of�

on any orbit-transversal is a full open seg-ment; in the third, it is a Cantor set; in the last, it is an isolated point. In the


third case, impossible if�

is real analytic,�

will be called an exceptionalminimal CR-invariant subset, similarly as in foliation theory. We shall seebelow that compactness of

� �¸� �imposes a strong restriction on the

possible�

’s.We mention that an analog of Proposition 4.14 holds for connected

generic submanifolds of codimension �B� � , provided one puts the re-strictive assumption that all its CR orbits are of codimension � = , the onlydifference being that CR orbits of codimension

=are not complex mani-

folds in this case.The presence of CR orbits of codimension �� in

�may produce min-

imal closed CR-invariant subsets with complicated transversal structure,even in the real analytic category ([BM1997]). Also, in a bounded stronglypseudoconvex boundary (see

a1.15(V) for background), there may exist

a CR orbit of codimension one whose closure constitutes an exceptionalminimal CR-invariant subset.

Theorem 4.15. ([Jo1999a]) There exists a bounded strongly pseudoconvexdomain � �Ë� � with � � boundary and a compact � � submanifold

� �Q � of dimension . which is generic in� � such that:2 � is � � foliated by CR orbits of dimension

�;2 � contains a compact exceptional minimal CR-invariant set, but

no compact CR orbit.

Summarized proof. The main idea is to start with an example due to Sack-steder, known in foliation theory, of a compact real analytic

�-dimensional

manifold � equipped with a � ¤ foliation & of codimension one which car-ries a compact exceptional minimal set, but no compact leaf. Accordingto [HH1983], there exists such a pair (� ' & � , together with a � � diffeo-morphism w·i � � ¦ ��ì_� i , where

�is some compact orientable � �

surface of genus � embedded ino � , and where

� i is the unit circle. Let� £ "Ìå¦ _ �" � ��Ä`� o � denote the � � unit outward normal vector field tosuch a

�¸� o � , and considero � to be embedded in

� � . For � ¢ 6 small,the map w � � �Ûìç� i £ �" ' × � å ó ¦ "1� _ �" � � � ×�� may be seen to be a totally real � � embedding. By results of Bruhat-Whitney and Grauert, one may approximate the � � totally real embeddingw � � w·i by a � ¤ embedding w � � ¦ � � which is arbitrarily close tow � � w·i in � i norm, hence totally real. Denote × � N wz(� �

. The trans-ported foliation

C � N w � !& � being real analytic, one may then proceedto an intrinsic complexification of all its totally real � -dimensional leaves,


getting some ' -dimensional real analytic hypersurface × � , containing × ,equipped with a foliation

C � , of × � , by � -dimensional complex manifolds,with

C � , G × N;C. This foliation

C � , is closed in some tubular neigh-borhood � of × in

� � , say � � N � µO� � � � )+*W�64±=µ ' × � 8 � � , with� ¢ 6 small. Since × is totally real, the boundary Q � is strongly pseudo-convex (Grauert) and is � � . Clearly, the intersection

� � N × � , G Q&� isa . -dimensional � � submanifold. The intersections of the � -dimensionalcomplex leaves of

C � , with Q&� show that�

is foliated by strongly pseudo-convex

�-dimensional boundaries, which obviously consist of a single CR

orbit. Thus a CR orbit of�

is just the intersection of a global leaf ofC � ,

with Q&� . In conclusion, letting aã�/� # be the minimal exceptional set ofSacksteder’s example,

�contains the exceptional minimal CR-invariant

set q!w � Wa1�/� # � t � , G Q&� and no compact CR orbit.

�4.16. Global minimality and laminations by complex manifolds. TheCR orbits being essentially independent bricks, it is natural to define theclass of CR manifolds which consist of only one brick.

Definition 4.17. A � ¤ , � � or � 3 � � CR manifold�

is called globally min-imal if if consists of a single CR orbit.

Each CR orbit of a CR manifold is a globally minimal immersed CRsubmanifold of

��. To simplify the overall presentation and not to expose

superficial corollaries, almost all the theorems of Parts V and VI in thismemoir will be formulated with a global minimality

�.

Lemma 4.18. ([Gr1968, Jo1995, BCH2005]) A compact connected �ã� hy-persurface in

� �is necessarily globally minimal.

Proof. Otherwise, the closure of a CR-orbit of codimension one in�

would produce a compact CR-invariant subset�

which is a union of im-mersed complex hypersurfaces, each dense in

�. Looking at a point of�

where the pluriharmonic functionµ � � NfË�y µ�� (or �� N«ª ,Jµ�� ) is max-

imal, the maximum principle entails thatµ � (or �l� ) is constant on

�, for% N = 'lklklkm' b , hence

� NO� ��¢ k � , contradiction.

�More generally, the same simple argument yields:

Corollary 4.19. Any Stein manifold cannot contain a compact set which islaminated by complex manifolds

In the projective space ¨ � �S� , one expects compact orientable con-nected �� hypersurfaces

�to be still globally minimal, but arguments

are far to be simple. In fact, there are infinitely many compact projective


algebraic complex hypersurfaces q in ¨ � �·� . However, they cannot becontained in such an

� � ¨ � �S� since, otherwise, their complex normalbundle ÄÌ¨ � �S�±J�b1H Ä�q , known to be never trivial, but equal to the com-plexification of the trvial bundle Ä �cJ�b&H Ä�q , would be trivialized.

Thus, according to Proposition 4.14 above, the very question is aboutnonexistence of closed CR-invariant sets

� ��laminated by complex

hypersurfaces which either coincide with�

or are transversally Cantorsets. If

� � ¨ � �S� is real analytic, it might only be Levi-flat.Nonexistence of orientable Levi-flat hypersurfaces in ¨ � �S� was ex-

pected, because they would divide the projective space in two smoothlybounded pseudoconvex domains. In the real analytic case, non-existencewas verified by Lins-Neto for b�� and by Ohsawa for b N � ; in thesmooth (harder) case, see [Si2000].

Open question 4.20. Is any compact orientable connected �®� hypersur-face of ¨ � �S� globally minimal ?

So, the expected answer is yes. In fact, the question is a particularcase of a deep conjecture stemming from Hilbert’s sixteen problem aboutphase diagrams of vector fields having polynomial coefficients on the two-dimensional projective space. This conjecture is inspired by the Poincare-Bendixson theory valid over the real numbers, according to which the clo-sure of each leaf of such a foliation over ¨ � ox� contains either a compactleaf or a singularity. In its most general form, it says that ¨ � �� cannotcontain a compact set laminated by ØbSó = � -dimensional complex manifold,unless it is a trivial lamination, viz just a compact complex projective alge-braic hypersurface; however, nontrivial laminations by Fb�ó�� -dimensionalcomplex manifolds may be shown to exist.

A related general open question is to find topologico-geometrical criteriaon open subsets of ¨ � �·� insuring the existence of nonconstant holomor-phic functions there.

4.21. Finite-typisation of generic submanifolds. Let�

be a connected��3 ( ��¼�� ) generic submanifold of� �

of codimension �ð� =and of

CR dimension ~ N b\óJ� � =. The distribution of subspaces ��å¦ Ää". �

of Ä � is of constant rank �<~ . We apply to the complex tangential bundleÄ " � the concept of finite-type.

Definition 4.22. A point �� is said to be of finite type if the system¯

of local sections of Ä " � defined in a neighborhood of � satisfies¯ 3»º� �·NÄ�. � . The smallest integer �YÁ� � �f¼ with

¯ K ^ .m_ Á� �rN Äv. � is called thetype of

�at � .


We want now to figure out how, in general, a generic submanifold of��must be globally minimal and in fact, of finite type at every point. We

equip with the strong Whitney topology the set 3#c �K � � of � 3 ( ��¼ �B� )connected generic submanifolds

� �0�S�of codimension �/� =

and ofCR dimension ~ N b@ó��Ë� =

. No restriction is made on the globaltopology.

As a model case, let ¼ �O� and consider�

to be rigid algebraic repre-sented by É � N ÊÉ �ã�È%�¨Ì�u=µ 'ZÊµ �·N ÊÉ ��¨% ù¡ � ¡ á ¡ < ¡ � 3 �Ó� � �H� < µ � Êµ < 'where : ' ; �fd � , where the polynomials ¨»� are real, � � � �H� < N � � � < � � andhave no pluriharmonic term, namely 6 ¨»�Ã=µ ' 6 � ¨Ì�Ã76 '<Êµ � , for � N= 'lklklkm' � . A basis of = ' 6 � and of Ø6 ' = � vector fields is given by

R Ü � N QQ µ Ü �¨% Kù �Ï¹»i ¨Ì� � ê z QQ É � and R Ü � N QQ Êµ Ü ó % Kù ��¹»i ¨�� Ïê z QQ ÊÉ � 'for Ú N = 'lklklkm' b . The Lie algebra

¯ 3 generated by all Lie brackets oflength � ¼ of the system

¯ � N � R Ü ' R Ü �²i�� 3 � � contains the subalgebra¯ 3 $&% � ´#� À � K generated by the only brackets of the formÝ R « � '�klklk�' Ý R « ' Ý R � � 'lklklk ' Ý R � � ' Ý R Ü � ' R ÜÏà Þ Þ klk�k Þ�Þ klklk Þwhere � �9Ê¬� "�� ¼ and where

= � q¥i 'lklk�k�' q&° ' Q i 'lk�klk�' Q � ' Úvi ' Ú � �©~ .One computes

Ý R Ü � ' R ÜÏà Þ N ó�% 8 Kù �Ï¹»i ¨�� ê z � Ïê z à QQ É � � Kù �Ï¹»i ¨�� ê z � Ïê z à QQ ÊÉ � A 'hence the above iterated Lie brackets are equal to

ó�% 8 Kù �Ï¹»i ¨Ì� � ê � � bcbcb ê � Ïê � � bcbcb Ïê � � ê z � Ïê z à QQ É � � Kù ��¹»i ¨�� ê � � bcbcb ê � Ïê � � bcbcb Ïê � � ê z � Ïê z à QQ ÊÉ � A kThis shows that all these brackets are linearly independent as functions ofthe jets of the ¨Ì� . In fact, the number of such brackets is exactly equal tothe dimension of the space of polynomials ¨��µ 'ZÊµ � of degree � ¼ havingno pluriharmonic term, namely equal to7�<~ï�@¼ � �<~±�¼» ó/� P~��@¼ � ~±Z¼» � = k


For a general � 3 submanifold�

(not necessarily rigid), one verifies thatthe same collection of brackets is independent in terms of the jets of thedefining equation of

�. Generalizing slightly Lemma 2.13, we see that in

the vector space of � ì ��r� ç � (real or complex) matrices, the subset ofmatrices of rank � ��ó = is a real algebraic set of codimension equal to ç � = �

. If we choose ¼ large enough so that the dimension of¯ 3 $&% � ´#� À � K

is � ��?)+*�, � N ��=~Û�>� �«N �Ãb (applying the previous assertion withç � N )�*�, � ), if we form the � ì �� ç Ô � , ç Ô¥� ç , matrix consisting of thecoordinates of the

½½ Î -part of brackets of length ��¼ as above, then the setwhere this matrix is of rank �©� ó = is of codimension ��)+*�, � � = inthe space of ¼ -th jets of the defining equations of

�. Consequently, the jet

transversality theorem applies.

Theorem 4.23. Let b � =, ~ � =

and �ï� =be integers satisfying~ï�@� N b and let ¼ be the minimal integer having the property that �<~��J¼ � �<~±Z¼» ó � P~��@¼ � ~±�¼M � = ��+P~��È� �SN �Ãb k

Then the set of � 3 connected generic submanifolds� �p�·�

of codimen-sion � and of CR dimension ~ that are of finite type � ¼ at every point isopen and dense in the set 3 c �K � � of all generic submanifolds.

In particular, a connected � ò (resp. � � , resp. �� ) hypersurface in� �

(resp. in� � , resp. in

��for bÛ� . ) is of finite type . (resp.

�, resp.� , or equivalently, is not Levi-flat) at every point after an arbitrarily small

perturbation.Similarly, if instead of the subalgebra

¯CR � rigid , one would have

considered the (smaller) subalgebra consisting of only the brack-ets Ý R « � 'lklk�k�' Ý R « ' Ý R Ü � ' R ÜÏà Þ�Þ k�klk Þ , where �/� Ê � ¼ and where= � qVi 'lklklkm' q�° ' Úvi ' Ú � � ¼ , one would have obtained finite type �©¼ , for¼ minimal satisfying �<~ ^ � á 3 i�_no� oÝ^ 3 i�_no �©~ � �(�� <~¶�F� � . We also mentionthat the same technique enables one to prove that, after an arbitrarilysmall perturbation,

�is finitely nondegenerate at every point and of finite

nondegeneracy type �J� , with � minimal satisfying �� ^ � á � _no� o � o �?.Hb�ó = . Inparticular, � N{� when ~ N � N =

while � N � suffices when ~ N =for

all ��(� . Details are left to the reader.To conclude, we state the analog of Open question 2.17 for induced CR

structures.

Open question 4.24. ([JS2004], [ P ]) Given a fixed generic submanifold�of class ��3 that is of finite type at every point, is it always possible


to perturb slightly a � 3 submanifold� i of

�that is generic in

��, of

codimension �+i � =and of CR dimension ~yi N bDó��i � =

, as a � 3submanifold

T� i of�

that is of finite type at every point ? If so, what isthe smallest regularity ¼ in terms of � , ~ , �&i , ~�i and of the highest type atpoints of

�?

4.25. Spaces of CR functions and of CR distributions. A � i function> � � ¦ �is called Cauchy-Riemann (CR briefly) if it is annihilated by

every section of ÄÌA�� i � . Equivalently:2 �¼> is�

-linear on Ä " � ;2 �¼>dC �Z¢Öi3C �� C �Z¢ � ÂÂ L N 6 ;2fe L > Q�� N 6 , for every � i form � of type Øb ' ~Oó = � in��

havingcompact support.

(Remind the local expression of µu' � � forms:{hg � i Ê g � i �Z¢ g C � Ê¢ i , whereV N P%êi 'lklk�k�' %Ø´ � and

d�N $�Hi '�klklk�' � � � .) A (only) continuous function> � � ¦ �is CR if the last condition e L > Q � N 6 holds. Fur-

ther, Lebesgue-integrable CR functions, CR measures, CR distributionsand CR currents may be defined as follows ([Trv1981, Trv1992, HM1998,Trp1996, Jo1999b]).

Thanks to graphing functions, one may equip locally�

with some (infact many) volume form, or equivalently, some deformation of the canoni-cal )+*-, � -dimensional Legesgue measure defined on tangent spaces. LetM be a real number with

= �4M¬�� . Since two such measures are multipleof each other, it makes sense to speak of R ¯�ß #" functions

� ¦ �. In this

setting, a R ¯�ß #" �B� function > is CR if e L > Q�� N 6 , for every � i form �of type Øb ' ~có = � in

� �having compact support.

A distribution i on�

is CR if for every section R of ÄäA�� i � defined inan open subset ¾ ��

and every õö�\� �" Y¾ '³�·� , one has j'i ' R§�õ �'kÌN 6 .A CR distribution of order zero on

�is called a CR measure. Equiv-

alently, a CR measure is a continuous linear map �xÔ4å¦ Q n�·Ô � from com-pactly supported forms on

�of maximal degree �<~ �è� to

�, that is CR in

the weak sense, namely Q Q�� N 6 , for every � i form � of type Fb ' ~�ó = �having compact support. Once a volume form )lAw� ÄFL is fixed on

�, the

quantity Q )lAw� Ä�L is a CR (Borel) measure on�

.

4.26. Traces of CR functions on CR orbits. A � i function > � � ¦�is CR on

�if and only if its restriction to every CR orbit of

�is

CR (obvious). If > is � A or Rnm �ß #" , a similar but nontrivial statement holds.


By “almost every CR orbit”, we shall mean “except a union of CR orbitswhose )+*-, � -dimensional measure vanishes”.

Theorem 4.27. ( � N =: [Jo1999b]; �ö� =

: [Po1997, MP1999]) Assumethat

�is at least � � and let > be a function in R ¯�! " �B� with

= �oM/�� . Then the restriction > J éqpsr is in R ¯�ß #" on � $&% , for almost every � $&% .Furthermore, > is CR if and only if, for almost every CR orbit � $&% of

�,

its restriction > J éqp1r is CR.

The theorem also holds for > continuous, with > J éqp1r being CR for everyCR orbit. Here, � � -smoothness is needed. Property (5) of Sussman’s orbitTheorem 1.21 together with a topological reasoning yields a covering byorbit-chart which is used in the proof.

Proposition 4.28. ([Jo1999a, Po1997, MP1999]) Assume�

is � � or � 3 � � ,with ¼/�Û� , 6��: � =

and let

� � N<�� o � J ��J 8 = � . There exists acountable covering

� Ü |m} ¾ Ü N � such that for each Ú , there exist ç Ü �ydwith 6�� ç Ü �F� and a ��3 i � � diffeomorphism:w Ü � � Ü ' ¢ Ü � £ � � � á�� z ì � K � z å ó ¦ w Ü � Ü ' ¢ Ü � �¹¾ Ü 'such that:2 w Ü � � � á�� z ìæ� ¢ �Ü � � is contained in a single CR orbit, for every

fixed ¢,�Ü � � K � z ;2 for each � � �, there exists Ú N Ú�. � d with � � ¾ Ü Ü ,

viz there exist � Ü Ü � . and ¢ Ü Ü � . with w Ü Ü � Ü Ü � . ' ¢ Ü Ü � . � N � , such thatw Ü Ü e � � � á�� z Ü ìæ� ¢ Ü Ü � .Ã� g is an open piece of the CR orbit of � , i.e.)+*-,�� $&% �(' � ��N �<~e� ç Ü Ü .In the proof of the theorem, � � -smoothness of the maps w Ü (hence � � -

smoothness of�

) is required to insure that the pull-back w}�Ü FÄ" �cJ�ò z � is� i . However, we would like to mention that if�

is �4�� with 6E8B:î8 =results of [Tu1990, Tu1994a, Tu1996] and Theorem 3.7(IV) insuring the� �� < -smoothness of local and global CR orbits, for every

; 8?: , this wouldyield orbit-charts w Ü of class �� < , and then the above theorem holds truewith

�of class �� .

4.29. Boundary values of holomorphic functions for functional spaces� 3 � � , t�Ô , R ¯�ß #" . Let�

be a generic submanifold of�·�

of codimention ��=and of nonnegative CR dimension ~ ��6 (we admit ~ N 6 ). Assume

�


is at least � i . In appropriate coordinates ¢ N �µ '-É �«N � �J%¶ ' ü��J%0û � ��\ìy� �centered at one of its points � :� N ý =µ ' É � �±� �� ìFe � K� � ì % o K g � û N wz ��' ¶ ' ü � � '

for some �+i ¢ 6 , with wzØ6 �äN 6 and �jwzØ6 �äN 6 . Let � be a real numberwith 6��9��9��i . The height function:�·F� � � N ,��u�¡ ¸ ¡ � ¡ ¹ ¡ � ¡ E ¡ � � J wz ��' ¶ ' ü �±Jis continuous and tends to 6 , as � tends to 6 . For every ��p��i and every� ¢ �·F� � , the boundary of

� G Ý � �� ìFe � K� ì % � K� g Þ is contained in theboundary Q e � K� ì � K� g of the horizontal space, times the vertical space% � K� .

Let�

be an open convex cone ino K

having vertex 6 . We shall assumeit to be salient, namely contained in one side of some hyperplane passingthrough the origin. Equivalently, its intersection

� G£� K i with the unitsphere of

o Kis open, contained in some open hemisphere and convex in

the sense of spherical geometry.A local wedge of edge

�at � directed by

�is an open set of the form:

(4.30)� N �¶Ø� ' � ' � � � N ý � �¨%¶ ' ü¬�¨%0û � �±� �� ì � K� ì % � K� �ûíó wz ��' ¶ ' ü � � � � '

for some � ' � ¢ 6 satisfying �9�¸�+i and � ¢ �·F� � . This type of openset is independent of the choice of local coordinates and of local definingfunctions; in codimension �E�O� , it generalizes the notion of local side ofa hypersurface. Notice that � is connected.

If there exists a functionC

that is holomorphic in � and that extendscontinuously up to the edge� � � NB� G ÝG� �� ì e � K� ì % o K g Þof the wedge � , then the limiting values of

Cdefine a continuous CR

function on� �

.A more general phenomenon holds. A function

C, holomorphic in the

wedge � , has slow growth up to�

, if there exist Ú�� d and�¶¢ 6 such

that JMC P¢ ��J � � J ûíó wz ��' ¶ ' ü ��J Ü ' ¢ N � �È%¶ ' üò�È%*û � �y� kEquivalently,

JMC P¢ �±J � � qÙ)+*V�54�p¢ '��B� t Ü , with the same Ú but a possiblydifferent

�. As in the cited references, we shall assume

�to be � � .


Theorem 4.31. ([BCT1983, Ho1985, BR1987, BER1999]) IfC � ��Ø�¶Ø� ' � ' � �� has slow growth up to�

, it possessesa boundary value " L�C which is a CR distribution on the edge� G Ý � �� ìFe � K� ì % � K� g Þ precisely defined by:uwv L�C«' õyx � N Ä *-,$Fz|{ Æ A ý~} ?� �B� C � C � �È%�¶ ' ü¬�¨%�wz ��' ¶ ' ü � �È%�� õ� ��' ¶ ' ü � � � �Ó¶}�Óü 'where õ N õx ��' ¶ ' ü � is a � � function having compact support in � �� ì � K� .

(i) The limit is independent of the way how �¬� � approaches 6 � o K .(ii) If

v LòCis � « � < , qÛ� 6 , 6 � ; � =

, thenC

extends as a � « � <function on ��Ô I;e¼� G Ý � �� ì>e � K� ì % � K� Ò g Þ g , for every wedge�ËÔ N ��ÔFØ� ' �VÔ ' � Ô � with �·F� � 8��VÔ®� � and with

� Ô G � K i �í�� G � K i .(iii) Finally,

Cvanishes identically in the wege � if and only if

v L¬Cvanishes on some nonempty open subset of the edge

� �.

The integration is performed on the translation� {� � N � � �ïØ6 ' %�� ,

drawn as follows.

A L��

� ^ � � � � $ _

} ?� �� C �� C�

æ ? � C $ $

L � ¹ L�� } ?� � ^ � C � � � � C� _��Boundary values of functions holomorphic in a wedge

The proof is standard for� (o��

([Ho1985]), the main argument goingback to Hadamard’s finite parts. With technical adaptations in the case ofa general generic

�, several integrations by part are performed on a thinØ)+*-, � � = � -dimensional cycle delimited by

� A� and� {�

, taking advantageof Cauchy’s classical formula, until the rate of explosion of

Cup to the

edge is dominated. The uniqueness property (iii) requires analytic discmethods (Part V).


Boundary values in the R ¯ sense requires special attention. At first, re-mind that a function

Cholomorphic in the unit disc � belongs to the Hardy

class � ¯ �� if the supremum:J JMCKJ J Q<� ^ } _ � N �� ÇA \ ´ \ i 9 ý�� ÂÂ C µ ç � ° � ÂÂ ¯ > i7è ¯ 89�is finite. According to Fatou and Privalov, such a function

Chas radial

boundary values >ã ç � ° � � N Ä *-,¬´ Æ � i C µ ç � ° � , for almost every ¢��pqºóS� ' �¥t ,so that the boundary value > belongs to R ¯ �qÁóS� ' �¥t � . Furthermore, if

= �M¬8�� : Ä *�,´ Æ � i ý � � ÂÂ C µ ç � ° � ó >1 ç � ° � ÂÂ ¯ N 6 kConsider a bounded domain

¸�� having boundary of class at least � � ,

defined by¸ N � µ/� �1� � �Y�µ � 8<6v� , with �?�� satisfying ��(ëN 6

on Q ¸ . For � ¢ 6 small, let¸d� � N<� µy� ¸ � �Y�µ � 8¶ó � � . The induced

Euclidean measure on Q ¸d� (resp. Q ¸ ) is denoted by �Ó� � (resp. �Ó� ). Thenthe Hardy space � ¯ ¸K� consists of holomorphic functions

C � �K ¸K�having the property that the supremum:J J CEJ J Qy� ^��M_ � N �� Ç�� A 9 ý ½ �F� JMC �µ �±J ¯ � � � =µ � > i7è ¯ 8��is finite. The resulting space does not depend on the choice of a defin-ing function � ([St1972]). Let _ ê be the outward-pointing normal to theboundary at µ ��Q ¸ .

Theorem 4.32. ([St1972]) IfC � � ¯ ¸K� , for almost all µB�ÛQ ¸ , the

normal boundary value >1=µ � � N Ä *�, � Æ � A C �µ ó � _ ê � exists and defines a

function > which belongs to R ¯ ØQ ¸K� . Furthermore, if= �4M¬89� :Ä *-,� Æ � A ý ½ � J C =µíó � _ ê � ó/>1�µ �±J ¯ � ��µ �·N 6 k

In arbitrary codimension, the notion of R ¯ boundary values may be de-fined in the local sense as follows. Let

�be generic, let �f� � and let� N �¶Ø� ' � ' � � be a local wedge of edge�

at � , as defined by (4.30). Aholomorphic function

C �f�KØ� �belongs to the Hardy space � ¯�ß #" Ø� �

iffor every cone

� Ô �Ëo K with� Ô GÃ� K i �r� � GÃ� K i and every � Ô 8B� ,

the supremum:�� Ç{ Ò | $ Ò ý�} ?� Ò �B� C � Ò J C � �È%�¶ ' üò�È%#wz ��' ¶ ' ü � �¨%�� Ô �±J ¯ � � C � ¶�C �Óü08 �


is finite. Up to shrinking cubes, polydiscs and cones, the resulting spaceneither depends on local coordinates nor on the choice of local definingequations.

Theorem 4.33. ( � N =: [St1972, Jo1999b]; � � � : [Po1997]) If

C �� ¯�ß #" F� �, for almost ��' ¶ ' ü��F%jwz ��' ¶ ' ü �� and for every cone

� Ôwith

� Ô Gs� K i �í� � Gt� K i , the boundary value:>1 ��' ¶ ' ü � � N Ä *�,$ Ò z�{ Ò Æ A C � �È%¶ ' üò�È%jwz ��' ¶ ' ¶ � �È%�� Ô �exists and defines a function > which belongs to R ¯�! "�� $&% � � � . Further-more, if

= �4M¬8�� , for every �ÕÔ»8?� :�� µ$ Ò z�{ Ò Æ A¡ } ?� Ò �B� C � Ò.¢¢�£�¤¦¥¨§ª©¬« U� §®©°¯ M ¥ U « U « P §ª©s± Ô�² ©© T M ¥ U « U� P ¢¢ ¯l³ ¥@´ ³ «.´ ³ b ¨ �4.34. Holomorphic extendability of CR functions in �43 � � , t Ô , R ¯�! " . InPart V, we will study sufficient conditions for the existence of wedges towhich CR functions and distributions extend holomorphically.

Definition 4.35. A CR function of class � 3 � � or R ¯�ß #" = �4Mò89� � or a CRdistribution > defined on

�is holomorphically extendable if there exists a

local wedge � N �¶Ø� ' � ' � � at � and a holomorphic functionC �D�KØ� �

whose boundary valuev L¬C

equals > on� �

in the � 3 � � , R ¯ or t�Ô sense.

4.36. Local CR distributions supported by a local CR orbit. Assumenow that

�, of class � � and represented as in

a4.3, is not locally minimal

at � . Equivalently, � �! "$&% �(' � � is of dimension �<~B� ç � �6~B� �ó = . In asmall neighborhood,

� � N � �ß #"$&% �(' � � is a closed connected CR subman-ifold of

�passing through � and having the same CR dimension as

�.

There exist local holomorphic coordinates =µ ' Éä�rN =µ ' É i ' É � � � � � ì� � ì�� K � vanishing at � in which�

is represented by û N wz ��' ¶ ' ü � and�

is represented by the supplementary (scalar) equation(s) ü � N q � ��' ¶ ' ü¥i � ,with w and q � of class � � satisfying wx76 �SN 6 , �jwz76 �·N 6 , q � Ø6 �SN 6 and� q � 76 �N 6 . According to Theorem 4.2, the assumption that

�is CR and

has the same CR dimension as�

may be expressed as follows.

Proposition 4.37. Decomposing w N w�i ' w � � and defining:ûÕi N w·i¥ ��' ¶ ' ü¥i ' q � ��' ¶ ' ü�i ��N � Q im ��' ¶ ' ü¥i � 'û � N w � ��' ¶ ' ü¥i ' q � ��' ¶ ' ü�i ��N � Q � ��' ¶ ' ü¥i � kthe map: � � ��' ¶ ' ü¥i � � N q � ��' ¶ ' ü�i � �È% Q � ��' ¶ ' ü¥i �


is CR on the generic submanifold ûji N Q im ��' ¶ ' ü¥i � of� � ì�� .

In a small neighborhood ¾ of � , the restrictions� µÃi ÂÂ�µ 'lklklkm' � µ � ÂÂ�µ ' � É i ÂÂ�µ 'lklk�k�' � É � ÂÂ�µspan an =~ï� ç � -dimensional subbundle of

� Ä � � . Denoting � µ � N � µHi<C�� CI� µ � , � Êµ � N � ÊµÃi�C �� C´� Êµ � and � É ÔS� N � É iC �� CÃ� É � , for õB�� " Y¾ '³�S� , consider the (localized) distribution defined by:u q ) t ' õyx � N�ý ò�� µ õ � � µLCî� É Ô Cî� Êµ kProposition 4.38. ([Trv1992, HT1993]) Then q ) t is a nonzero local CRmeasure supported by

��G ¾ .

Proof. It is clear that q ) t is supported by��G ¾ and is of order zero.

Let the 76 ' = � vector fields R Ü and the complex-transversal ones �E� beas in

a4.3. Reminding �Óõ N { � Ü ¹»i R Ü �õ � � µ Ü � { � Ü ¹»i R Ü �õ � � Êµ Ü �{ K�Ï¹»i �Ç�u�õ � � É � ÂÂ L , we observe:R Ü �õ � � µ�CE� É Ô C�� Êµ N4¶ � û õ � � µLCî� É Ô C � ÊµÃi2C �� C4·� Êµ Ü C �� C � Êµ � ü k

Replacing this volume form in the integrand:j R Ü q ) t ' õ k � N ó j q ) t ' R Ü �õ � k N ó ý µ � ò R Ü �õ � � µLCî� É Ô Cî� ÊµN�¶ ý µ �½ò � û õ � � µ�C � É Ô C � ÊµÃi2C �� C ·� Êµ Ü C �� Cî� Êµ � üand applying Stokes’ theorem, we deduce j R Ü q ) t ' õ k N 6 , i.e. q ) t is CR.

�The last assertion of Theorem 4.31 and the vanishing of q � t on the dense

open set ¾�A& �\G ¾ � entails the following.

Corollary 4.39. ([Trv1992, HT1993]) The nonzero local CR measure q ) tdoes not extend holomorphically to any local wedge of edge

�at � .

By means of this wedge nonextendable CR measure, one may constructnon-extendable CR functions of arbitrary smoothness. Indeed, let

�be a

local generic submanifold with central point � , as represented ina4.3 and

let �Ç� be the complex-transversal vector fields satisfying �E� � É � à ��N � � �#� � àand qÁ�Å� � ' �Ç� à t N 6 .


Proposition 4.40. ([BT1981, Trv1981, BR1990, Trv1992, HT1996,BER1999]) For every CR distribution i on

�and every ¼p�ïd , there

exist an integer Q � d and a local CR function > of class � 3 defined insome neighborhood of � such that:i Nfe � �i � �� È� �K g � > k

Then with i � N q ) t and for ¼\�yd , an associated CR function > of class� 3 is also shown to be not holomorphically extendable to any local wedgeof edge

�at � . A Baire category argument ([BR1990]) enables to treat

the � � case.

Theorem 4.41. ([BR1990, BER1999]) If�

is not locally minimal at � ,then for every ¼ N 6 ' = ' � 'lk�klkm' � , there exists a CR function � of class � 3defined in a neighborhood of � which does not extend holomorphically toany local wedge of edge

�at � .

Open problem 4.42. Find criteria for the existence of CR distributions orfunctions supported by a global CR orbit.

In [BM1997], this question is dealt with in the case of CR orbits ofhypersurfaces which are immersed or embedded complex manifolds.

To conclude this section, we give the general form of a CR distributionsupported by a local CR orbit

� N ��ß #"$&% �(' � � . After restriction to�

, thecollection � µ � N �� á i 'lklk�k�' � K � of vector fields spans the normal bundleto�

in�

, in a neighborhood of � . Let i be a local CR distribution definedon�

that is supported by�

.

Theorem 4.43. ([Trv1992, BCH2005]) There exist an integer ¼��5d , andfor all

; �yd K � withJ ; J �(¼ , local CR distributions i µ< defined on

�such

that: u i ' õyx N ù< |m} C # � � ¡ < ¡ � 3 j'i µ< ' =� µ � < �õ � ÂÂ µ k1ka5. APPROXIMATION AND UNIQUENESS PRINCIPLES

5.1. Approximation of CR functions and of CR distributions. Let�

be a generic submanifold of�·�

. The following approximation theoremhas appeared to be a fundamental tool in extending CR functions holomor-phically (Part V) and in removing their singularities (Part VI). It is alsoused naturally in the proof of Theorem 4.43 just above as well as in theCauchy uniqueness principle Corollary 5.4 below. The statement is validin the general context of locally integrable structures

é, but, as explained


in the end of Section 3, we decided to focus our attention on embeddedCauchy-Riemann geometry.

Theorem 5.2. ([BT1981, HM1998, Jo1999b, BCH2005]) For every �(��, there exists a neighborhood ¾¥. of � in

�such that for every function >

or distribution i as defined below, there exists a sequence of holomorphicpolynomials 7¨ Ü =µ �� Ü |�} with:2 if

�is ��3 á+�� , with ¼e� 6 , 6�� : � =

, if > is a CR functionof class � 3 � � on

�, then

Ä *�, Ü Æ � J J ¨ Ü ó > J J ¸ � l � ^ ò Ü _ ¦ 6 ; in par-ticular, continuous CR functions on a �� generic submanifold areapproximable sharply by holomorphic polynomials;2 if�

is at least �� , if > is a R ¯�ß #" CR function = �¹M\8e� � , thenÄ *-, Ü Æ � J J ¨ Ü ó/> J J � �� º , ^ ò Ü _ ¦ 6 ;2 if�

is ��3 á+� , if i is a CR distribution of order � ¼ on�

, thenÄ *-, Ü Æ � u ¨ Ü ' õyx N u i ' õyx for every õö�\� �" Y¾». � .In [HM1998, BCH2005], convergence in Besov-Sobolev spaces R ¯� � �ß #"and in Hardy spaces � ¯ , frequently used as substitutes for the R ¯ spaces

when 6ö8»M58 =, is also considered, in the context of locally integrable

structure.

Proof. Let us describe some ideas of the proof, assuming for simplicitythat

�is � � and > is � i . In coordinates P¢³i '�klklk�' ¢ � � vanishing at � , choose

a local maximally real �� submanifold Í A contained in�

, passing through� and satisfying Ä�.lÍ A N �6Ë�y ¢ N 6v� . Let åÕ. be a small neighborhood of� , whose projection to Ä+. � is a �6~Ë� � � -dimensional open ball. We mayassume that Í A is contained in åj. with boundary

� A � N Í A G QKåÕ. beingdiffeomorphic to the Øb5ó = � -dimensional sphere. Consider a parameterüË� o K satisfying

J ü J 8 �, with

� ¢ 6 small. We may include Í A in afamily FÍ¡E � ¡ E ¡ \ � of maximally real �� submanifolds of ¾V. with Í�E ÂÂ Em¹ A NÍ A , whose boundary is fixed: Q&Í¼E Q&Í A N½� A , such that the Í¡E foliatesa small neighborhood ¾V. of � in

�. For ¢ò�u¾». , there exists a ü N ü®P¢ �

such that ¢ belongs to Í¼EÝ^ ° _ . We then introduce the entire functions:C Ü P¢ � � N 9zÚ� > � è � ý�¾B¿ ú ß û ç Ü ^ ° Û:_ à >1 Ö � � Ö i=C �� Cî� Ö � 'where P¢Só Ö � � � N { ��¹»i P¢=�ó Ö � � � and where Ú@�9d . Shrinking åj. and¾V. if necessary, we may assume that

J ª , P¢·ó Ö �±J � i� J Ë�y p¢·ó Ö �±J for all¢ ' Ö �BÍ¡E G ¾». and allJ ü J 8 �

. Here, the � � -smoothness assumption is


used. With this inequality, the above multivariate Gaussian kernel is easilyseen to be an approximation of the Dirac distribution at Ö N ¢ on Í EÝ^ ° _ .Consequently

C Ü P¢ � tends to >1P¢ � as Ú ¦ � . Moreover, the convergenceis uniform and holds in ��A±¶¾V. � .

We claim that the assumption that > is CR insures thatC Ü p¢ � has the

same value if the integration is performed on Í A :(5.3)

C Ü p¢ �SN 9xÚ� > � è � ý�¾ r ç Ü ^ ° ÛB_ à >1 Ö � � Ö i3C �� C � Ö � kIndeed, Í¡E�^ ° _ and Í A bound a Øb�� = �

-dimensional submanifold § ° con-tained in å�. with Q § ° N Í�EÝ^ ° _4ófÍ A . Since ç Ü ^ ° ÛB_ à is holomorphic withrespect to Ö and since �¼>1 Ö � C�� Ö i�C �� C�� Ö � ÂÂ L N 6 , because > is � i and CR,

the Øb ' 6 � form � � N ç Ü ^ ° ÛB_ à >1 Ö � � Ö N 6 is closed: �� N 6 . By an appli-cation of Stokes’ theorem, it follows that 6 N eKÀ ß �� N e ¾%¿ ú ß û �@ó e ¾ r � ,which proves the claim.

Finally, to approximate > by polynomials on ¾¥. in the �¥A topology, inthe above integral (5.3) that is performed on the fixed maximally real sub-manifold Í A , it suffices to develop the exponential in Taylor series and tointegrate term by term. In other functional spaces, the arguments have tobe adapted.

�As a consequence, uniqueness in the Cauchy problem holds. It may

be shown ([Trv1981, Trv1992]) that the trace of a CR distribution on amaximally real submanifold always exists, in the distributional sense.

Corollary 5.4. ([Trv1981, Trv1992]) If a CR function or distribution van-ishes on a maximally real submanifold Í of

�, there exists an open neigh-

borhood ¾ ¾ of Í in�

in which it vanishes identically.

Since every submanifold � of�

which is generic in�S�

contains smallmaximally real sumanifolds passing through every of its points, the corol-lary also holds with Í replaced by such a � .

Proof. It suffices to localize the above construction in a neighborhood ofan arbitrary point �y�DÍ and to take for Í A a neighborhood of � in Í . Theintegral (5.3) then vanishes identically.

�Corollary 5.5. ([Trv1981, Trv1992]) The support of a CR function or dis-tribution on

�is a closed CR-invariant subset of

�.

Proof. By contraposition, if a CR function or distribution vanishes in aneighborhood ¾V. of a point � in

�, it vanishes identically in the CR-

invariant hull of ¾V. , viz the union of CR orbits of all points Ñ\� ¾¥. . The


CR orbits being covered by concatenations of CR vector fields, neglectingsome technicalities, the main step is to establish:

Lemma 5.6. Let � � �, let R be a section of Ä " � and let Ñ�� Ny � Ç ³ �ÖR � º� � for some

³ � � o . If a CR function or distribution vanishesin a neighborhood of � , it vanishes also in a neighborhood of Ñ .

Indeed, we may construct a one-parameter family *� ² � A � ² � ² � of � � hy-persurfaces of

�with Ñ��ò�$� ² � and with � A contained in a small neigh-

borhood of � at which the CR function of distribution vanishes already.As illustrated by the following diagram, we can insure that at every pointÑ ² N(y � Ç ³ R � º� � , the vector RÌFÑ ² � is nontangent to � ² .

.�integral curves of �

Á Ù�ÂÄÃOÅwÆ ËÈÇ Ù ÐIË�Ï5ÐÁ Ê ÂÄÃOÅwÆ ËÈÇQÐIË�Ï5ÐPROPAGATION OF VANISHING ALONG THE INTEGRAL CURVE OF A CR VECTOR FIELD

Q r Q Ü Q Ü ÙIt follows that the hypersurfaces � ² are generic in

� �, for every

³. Then

the phrase after Corollary 5.4 applies to each � ² from � A up to � ² � , show-ing the propagation of vanishing.

�5.7. Unique continuation principles. At least three unique continuationproperties are known to be enjoyed by holomorphic functions � of severalcomplex variables defined in a domain

¸¸��·�. Indeed, we have � 6 in

either of the following three cases:

(ucp1) the restriction of � to some nonempty open subset of¸

vanishesidentically;

(ucp2) the restriction of � to some generic local submanifold Í of¸

van-ishes identically;

(ucp3) there exists a point �� ¸ at which the infinite jet of � vanishes.

In Complex Analysis and Geometry, the (ucpi) have deep influence onthe whole structure of the theory. Finer principles involving tools fromHarmonic Analysis appear in [MP2006b].


Problem 5.8. Find generalizations of the (ucpi) to the category of embed-ded generic submanifolds

�.

Since a domain¸

of�·�

trivially consists of a single CR orbit, it is nat-ural to assume that the given generic manifold

�is globally minimal (al-

though some meaningful questions arise without this assumption, we pre-fer not to enter such technicalities). In this setting, Corollaries 5.4 and 5.5provide a complete generalization of (ucp1) and of (ucp2).

A version of (ucp3) with the point � in the boundary Q ¸ does not hold,even in complex dimension one. Indeed, the function

y � Ç e ç � � � è ò H · É g is

holomorphic in É á � N ��É � � � Ë�yMÉ ¢ 6v� , of class � � on É á andflat at

ÉôN 6 . The restriction of this function to the Heisenberg sphereË�yMÉ�N µ Êµ of� � provides a CR example.

To generalize rightly (ucp3), let�

be a � i generic submanifold of codi-mension � � =

and of CR dimension ~ � =in� �

, with b N ~ï�>� . Letq be a � i submanifold of�

satisfying:Ä "Î � ã ÄVÎ q N Ä»Î �(' Ñ��Iq kHere, q plays the role of the point � in (ucp3). Denote by � b$&% the unionof CR orbits of points of q , i.e. the CR-invariant hull of q . It is an opensubset of

�. We say that a CR function > � � ¦ �

of class � i vanishes toinfinite order along q if for every �\�Iq , there exists an open neighborhood¾V. of � in

�such that for every ] �yd , there exists a constant

��¢ 6 withJ ��P¢ ��J � � qÙ)�*V�64�P¢ ' q � t [ ' ¢z�¹¾». kTheorem 5.9. ([Ro1986b, BT1988], [ P ]) Assume that q is the intersectionwith

�of some � -dimensional holomorphic submanifold of

�S�. If a CR

function of class � i vanishes to infinite order along q , then it vanishesidentically on the globally minimal generic submanifold

�.

Assuming that q is only a conic � -codimensional holomorphic subman-ifold entering a wedge to which all CR functions of

�extend holomor-

phically (Theorem 3.8(V)), the proof of this theorem may be easily gener-alized.

Open question 5.10. ([Ro1986b, BT1988]) Is the above unique continua-tion true for q merely � i ?

To attack this question, one should start with�

being unit sphere� � �� and q �u� � being any ÄÌ" � � -transversal real segment which is nowhere

locally the boundary of a complex curve lying inside the ball.


IV: Hilbert transform and Bishop’s equation inHolder spaces

Table of contents

1. Holder spaces: basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2. Cauchy integral, Sokhotskiı-Plemelj formulas and Hilbert transform . . . . . 114.3. Solving a local parametrized Bishop equation with optimal loss of smoothness129.4. Appendix: proofs of some lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.

[1 diagram]

In complex and harmonic analysis, the spaces ® 3 � � of fractionally differen-tiable maps, called Holder spaces, are very flexible to generate inequalities andthey yield rather satisfactory norm estimates for almost all the classical singularintegral operators, especially when ¨?ÊÌË�Ê F

. For instance, the Cauchy integralof a ® 3 � � function T csÍÏÎ ` defined on a ® 3 á i � � Jordan curve Í of the complexplane produces a sectionally holomorphic function, whose boundary values fromone or the either side are ® 3 � � on the curve. The Sokhotskiı-Plemelj formulasshow that the arithmetic mean of the two (in general different) boundary values ata point of the curve is given by the principal value of the Cauchy integral at thatpoint.

Harmonic and Fourier analysis on the unit disc Ð is of particular interest forgeometric applications in Cauchy-Riemann geometry. According to a theoremdue to Privalov, the Hilbert transform Ñ is a bounded linear endomorphism of® 3 � � M!J Ð U � P with norm Ò Ò Ò Ñ¼Ò Ò Ò 3 � � equivalent to

$� ^-i � _ , for some absolute constantÓÕÔ ¨ . This operator produces the harmonic conjugate Ñ of any real-valuedfunction c J Ð Î � on the unit circle, so that §Ö© Ñ always extends holo-morphically to Ð . Bishop (1965), Hill-Taiani (1978), Boggess-Pitts (1985) andTumanov (1990) formulated and solved a functional equation involving Ñ in orderto find small analytic discs with boundaries contained in a generic submanifold _of codimension ³ in ` � .

In a general setting, this Bishop-type equation is of the form:× M¦Ø � { P b × A ©ÙÑ ÈÚ M × M�Û P U Û URÜ P ¡ M¦Ø � { P Uwhere× AÞÝ � K is a constant vector, where Ú b Ú M KU Ø � { URÜ P is an � K -valued® 3 � � map, with ßf[ F

and ¨àÊáËhÊ F, where Ý � K , where Ø � { Ý J Ð and

where Ü Ý � � is an additional parameter which is useful in geometric applications.Under some explicit assumptions of smallness of

× A and of the first order jet ofÚ , the general solution× b × M¦Ø � { URÜ U × A P is of class ® 3 � � with respect to Ø � { and


in addition, for everyG

with ¨âÊ G ÊãË , it is of class ® 3 � < with respect to all thevariables M¦Ø � { URÜ U × A P . These smoothness properties are optimal.

1. HOLDER SPACES: BASIC PROPERTIES

1.1. Background on Holder spaces. Let ä Ý9å with ä¸[ Fand let æ bM æ i U��U æ � P Ý � � . On the vector space � � , we choose once for all the maxi-

mum norm Ò æ�Ò cÌb µ@ç�è+i�� Ò æ � Ò and, for any “radius” é satisfying ¨�Êêéìëîí ,we define the open cube ï �� cÌb �>æ Ý � � c Ò ¥ Ò~Ê�é � as a fundamental, concreteopen set. For é b í , we identify ï � � with � � .

Let ß Ýðå and let Ë Ý � with ¨�ëñËãë F. If ò b � or ` , a scalar functionT c ï �� Î ò belongs to the Holder class ® 3 � � M ï �� U ò P if, for every multiindex ó bM ó i U��U ó � P Ýôå � of length Ò ó°Ò�ëÖß , the partial derivative T ¾�õ M æ P cÌb ½Kö õ öc�½�¾ õ �>÷�÷�÷ ½�¾ õ ~ is

continuous in ï �� and if, moreover, the quantity:Ò Ò T Ò Ò 3 � � cÌb øA � ¡ � ¡ � 3¨ùwú%û¾ | � ~� Ò T ¾�õ M æ P Ò § ø¡ � ¡ ¹ 3 ùwú%û¾ Ò ÒOü¹ ¾ Ò | � ~� Ò T ¾ õ M æ Ô Ô P © T ¾ õ M æ Ô P ÒÒ æ Ô Ô ©Ïæ Ô Ò �is finite (if Ë b ¨ , it is understood that the second sum is absent). In case T bM T i U��U T � P is a ò � -valued mapping, with ý�[ F

, we simply define Ò Ò T Ò Ò 3 � � cÌbµ?ç�è i��<�� Ò Ò T � Ò Ò 3 � � . This is coherent with the choice of the maximum norm Ò þ�Ò cÌbµ?ç�è i�� Ò þ � Ò on ò � . For short, such a map will be said to be ® 3 � � -smooth or ofclass ® 3 � � and we write T Ý ® 3 � � . One may verify Ò Ò T i T � Ò Ò 3 � � ë�Ò Ò T i Ò Ò 3 � � Û Ò Ò T � Ò Ò 3 � �and of course Ò Ò � i T i § � � T � Ò Ò 3 � � ëîÒ � i Ò�Ò Ò T i Ò Ò 3 � � § Ò � � Ò�Ò Ò T � Ò Ò 3 � � . If ß b ¨ and Ë b F

,

the map T is called Lipschitzian. The condition ¢¢ T M¦Ø � { Ò Ò P © T M¦Ø � { Ò P ¢¢ ë Ó Û Ò ± Ô Ô © ± Ô Òon the unit circle was first introduced by Lipschitz in 1864 as sufficient for thepointwise convergence of Fourier series.

Thanks to a uniform convergence argument, the space ® 3 � � M ï �� U ò P is shownto be complete, hence it constitutes a Banach algebra. The space of functionsdefined on the closure ï �� also constitutes a Banach algebra. If Ë is positive,thanks to a prolongation argument, one may verify that ® 3 � � M ï �� U ò P identifieswith the restriction ® 3 � � ¤ ï �� U ò ² ¢¢ � ~� .Holder spaces may also be defined on arbitrary convex open subsets. Moregenerally, on an arbitrary subset ÿ ¢�� , it is reasonable to define the Holdernorms Ò Ò Û Ò Ò 3 � � , ¨@ÊfËÞë F

, only if � � ù � � M æ Ô Ô U æ Ô P ë Ó Û Ò æ Ô Ô © æ Ô Ò for every two pointsæ Ô Ô U æ Ô Ý ÿ . This is the case for instance if ÿ is a domain in � � having piecewise® i � A boundary.Introducing the total order M ß i�U Ë i P ë M ß U Ë P defined by: ß i Ê+ß , or: ß i b ß

and Ë i ëfË , we verify that ® 3 � � is contained in ® 3 � � � � and that:� Ò Ò T Ò Ò 3 � A ë�Ò Ò T Ò Ò 3 � � for all Ë with ¨@ÊÖËðë Fand for all ß Ýdå ;� Ò Ò T Ò Ò 3 � �H� ë��Ò Ò T Ò Ò 3 � � à for all Ë i�U Ë � with ¨�ÊáË i ÊáË � ë F

and for allß Ýôå ;

HOLOMORPHIC EXTENSIONS AND REMOVABLE SINGULARITIES 113� Ò Ò T Ò Ò 3 � i ë�Ò Ò T Ò Ò 3 á i � A , for all ß Ýôå .

The first inequality above is trivial while the third follows from (1.3) below. Weexplain the factor � in the second inequality. Since Ò æ Ô Ô © æ Ô Ò �H� ë Ò æ Ô Ô © æ Ô Ò � àonly if Ò æ Ô Ô © æ Ô ÒBë F

, we may estimate:

ùwú%ûA \ ¡ ¾ Ò Ò ¾ Ò ¡ �&i Ò T M æ Ô Ô P © T M æ Ô P ÒÒ æ Ô Ô ©Ïæ Ô Ò �H� ë ùwú%ûA \ ¡ ¾ Ò Ò ¾ Ò ¡ �&i Ò T M æ Ô Ô P © T M æ Ô P ÒÒ æ Ô Ô ©Ïæ Ô Ò � à ë�Ò Ò T Ò Ò A�� à �On the other hand, if Ò æ Ô Ô © æ Ô Ò Ô F

, we simply apply the (not fine) inequalities:Ò T M æ Ô Ô P © T M æ Ô P ÒÒ æ Ô Ô © æ Ô Ò �H� ë ¢¢ T M æ Ô Ô P © T M æ Ô P ¢¢ ë5]�Ò Ò T Ò Ò A�� A ë5]�Ò Ò T Ò Ò A�� à �Consequently:Ò Ò T Ò Ò A�� H� b Ò Ò T Ò Ò A�� A § ùRú%û¾ Ò Ò ü¹ ¾ Ò Ò T M æ Ô Ô P © T M æ Ô P ÒÒ æ Ô Ô © æ Ô Ò � � ë��Ò Ò T Ò Ò A�� à Uwith a factor � . For general ß�[ F

, the desired inequality follows:Ò Ò T Ò Ò 3 � �H� b Ò Ò T Ò Ò 3 i � A § ø¡ � ¡ ¹ 3 Ò Ò T ¾ õ Ò Ò A�� H� ë�Ò Ò T Ò Ò 3 i � A § � ø¡ � ¡ ¹ 3 Ò Ò T ¾ õ Ò Ò A�� àë��Ò Ò T Ò Ò 3 � � à �In the sequel, sometimes, we might abbreviate ® 3 � A by ® 3 , a standard notation.

However, we shall never abbreviate ® A�� by ® � , in order to avoid the unpleasantambiguity ® i � A ] ® i ] ® A�� i . Without providing proofs, let us state some funda-mental structural properties of Holder spaces. Some of them are in [Kr1983].� The inclusions ® « � < ¢s® 3 � � for M �/U G P Ô M ß U Ë P are all strict. For instance, on� , the function � 3 � � b � 3 � � M æ P equal to zero for æ¨ëç¨ and, for æä[Ã¨ :� 3 � � M æ P b � æ 3 á+��U �� ¨@ÊÖËÞë F Uæ 3� log æ U �� Ë b ¨ Uis ® 3 � � in any neighborhood of the origin, not better.� If ¨ÙÊ�Ë i Ê�Ë , any uniformly bounded set of functions in ® 3 � � contains asequence of functions that converges in ® 3 � �H� -norm to a function in ® 3 � �H� . This isa Holder-space version of the Arzela-Ascoli lemma.� For ¨@ÊÖËðë F

, define the Holder semi-norm (notice the wide hat):Ò Ò T Ò Ò �A�� cÌb ùwú%û¾ Ò ÒOü¹ ¾ Ò | � ~� Ò T M æ Ô Ô P © T M æ Ô P ÒÒ æ Ô Ô © æ Ô Ò � �The constants satisfy Ò Ò �#Ò Ò �A�� b ¨ and, of course, we have Ò Ò T Ò Ò A�� ] Ò Ò T Ò Ò A�� A § Ò Ò T Ò Ò �A�� .As a function of Ë , the semi-norm is logarithmically convex:Ò Ò T Ò Ò �A�� ° � � á ^-i ° _ � à b R Ò Ò T Ò Ò � A�� H� \ ° Û R Ò Ò T Ò Ò � A�� à \ i ° �Here, ¨¨ÊÖË i ÊfË � ë F

and ¨@ë X ë F.

114 JOEL MERKER AND EGMONT PORTEN� Importantly, if T is ò � -valued, ifF ë��ë¹ý , from the Taylor integral

formula:

(1.2) T � ¤ æ Ô Ô�² © T � ¤ æ Ô�² b iA �ø �º¹»i J T �J æ � ¤ æ Ô §�� M æ Ô Ô ©Ïæ Ô P ²�� æ Ô Ô� ©Ïæ Ô�� ³ � Ufollows the mean value inequality:

(1.3) ¢¢ T ¤ æ Ô Ô ² © T ¤ æ Ô ² ¢¢ b µ@ç�èi�� ¢¢ T � ¤ æ Ô Ô ² © T � ¤ æ Ô ² ¢¢ë�Ò Ò T Ò Ò �i � A Û ¢¢ æ Ô Ô © æ Ô ¢¢ Uwhere æ Ô Ô U æ Ô Ý ï �� are arbitrary, and whereÒ Ò T Ò Ò �i � A cÌb µ@ç�èi�� øÜ ¹»i ùwú%û¡ ¾ ¡ \ � Ò T �Á� ¾ z M æ P Ò �This useful inequality also holds (by definition) if T is merely Lipschitzian, withÒ Ò T Ò Ò �i � A replaced by Ò Ò T Ò Ò �A�� i .� If a function T is ® 3 � A , then for every multiindex ó ÝÞå � of length Ò óBÒ ëîß ,the partial derivative T ¾�õ is ® 3 Y¡ � ¡ � A and Ò Ò T ¾�õ Ò Ò 3 Y¡ � ¡ � A ëîÒ Ò T Ò Ò 3 � A .�

2. CAUCHY INTEGRAL, SOKHOTSKII-PLEMELJ FORMULAS AND HILBERT

TRANSFORM

2.1. Boundary behaviour of the Cauchy integral. Let ÿ be a domain in ` , letª Ý ÿ and let Í be a ® i -smooth simple closed curve surrounding ª and orientedcounterclockwise. Assume that its interior domain (to which ª belongs) is entirelycontained in ÿ . In case Í is a circle, Cauchy ([Ca1831]) established in 1831 thecelebrated representation formula:T M ª P b F]�� © 1 T M!O P ³ OO ©�ª Uvalid for all functions T Ý�� M ÿ P holomorphic in ÿ . Remarkably, Í may bemodified and deformed without altering the value T M ª P of the integral.

The best proof of this formula is to derive it from the more general Cauchy-Green-Pompeiu formula, itself being an elementary consequence of the Green-Stokes formula, which is valid for functions T of class only ® i defined on theclosure of a domain ÿ£¢ ` having ® i -smooth oriented boundary J ÿ ([Ho1973]):T M ª P b F]�� © ½ � T M!O P ³ OO ©�ª § F]�� © ? � J T J��OO ©�ª ³ O ´ ³ �O �Indeed, for holomorphic T , one clearly sees that the “remainder” double integraldisappears.


The holomorphicity of the kerneliï ê enables then to build concisely the funda-

mental properties of holomorphic functions from Cauchy’s formula: local conver-gence of Taylor series, residue theorem, Cauchy uniform convergence theorem,maximum principle, etc. ([Ho1973]). Studying the Cauchy integral for itself ap-peared therefore to be of interest and became a thoroughly investigated subject inthe years 1910–1960, under the influence of Privalov.

If ª Ý ÿ belongs to the exterior of Í , i.e. to the unbounded component of`! Í , by a fundamental theorem also due to Cauchy, the integral vanishes: ¨ bi� � �#" 1 � ^ ï _ K ïï ê . Thus, fixing the countour Í , as ª moves toward Í , the Cauchyintegral is constant, either equal to T M ª P or to ¨ . What happens when ª hits thecurve Í ?

Denote by O A a point of Í and by Ð M!O A U%$ P the open disc of radius $ Ô ¨ centeredat O A . If Í � denotes the complement Í |Ð M!O A U%$ P , introducing an arc of small circlecontained in J Ð M!O A U%$ P to join the two extreme points of Í � , it may be verified that

(2.2)F] T M!O A P b �� µ� Æ A F]�� © 1 � T M!O P ³ OO © O A �Geometrically speaking, essentially one half of the circle J Ð M!O A U%$ P of radius $

centered at O A is contained in the domain ÿ . Consequently, the “correct value” ofthe Cauchy integral at a point O A of the curve Í is equal to the arithmetic mean:F] ` � � µê Æ ï r � ê W�·-O�W L X § �� µê Æ ï r � ê t'& � O�W L X a b F] M T M!O A P § ¨ P �Let us recall briefly why the excision of an $ -neighborhood of O A in the domainof integration is necessary to provide this “correct” average value. Parametriz-ing Í by a real number, the problem of giving a sense to the singular integral" 1 � ^ ï _ K ïï ï r amounts to the following classical definition of the notion of principalvalue ([Mu1953, Ga1966, EK2000]).

2.3. Principal value integrals. Let ( U*) Ý � with (hÊ ) and let T be a ® i -smooth real-valued function defined on the open segment M ( U*) P . Pick æ Ý � with(�Ê�æ Ê ) and consider the integral " �° K óó ¾ whose integrand is singular. The twointegrals avoiding the singularity from the left and from the right, namely:

¾ � �° ³ þþ ©Ïæ b �+-, M $ i P © �.+-, M æ ©/( P ç-0 � �¾ á � à ³ þþ ©Ïæ b �+-, M ) © æ P © �+-, M $ � P

tend to ©¡í , as $ i Î ¨ á , and to § í as $ � Î ¨ á . Clearly, if $ � b $ i (or moregenerally, if $ i and $ � both depend continuously on an auxiliary parameter $ Ô ¨with

F b �� µ � Æ A � � à ^ � _� � ^ � _ ), the positive and the negative parts compensate, so that


the principal value:

û �21=� �° ³ þþS©�æ cÌb �� µ� Æ A � ` ¾ �° § �¾ á � a b �+-, ) © ææS©3(exists. Briefly, there is a key cancellation of infinite parts, thanks to the fact thatthe singular kernel

ió is odd. This is why in (2.2) above, the integration wasperformed over the excised curve Í � .

Generally, if � c ( U*) ¡ Î � is a real-valued function the principal value integral,defined by:

û �21=� �° � M þ P ³ þþ © æ cÌb �� µ� Æ A � ` ¾ �° § �¾ á � ab �° � M þ P © � M æ Pþ © æ ³ þ § � M æ P û �21=� �° ³ þþS© æ ³ þb �° � M þ P © � M æ Pþ © æ ³ þ § � M æ P �+-, ) ©Ïææ ©4(exists whenever the quotient À ^ ó _ À ^ ¾ _ó ¾ is integrable. This is the case for instance

if � is of class ® i � A or of class ® A�� , with Ë Ô ¨ , since " iA þ � i ³ þ Êãí . More istrue.

Theorem 2.4. ([Mu1953, Ve1962, Dy1991, SME1988, EK2000], [ 5 ]) Let �uc ( U*) ¡ Î � be ® 3 � � -smooth, with ß�[ç¨ and ¨@ÊfËÞÊ F. Then for every æ Ý M ( U*) P ,the principal value integral 6

M æ P cÌb û �21=� �° � M þ P ³ þþ © æexists. In every closed segment ( Ô U*) Ô ¡ contained in M ( U*) P , the func-tion

6M æ P becomes ® 3 � � -smooth and enjoys the norm inequalityÒ Ò 6 Ò Ò ¸ � l �87 °¼Ò � �hÒ:9 ë $� ^-i � _ Ò Ò � Ò Ò ¸ � l �87 ° � ��9 , for some constant

Ó b Ó M ß U ( U*)�U ( Ô U*) Ô P . If �together with its derivatives up to order ß vanish at the two extreme points ( and) , the function

6M æ P is ® 3 � � -smooth over ( U*) ¡ and enjoys the norm inequalityÒ Ò 6 Ò Ò ¸ � l �87 ° � ��9 ë $� ^-i � _ Ò Ò � Ò Ò ¸ � l �87 ° � ��9 , for some constant

Ó b Ó M ß U ( U*) P .Notice the presence of the (nonremovable) factori� ^Ói � _ .

2.5. General Cauchy integral. Beginning with works of Sokhotskiı [So1873],of Harnack [Ha1885] and of Morera [Mo1889], the Cauchy integral transform:

£ M ª P cÌb F]�� © 1 T M!O P ³ OO ©�ªhas been studied for itself, in the more general case where Í is an arbitrary closedor non-closed curve in ` and T is an arbitrary smooth complex-valued function


defined on Í , not necessarily holomorphic in a neighborhood of Í (precise rig-orous assumptions will follow; historical account may be found in [Ga1966]).In Sokhotskiı’s and in Harnack’s works, the study of the boundary behaviour ofthe Cauchy integral was motivated by physical problems; its boundary propertiesfind applications to mechanics, to hydrodynamics and to elasticity theory. Let usrestitute briefly the connection to the notion of logarithmic potential ([Mu1953]).

Assuming Í and T c`Í+Î � to be real-valued and of class at least ® i � A , pa-rametrize Í by arc-length O b O M Ü P , denote ; M Ü P cÌb O¬M Ü P ©çª the radial vectorfrom ª to O M Ü P , denote Z b Z M Ü P b Ò ; M Ü P Ò its euclidean norm, denote < M Ü P cÌb K%=K �the unit tangent vector field to Í and denote > M Ü P cÌb K%=K � ¢¢ K%=K � ¢¢ the unit normalvector field to Í . Puting ª b ¥�§+©¬« and decomposing the Cauchy transform£ M ª P b × M ¥ U « P §®©@? M ¥ U « P in real and imaginary parts, the two functions

×and

? are harmonic in `! Í , since £ is clearly holomorphic there. After elementarycomputations, one shows that

×may be expressed under the form:× M ¥ U « P b F]�� 1 TBA + ù M ; U > PZ ³ Ü U

which, physically, represents the potential of a double layer with moment-density�� . Also, ? may be expressed under the form:

? M ¥ U « P b F]�� 1 ³ T³ Ü �+-, Z ³ Ü Uwhich, in the case where Í consists of a finite number of closed Jordan curves,represents the potential of a single layer with moment-density © i� � K �K � .2.6. The Sokhotskiı-Plemelj formulas. Coming back to the mathematical studyof the Cauchy integral, we shall assume that the curve Í over which the integrationis performed is a connected curve of finite length parametrized by arc length ( U*) ¡DC ÜFE © Î O M Ü P Ý Í Uwhere (ÖÊ ) , where O¬M Ü P is of class ® 3 á i � � over the closed segment ( U*) ¡ , andwhere ß [�¨ , ¨�ÊáË�Ê F

. Topologically, we shall assume that Í%b O ( U*) ¡ iseither:� a Closed Jordan arc, namely O c ( U*) ¡ Î ` is an embedding;� or a Jordan contour, namely O c M ( U*) P Î ` is an embedding, O¬M ( P bO¬M ) P , O extends as a ® 3 á i � � -smooth map on the quotient ( U*) ¡ M (�G ) P

and ÍÁb O ( U*) ¡ is diffeomorphic to a circle.

Various more general assumptions can be made: Í consists of a finite numberof connected pieces, Í is piecewise smooth (corners appear), Í possesses certaincusps, Í is only Lipschitz, the length of Í is not finite, T is � ¯ -integrable, T is� ¯� , i.e. T Ý � ¯ M Í P and " 1 Ò T M Ü § Z P © T M Ü P Ò ¯ ëIH � ² Ò Z Ò � , T belongs to certainSobolev spaces, T M!O P ³ O is replaced by a measure ³ � M!O P , etc., but we shall notreview the theory (see [Mu1953, Ve1962, Ga1966] and especially [Dy1991]).


The natural orientation of the segment ( U*) ¡ induced by the order relation on� enables to orient the two semi-local sides of Í in ` : the region on the leftto Í will be called the positive side (“ § ”), while the region to the right will becalled negative (“ © ”). In the case where Í is Jordan contour, we assume that Í isoriented counterclockwise, so that the positive region coincides with the boundedcomponent of `J Í .

Theorem 2.7. ([Mu1953, Ve1962, Ga1966, Dy1991, SME1988, EK2000], [ 5 ])Let Í be a ® 3 á i � � -smooth closed Jordan arc or Jordan contour in ` and let T cÍìÎ ` be a ® 3 � � -smooth complex-valued function.

(a) If Í Ô is any closed portion of Í having no ends in common with those ofÍ , then for every O i Ý Í Ô , the Cauchy transform £ M ª P cÌb i� � � " 1 � ^ ï _ K ïï êpossesses (a priori distinct) limits £ á M!O i P and £ M!O i P , when ª tends toO i from the positive or from the negative side.

(b) These two limits £ á and £ are of class ® 3 � � on Í Ô with a norm estimateÒ Ò £ � Ò Ò ¸ � l � ^ 1 Ò _ ë $ ^ 3 � 1 Ò � 1 _� ^-i � _ Ò Ò T Ò Ò ¸ � l � ^ 1 _ , for some positive constant whereÓ M ß U Í Ô U Í P .(c) Furthermore, if K Ôá and K Ô denote an upper and a lower open one-sided

neighborhood Í Ô in ` , the two functions £ � c K Ô� Î ` defined by:

� £ � M ª P cÌb £ M ª P �� ª Ý K Ô� U£ � M ª P cÌb £ � M!O i P �� ª b O i Ý Í Ô Uare of class ® 3 � � in K Ô�ML Í Ô , with a similar norm estimateÒ Ò £ � Ò Ò ¸ � l � ^ ¤ ÒNPO 1 Ò _ ë $ � ^ 3 � 1 Ò � 1 _� ^Ói � _ Ò Ò T Ò Ò ¸ � l � ^ 1 _ .

(d) Finally, at every point O A of the curve Í not coinciding with its ends, £ áand £ satisfy the two Sokhotskiı-Plemelj formulas:

QR2S £ á M!O A P © £ M!O A P b T M!O A P UF] � £ á M!O A P §®£ M!O A P � b û �21=� F]�� © 1 T M!O PO © O A ³ O �


T #T �

UV T �T #VU

æ æ1 1

ï ^s°,_ï ^��P_ WYX

ááZ 3P[]\_^ 3` ^ 3 Í

a ` # a Í ÒcbÆed fgd \hji*k�l monqp [srqtu[[ Í [ X

TWO DIAGRAMS FOR THE SOKHOTSKI vI-PLEMELJ FORMULAS

Sometimes, £ is called sectionnally holomorphic, as it is discontinuous acrossÍ . Its jump across Í is provided by the first formula above, while the arithmeticmean

T � á T #� is given by the value of the Cauchy (singular) integral at O A Ý Í .Morera’s classical theorem ([Mo1889]) states that if £ á and £ match up on theinterior of Í , then the Cauchy integral is holomorphic in ` minus the endpoints ofÍ . As is known ([Sh1990]), this theorem is also true for an arbitrary holomorphicfunction £ Ýw� M `! Í P which is not necessarily defined by a Cauchy integral.

2.8. Less regular boundaries. The boundary behaviour of the Cauchy transformat the two extreme points x M ( P and x M ) P of a Jordan arc is studied in [Mu1953]. Werefer to [Dy1991] for a survey presentation of the finest condition on Í (namely, itto be a Carleson curve) which insures that the Cauchy integral exists and that theSokhotskiı-Plemelj formulas hold true, almost everywhere. Let us just mentionwhat happens with the Cauchy integral £ M ª P in the limit case Ë b ¨ .

If Í is (only) ® i � A , if T is (only) ® A�� A , then for O i in the interior of Í , thelimit £ M!O i P exists if and only if the limit £ á M!O i P exists ([Mu1953]). However,generically, none limit exists.

A more useful statement, valid in the case Ë b ¨ , is as follows. AssumeÍ to be ® 3 á i � A with ßî[ ¨ and let Í Ô be a closed portion of the interior of Í .Parametrize Í Ô by a ® 3 á i � A map O Ô c ( Ô U*) Ô ¡ Î Í Ô . Extend O Ô b O Ô M Ü P as a a® 3 á i � A embedding O Ô M ÜeU%$ P defined on ( Ô U*) Ô ¡�y M © $ A U%$ A P , where $ A Ô ¨ , withO Ô M ÜeU ¨ Pä] O Ô M Ü P and with O Ô M ÜeU%$ P in the positive side of Í Ô for $ Ô ¨ . The familyof curves Í Ô� cÌb O Ô M ( Ô U*) Ô ¡zyÁ� $ � P foliates a strip thickening of Í Ô .Theorem 2.9. ([Mu1953]) For every choice of a ® 3 á i � A extension O Ô M Ü U%$ P , andevery T Ý ® 3 � A M Í U ` P , the difference from either side of the Cauchy transform£ Ò 1 Ò � © £ Ò 1 Ò # � Ò tends to T Ò 1 Ò in ® 3 � A norm as $ Î ¨ :�� µ� Æ A ùwú%û� | 7 ° Ò � � Ò 9 ¢¢ ¢¢ £ M!O Ô M Ü U%$ PYP © £ M!O Ô M ÜeU © $ PYP © T M!O Ô M Ü PYP ¢¢ ¢¢ 3 � A b ¨ �

To conclude, we state a criterion, due to Hardy-Littlewood, which insures ® 3 � � -smoothness of holomorphic functions up to the boundary.


Theorem 2.10. ([Mu1953, Ga1966], [ 5 ]) Let Í be a ® 3 á i � � -smooth Jordan con-tour, divinding the complex plane in two components ÿ á (bounded) and ÿ (un-bounded). If T Ý{� M ÿ � P satisfies the estimate Ò J 3ê T M ª P Ò2ë Ó M F ©ÖÒ ª�Ò P i � , forsome ß Ý�å , some Ë with ¨DÊÌËàÊ F

, and some positive constantÓ^Ô ¨ , then T

is of class ® 3 � � in the closure ÿ � b ÿ � L Í .

2.11. Functions and maps defined on the unit circle. In the sequel, ÿ will bethe unit disc Ð cÌb � O Ý ` c Ò O ÒlÊ F � having as boundary the unit circle J Ð cÌb� O Ý ` c Ò O Ò b F � . Consider a function T c J Ð Î ò , where ò b � or ` .Parametrizing J Ð by O b Ø � { with ± Ý � , such an T will be considered as thefunction �|C ± E © Î T M¦Ø � { P Ý ò �For � Ýdå , we shall write T { � cÌb K � �Kw{ � .

Let Ë satisfy ¨�ÊñËãë Fand assume that T Ý ® A�� . We define its ® A�� semi-

norm (notice the wide hat) precisely by:Ò Ò T Ò Ò �A�� cÌb ùwú%û{ Ò Ò¬ü¹ { Ò Ò T M¦Ø � { Ò Ò P © T M¦Ø � { Ò P ÒÒ ± Ô Ô © ± Ô Ò � �Thanks to ]�� -periodicity, ùwú%û { Ò Ò¬ü¹ { Ò may be replaced by ùwú%û A \ ¡ { Ò Ò { Ò ¡ � � . Accord-ing to the definition of

�1.1, the function T is ® 3 � � if the quantityÒ Ò T Ò Ò 3 � � cÌb øA �<�³� 3 Ò Ò T { � Ò Ò A�� A § Ò Ò T { � Ò Ò �A�� ÊÖí

is finite. Besides Holder spaces, we shall also consider the Lebesgue spaces� ¯ M!J Ð P , with } Ý � satisfyingF ë~}àë í . As J Ð is compact, the Holder

inequality entails the (strict) inclusions � � M!J Ð P ¢ � ¯ Ò M!J Ð P ¢ � ¯ M!J Ð P ¢� i M!J Ð P , forF Ê{}�Ê{} Ô ÊÖí .

2.12. Fourier series of Holder continuous functions. If T is at least of class � ion J Ð , let �T Ü cÌb F]�� © ½ } O Ü T M!O P ³ OOdenote the � -th Fourier coefficient of T , where � Ý�� . Given ä Ý�å , consider theä -th partial sum of the Fourier series of T :� � T M¦Ø � { P cÌb ø � � Ü � �

�T Ü Ø � Ü { �We remind that Dini’s (elementary) criterion:

�A ¢¢ T M¦Ø �X^ { á ° _ P § T M¦Ø �ý^ { ° _ P © ] T M¦Ø � { P ¢¢X ³ X ÊÖí


insures the pointwise convergence� � µ �@Æ � � � T M¦Ø � { P b T M¦Ø � { P . If T is ® A�� onJ Ð , with ¨âÊêËÖë F

, the above integral obviously converges at every Ø � { Ý J Ð ,so that we may identify T with its (complete) Fourier series:T M¦Ø � { P b � T M¦Ø � { P cÌb øÜ |�� T Ü Ø � Ü { �In fact ([Zy1959]), if T Ý ® 3 � � with ß Ýáå and ¨/ëÕË¹ë F

, then ¢¢�T Ü ¢¢ ë� � � �¡ Ü ¡ � � � Ò Ò T Ò Ò �3 � � for all � Ý�� e��¨ � . Also, if T Ý ® A�� , then �� ¢¢

�T � ¢¢2� converges

for � Ô ��'��D� . In 1913, Bersteın proved absolute convergence of �� ¢¢�T � ¢¢ forË Ô F �� .

2.13. Three Cauchy transforms in the unit disc. In the case ÿ b Ð , our goal isto formulate Theorem 2.7 with more precision about the constant

Ó M ß U J ÿ P . For� Ý J Ð in the unit circle and T Ý ®z�� M!J Ð U%� P with ß��u¨ , ¨ìÊ�Ë�Ê F, as in�

2.6, we define: � � T M � P cÌb ��.�� Æ �'� F� � © 8� } T M!O PO ©4� � ³ O U�¡ T M � P cÌb û �21=� F� � © ¢� } T M!O PO © � ³ O U�D£ T M � P cÌb ��.�� Æ � � F� � © ¢� } T M!O PO ©3� � ³ O �The Sokhotskiı-Plemelj formulas hold: T M � P b � � T M � P © � £ T M � P and

� T M � P b�� T M � P § � £ T M � P ¡ . A theorem due to Aleksandrov6 enables to obtain a preciseestimate of the ®¤�� norms of these Cauchy operators. To describe it, define:¥ � cÌb§¦ T Ý ® � � M!J Ð U%� P c �T b ¨©¨ �Then Ò Ò Û Ò Òuª � � is a norm on

¥ � , since only the constants � satisfy Ò Ò �KÒ Ò'ª � � b ¨ . For« U%¬ Ý � with ¨�Ê « U¬ Ê F

, recall the definition ® M « U%¬ P cÌb " � ¥°¯£ � M F ©¥ P@± £ � ³ ¥ of the Euler beta function.

Theorem 2.14. ([Al1975]) The operator

� T M � P cÌb û �21=� ��³²�´ " � }¶µ�·¹¸*º¸ £» ³ O is abounded linear endomorphism of

¥ � having norm:

¢¢ ¢¢ ¢¢�¡ ¢¢ ¢¢ ¢¢ ª � � b F� � ® ` Ë � U F ©ðË� a �

One may easily verify the two equivalences ® ¤ � � U � £ �� ² G �� as Ë Î ¨ and® ¤ � � U � £ �� ² G �� £ � as Ë Î Fas well as the two inequalities:FË M F ©ÙË P ë�® ` Ë � U F ©ðË� a ë ¼Ë M F ©ðË P �6We are grateful to Burglind Joricke who provided the reference [Al1975].


Thus, the nonremovable factor�� · � £ � º shows what is the precise rate of explosion

of the norm ¢¢ ¢¢ ¢¢� ¢¢ ¢¢ ¢¢ ª � � as Ë Î ¨ or as Ë Î F

.

Further, if T Ý ® � � does not necessarily belong to¥ �

, it is elementary tocheck that ¢¢ ¢¢

� T ¢¢ ¢¢ � ë¾½� Ò Ò T Ò Ò � � , for some absolute constantÓ Ô ¨ . It follows

that the (complete) operator norm ¢¢ ¢¢ ¢¢� ¢¢ ¢¢ ¢¢ � � behaves like ½� · � £ � º .

In conclusion, thanks to the Sokhotskiı-Plemelj formulas

� � T b �� M � T § T Pand

� £ T b �� M � T © T P , we deduce that there exists an absolute constantÓ � Ô F

such that: F Ó �Ë M F ©ðË P ë ¢¢ ¢¢ ¢¢�z¿ ¢¢ ¢¢ ¢¢ � � ë Ó �Ë M F ©ÞË P Uwhere ¨@ÊfËªÊ F

and where À b © U ¨ U § .Next, what happens with T Ý ® �� , for ß Ý å arbitrary ? For À b © U ¨ U § , the� ¿are bounded linear endomorphisms of ® �� and similarly:

Theorem 2.15. There exists an absolute constantÓ � Ô F

such that if ß Ý�å and¨?ÊfËªÊ F, for À b © U ¨ U § :F Ó �Ë M F ©ÙË P ë ¢¢ ¢¢ ¢¢

� ¿ ¢¢ ¢¢ ¢¢ �� ë Ó �Ë M F ©ÞË P �In other words, the constantÓ � is independent of ß . To deduce this theorem

from the estimates with ß b ¨ (with different absolute constantÓ � ) we proceed

as follows, without exposing all the rigorous details.Inserting the Fourier series

� M T U Ø ´ { P in the integrals definingÓ £

,Ó

,Ó �

andintegrating termwise (an operation which may be justified), we get:QÁÁÁÁÁÁÁÁR ÁÁÁÁÁÁÁÁS

� £ T M¦Ø ´ { P b © ø� \ �T � Ø ´ � { U� T M¦Ø ´ { P b © F� ø� \ �T � Ø ´ � { § F� �T § F� ø� � �T � Ø ´ � { U� � T M¦Ø ´ { P b �T § ø� � �T � Ø ´ � { �

If ß/� F, by differentiating termwise with respect to ± these three Fourier repre-

sentations of the

� ¿, we see that these operators commute with differentiation.

Lemma 2.16. For every � Ýôå with ¨¨ë � ëfß and for À b © U ¨ U § , we have:�z¿ M T { Â P b R �z¿ T \ { Â �Dealing directly with the principal value definition of

� T , another proof ofthis lemma for

� would consist in integrating by parts, deducing afterwards that� £

and

� �enjoy the same property, thanks to the Sokhotskiı-Plemelj formulas.


To establish Theorem 2.15, we introduce another auxiliary ® �� norm:Ò Ò T Ò Ò Ã�� cÌb ø ÅÄ°ÆgÄ � Ò Ò T { Â Ò Ò � � b Ò Ò T Ò Ò �-� � § ø ÅÄ°ÆÇÄ � £ � Ò Ò T { Â Ò Ò ª � � Uwhich is equivalent to Ò Ò Û Ò Ò �-� � , thanks to the elementary inequalities ([ 5 ]):Ò Ò T Ò Ò �-� � ëîÒ Ò T Ò Ò Ã�� ë|È³É § �zÊ~Ò Ò Ë2Ò Ò �� Notice that Ò Ò Û Ò Ò Ã � � b Ò Ò Û Ò Ò � � . The next lemma applies to Ì b � £ÎÍ � �Í � �

and toÌ b Ñ , the Hilbert conjugation operator defined below.

Lemma 2.17. ([ 5 ]) Let Ì be a bounded linear endomorphism of all the spaces® �� ÈsÏlÐ Í � Ê with ß Ýñå , ¨ãÊ Ë½ÊÐÉ , which commutes with differentiations,namely Ì�ÈuË { Â Ê b È'Ì-Ë¤Ê { Â , for � Ýdå . Assume that there exist a contant

Ó � È¦ËÑÊ Ô Édepending on Ë such that

Ó � È¦ËÒÊ £ � ë�Ò Ò Ò¹Ì�Ò Ò Ò � � ë Ó � È¦ËÒÊ . Then for every ß ÝDå :� Ó � È¦ËÒÊ £ � ëÞÒ Ò ÒÓÌ�Ò Ò Ò Ã�� ë Ó � È¦ËÒÊ ÍÈ³É § �zÊ£ � Ó � È¦ËÒÊ £ � ëÞÒ Ò ÒÓÌ�Ò Ò Ò �� ë|È³É § �zÊ Ó � È¦ËÑÊ �Proof. Indeed, if Ë Ý ® �� , we develope a chain of (in)equalities:Ò Ò¹Ì-Ë2Ò Ò �� ë+Ò ÒÓÌ-Ë2Ò Ò2Ã�� b ø ÅÄ°ÆgÄ � Ò Ò¹È³Ì-Ë¤Ê { Â Ò Ò � � b ø ÅÄ°ÆgÄ � Ò ÒÓÌÔÈsË { Â Ê�Ò Ò � �ë Ó � È¦ËÒÊ ø ÅÄ°ÆgÄ � Ò Ò Ë { Â Ò Ò � � b Ó � È¦ËÒÊ.Ò Ò Ë2Ò Ò Ã�� ëÕÈ³É § �zÊ Ó � È¦ËÑÊ�Ò Ò Ë2Ò Ò �� This yields the two majorations. Minorations are obtained similarly. ï

To conclude this paragraph, we state a Tœplitz type theorem about

� �, which

will be crucial in solving Bishop’s equation with optimal loss of smoothness, aswe will see in Section 3. A similar one holds about

� £, assuming Ö ÝØ×3Ù È � ÐÚÊ

instead, where � is the Riemann sphere.

Theorem 2.18. ([Tu1994b], [ 5 ]) There exists an absolute constantÓ � Ô É such

that for all Ë Ý ®Û�-� � , ß Ýdå , ¨¨ÊfËªÊÕÉ , and all Ö ÝØÜ Ù È¦ÐÚÊoÝßÞ � È¦ÐàÊ�¦³� Ù È¦ÐÚÊ :¢¢ ¢¢� � ÈsË Ö¡Ê ¢¢ ¢¢ �� ë Ó �ËcÈ³É ©ðËÒÊ Ò Ò Ë2Ò Ò �� Ò Ò2Ö=Ò Ò á8â �

Closely related to the Cauchy transform are the Schwarz and the Hilbert trans-forms.

2.19. Schwarz transform on the unit disc. Let Ý � � ÈsÏlÐ Í �cÊ be real-valued.The Schwarz transform of is the function of ª Ý Ð defined by:ã ÈIª¢ÊcÝßÞ É� � © � } Èsä°Ê�å ä §çæä�è æ

é ³ ääëê


Thanks to the holomorphicity of the kernel,ã È æ Ê is a holomorphic function of

æ Ý Ð . Decomposing it in real and imaginary parts:ã È æ ÊJÞíì È æÍ�æ Ê §ª© Ñ È æ

Í�æ ÊÍ

we get the Poisson transform of :ì È æÍ�æ Ê�ÝßÞ É� � © ¢� } Èsä°Ê¢îðï å ä §çæä�è æ

é ³ ää Ítogether with the Hilbert transform of :Ñ È æ

Í�æ ÊcÝßÞ É� � © ¢� } Èsä°Ê°ñ � å ä §�æäòè æ

é ³ ää êThanks to the harmonicity of the two kernels, ì and Ñ are harmonic in Ð . Thepower series of

� , of ì and of Ñ are given by:QÁÁÁÁÁÁÁÁR ÁÁÁÁÁÁÁÁSã È æ ÊJÞôó § � ø� � ó � æ �

Íì È æ

Í�æ ÊJÞ ø� \ ó � �æ � § ó § ø� � ó � æ �

ÍÑ È æ

Í�æ ÊJÞ É ©

õ èëö��÷ óø �úùæ �üû ö��ý óø � æ ��þÍ

where óø � is the � -th Fourier coefficient of ø . These three series converge normallyon compact subsets of ÿ .

2.20. Poisson transform on the unit disc. Let us first summarize the basic prop-erties of the Poisson transform ([Ka1968, DR2002]). Setting æ Þ � � ´ { with�� É and ä Þ � ´�� , computing îðï ¸ ��¸ £ �� and switching the convolutionintegral, we obtain:ìPøÑÈ�� ´ { ÊJÞ É�� ²£ ²�� È��%Ê°øÑÈ � ´ · { £ � º Ê��JÞ � �� ø È � ´ { Ê Íwhere � � È��%ÊcÝßÞ Éúè3� �Éúè � �� û � �is the Poisson summability kernel. It has three nice properties:! � �" �

on ÏPÿ for�#� �$�|É ,! ��³²&% ²£ ² � � È��%Ê��!ÞíÉ for

�#� ��ÕÉ , and:!('*) � �,+ �'� � � È��'ÊJÞ �for every �.-0/¹è � Í �21�3�4 ��5 .

Consequently, � � is an approximation of the Dirac measure 6 � at É7- Ï©ÿ . Forthis reason, the Poisson convolution integral possesses excellent boundary valueproperties.


Lemma 2.21. ([Ka1968, DR2002]) Convergence in norm holds:

(i) If ø8-:9.; with É �=< �?> or « Þ=> and ø is continuous, then'*) � �,+ �'�A@ @ � �� øàè4ø @ @ á�BoÞ �.

(ii) If øC-ED �� with FC-HG and�#�JIK� É , including

I Þ �and

I Þ É , then'*) � �,+ �'�A@ @ � �� øàè4ø @ @ �� Þ �.

In DÛ�� , the pointwise convergence '*) � �,+ �'� � �� øÒÈ � ´ { ÊML�øÑÈ � ´ { Ê follows ob-viously. However, in 9 ; , from convergence in norm one may only deduce point-wise convergence almost everywhere for some sequence � � L É which dependson the function. In 9 ; , almost everywhere pointwise convergence was proved byFatou in 1906.

Theorem 2.22. ([Fa1906, Ka1968, DR2002]) If øN-K9O; with É �P<E� > , thenfor almost every � ´ { -�ÏPÿ , we have:'*) ��,+ �'� � �� ø È � ´ { ÊJÞôøÒÈ � ´ { Ê ê

In summary, the Poisson transform ìPø yields a harmonic extension to ÿ ofany function ø(-C9 ; ÈsÏ©ÿ ÍRQ Ê or ø(-HD �� ÈsÏ©ÿ ÍRQ Ê , with expected boundary valueÀ � } È³ì øÛÊJÞôø on ÏPÿ .

2.23. Hilbert transform on the unit disc. Next, we survey the fundamental prop-erties of the Hilbert transform. Again, ø is real-valued on Ï©ÿ . Setting æ ÞI� � ´ {with

�� S� É and ä Þ � ´T� , computing ñ � ¸ ��¸ £ �� and switching the convolutionintegral, we obtain: U øÑÈ�� ´ { Ê!Þ É�� ²£ ²#V � È��'Ê°øÒÈ � ´ · { £ � º Ê�� Íwhere V � È��%ÊoÝßÞ � �#� )*W �Éüè � �� û � �is the Hilbert kernel. It is not a summability kernel, being positive and negativewith 9 � norm tending to > as �AL É £ ; for this reason, the Hilbert transformdoes not enjoy the same nice boundary value properties as the Poisson transform:Holder classes are needed.

Setting ��ÞÐÉ , the Poisson kernel ì � È��%Ê vanishes identically and the Hilbertkernel tends to

�YX�Z\[]�� £ �Y^`_RXa� Þ ^b_RXa�dc³�X�Z\[]�dc³� . Near �àÞ �, the function ��fe�È��,g � Ê behaves

like the function � gh� , having infinite 9 � norm. For øJ-iD � � ÈsÏPÿ ÍRQ Ê , it may beverified that, as æ L � ´ { -�ÏPÿ , the Hilbert transform

U øÑÈ æ Ê tends toU øÑÈ � ´ { Ê�ÝßÞij ê\k¡ê É�� ²£ ² øÑÈ � ´ · { £ � º Êe,l W È��,g � Ê ��Þij ê\k¡ê É��m � ²£ ² øÒÈsä°Ê°ñ �on ä û � ´ {ä�è � ´ {qp �°ääëê


Since îðï ¸ ��rtsù¸ £ r sù #v �for äØÞ � ´w� -çÏ©ÿ , we get ñ � ¸ ��rxsbu¸ £ r sbu Þ � ´ ¸ ��rxsbu¸ £ r sbu so that

we may rewrite m U øÑÈ � ´ { ÊJÞyj ê\k¡ê É��m � ²£ ² øÑÈsä¢Ê ä û � ´ {ä�è � ´ { �°ää êSetting ì ø�ÝßÞ ��³²�´ % � } øÑÈsä¢Ê{z ¸¸ Þ�óø , the algebraic relation

�¸ £ r sù è �¸ Þ ¸ ��rxsbu¸ £ r sbu �¸gives a fundamental relation between

� and

U:� � è�ì Þ m U ê

From Theorem 2.15, we deduce ( ì is innocuous):

Theorem 2.24. ([Pri1916, HiTa1978, Bo1991, BER1999], [ � ]) There exist anabsolute constant | � " É such that if F}-HG and

� � I �ÕÉ :É~gf| �I È³Éðè I Ê � @ @ @ U @ @ @ �� | �I È³Éúè I Ê8êIt follows that at the level of Fourier series,

Utransforms øÑÈ � ´ { Ê!Þ � øÒÈ � ´ { ÊcÞ�{�e�{�Bóø � � ´ � { toU øÒÈ � ´ { ÊcÝßÞ É m õ è ö�e÷ ó ø � � ´ � { û ö�eý ó ø � � ´ � { þ ê

Notice that È��U ø¤Ê Þ �. In fact, this formula coincides with the series� ´ è �ô��÷ óø �Pùæ � û ��ý óø � æ � , written for æ L � ´ { , the limit existing pro-

vided� � I �ÕÉ .

By termwise differentiation of the above formula,

U È�ø { Â Ê Þ~È U ø¡Ê { Â for�A��S� F , if ø�-�DÛ�� (some integrations by parts in the singular integral defining

U øwould yield a second proof of this property).

The Poisson transform ìPø of øJ-KD � � having boundary value À � } È³ì øÛÊ Þ øand the Schwarz transform being holomorphic in ÿ , we see that the functionø û m U ø on Ï©ÿ extends holomorphically to ÿ as �ÔøÑÈ æ Ê . So

U ø on ÿ is one ofthe Harmonic conjugates of ø . In general, these conjugates are defined up to aconstant. The property È �U ø¡Ê Þ �

means that

U øÒÈ � ÊJÞ �.

Lemma 2.25. The Hilbert transform

U ø on ÏPÿ is the boundary value on Ï©ÿ ofthe unique harmonic conjugate in ÿ of the harmonic Poisson extension ìPø , thatvanishes at

� -�ÿ .

For ø�-ED �� ÈsÏPÿ ÍRQ Ê Í ø û m U ø extends holomorphically to ÿ êFurthermore,

U È U ø¡Ê!Þíèüø û óø .


2.26. Hilbert transform in 9 ; spaces. It is elementary to show that the study

of the principal value integral j ê\k©ê ��³² % ²£ ²0� · r s��\u �f�� º�� [ · �dc³� º �� is equivalent to the studyof the same singular convolution operator, in which ��fe�È��Rg � Ê is replaced by � gh� .Similarly, one may define the Hilbert transform on the real line:� ËÎÈd��ÊcÝßÞ�j ê\k¡ê �� ËÎÈ��Ê� è�� f� êIf Ë is D � � on

Qand has compact support or satisfies % � @ Ë @ ��> , replacingËÎÈ��Ê in the numerator by /2ËÎÈ��Ê èÕËÎÈd��Ê 1 û ËÎÈd��Ê and reasoning as in � 2.3, one

straightforwardly shows the existence of the above principal value.Privalov showed that

� ËÎÈd��Ê exists for almost every ��- Qif Ë�-�9 � È Q Ê . A

theorem due to M. Riesz states that the two Hilbert transforms�

on the real lineand

Uon the unit circle are bounded endomorphisms of 9O; , for ÉK� < ��> ,

namely if Ë�-}9 ; È Q Ê and øC-}9 ; ÈsÏPÿÚÊ , then:@ @ � Ë @ @ á B · � º � | ; @ @ Ë @ @ á B · � º l W�� @ @ U ø @ @ á B · � } º � | ; @ @ ø @ @ á B · � } º Íwhith the same constant | ; ([Zy1959], Chapters VII and XVI). In [Pi1972], Zyg-mund’s doctoral student Pichorides obtained the best value of the constant | ; :for É0� <i� � , | ; Þ�e,l W ²� ; , while, by a duality argument, | ; Þ��fe ²� ; for� ��< �8> . The two elementary bounds e,l W ²� ; � ;; £ � for É0� <i� � and��fe ²� ; ��<

for � ��< ��> , yield:(2.27)@ @ � Ë @ @ á�B · � º � <��< è�É @ @ Ë @ @ á�B · � º l W � @ @ U Ë @ @ á�B · � } º � <��< è É @ @ Ë @ @ á�B · � } º Ífor É�� < �o> . In 9 � , the Hilbert transform is unbounded but, according to atheorem due to Kolmogorov ([Dy1991, DR2002]), it sastisfies a weak inequality:� 4 � ËÎÈd��Ê "N� 5�� | � @ @ Ë @ @ á�� Ífor every � - Q

with �H" �, where � is the Lebesgue measure and where | " �

is some absolute constant.

2.28. Pointwise convergence of Fourier series. The boundedness of the Hilberttransform in 9 ; has a long history, closely related to the problem of pointwise con-vergence of Fourier series. In 1913, before M. Riesz proved the estimates (2.27),using complex function theory and the Riesz-Fischer theorem, Luzin showed that�

is bounded in 9 � and formulated the celebrated conjecture that Fourier seriesof 9 � functions converge pointwise almost everywhere. This “hypothetical theo-rem” was established by Carleson ([Ca1966]) in 1966 and slightly later by Hunt([Hu1966]) in 9 ; for É$� < ��> . A complete self-contained restitution of theseresults is available in [DR2002]. Let us survey the main theorem.


The -th partial sum of the Fourier series of a function Ë on Ï©ÿ is given by:��¡ ËÎÈ � ´ { ÊJÞ É�� ²£ ²�¢ ¡ È��%Ê8ËÎÈ � ´ · { £ � º Ê�� Íwhere ¢ ¡ È��%ÊcÝßÞ � )dW È� û É~g � Êx�� )*W �,g �is the Dirichlet kernel, having unbounded 9 � norm @ @ ¢ ¡ @ @ á��£ ¤²�¥ ' �f¦§ . It iselementary to show that the behaviour of this convolution integral, as &L¨> , isequivalent to the behaviour of the integral:� ²£ ² � )*W ©�� ËÎÈ � ´ · { £ � º Ê�� êWithout loss of generality, Ë is assumed to be real-valued, so that the above inte-gral is the imaginary part of the Carleson integral:� ¡ ÈsË Í � ´ { ÊoÝßÞ�j ê\k©ê � ²£ ² � ´ ¡ �� ËÎÈ � ´ · { £ � º Ê�� êIn, chapters 4, 5, 6, 7, 8, 9 and 10 of [DR2002], the main proposition is to provethat the Carleson maximal sublinear operator:�«ª ËÎÈ � ´ { ÊcÝßÞ��¬]j¡ �hA®®® � ¡ ÈsË Í � ´ { Ê ®®®is bounded from 9¯; to 9.; . The proof involves dyadic partitions, changes of fre-quency, microscopic Fourier analysis of Ë , choices of allowed pairs and seven ex-ceptional sets. By an elementary argument, one deduces that the maximal Fourierseries sublinear operator: � ª ËÎÈ � ´ { Ê�ÝßÞ°�,¬]j¡ �{K®®® � ¡ ËÎÈ � ´ { Ê ®®®is bounded from 9¯; to 9�; .Theorem 2.29. ([Ca1966, Hu1966, DR2002]) If Ë0-79 ; with É$� < ��> , thereexists an absolute constant | " É such that:@ @ � ª Ë @ @ á B � | < ¤È < è�ÉeÊt± @ @ Ë @ @ á B ê

Then by a standard argument, '*) �#¡ + Ù ��¡ ËÎÈ � ´ { ÊÎÞ{ËÎÈ � ´ { Ê almost everywhere.

2.30. Transition. Since the grounding article [HiTa1978], the nice behaviour ofthe Hilbert transform in the Holder classes (Theorem 2.24) is the main reasonwhy Bishop analytic discs have been constructed in the category of D �-� � genericsubmanifolds of � ¡ ([BPo1982, BPi1985, Tu1990, Trp1990, Bo1991, BRT1994,Tu1994a, Me1994, Trp1996, Jo1996, BER1999]). Perhaps it is also interesting toconstruct Bishop analytic discs in the Sobolev classes.

HOLOMORPHIC EXTENSIONS AND REMOVABLE SINGULARITIES 129� 3. SOLVING A LOCAL PARAMETRIZED BISHOP EQUATION WITH OPTIMAL

LOSS OF SMOOTHNESS

3.1. Analytic discs attached to a generic submanifold of � ¡ . As in Theo-rem 4.2(III), let ² be a D¤�-� � local graphed generic submanifold of equation³ ÞµćÈ�¶ ÍR·©Í ø¡Ê , where ´ is defined for @ ¶ û m · @ ��¸ � , @ ø @ �J¸ � , for some ¸ � " �and where ćÈ � ÊoÞ �

, ��ćÈ � Ê!Þ �and @ ´ @ �i¸ � .

Definition 3.2. An analytic disc is a mapÿº¹wä¼» è2L¾½ Èsä¢ÊJÞ È`¿ Èsä°Ê ÍwÀ Èsä¢Ê'Ê.- �ÂÁPÃw� zwhich is holomorphic in the unit disc ÿ and at least D � in ÿ . It is attached to ²if it sends ÏPÿ into ² .

Thus, suppose that È`¿ Èsä°Ê ÍwÀ Èsä°Ê'Ê is attached to ² and sufficiently small,namely @ / Ä û m�Å�1 È � ´ { Ê @ ��¸ � , @ÇÆ È � ´ { Ê @ ��¸ � and @\È È � ´ { Ê @ ��¸ � on ÏPÿ , where¿ Èsä¢ÊoÞ�Ä�Èsä¢Ê û mxÅ Èsä¢Ê and

À Èsä°ÊoÞ Æ Èsä°Ê û m È Èsä°Ê . Then clearly, the disc sendsÏ©ÿ to ² if and only ifÈ È � ´ { ÊÎÞ�´ n ÄçÈ � ´ { Ê Í Å È � ´ { Ê Í Æ È � ´ { Ê p Ífor every � ´ { - ÏPÿ . Thanks to the Hilbert transform, we claim that we mayexpress analytically the fact that the disc is attached to ² .

At first, in order to guarantee the applicability of the harmonic conjugation op-erator

U, all our analytic discs will D �� on ÿ , with FÉ-ÊG and

� � I �ÐÉ .We let

Uact componentwise on maps Æ Þ È Æ � Í êÇêÇê Í Æ z Ê�-8D¡�� ÈsÏ©ÿ ÍRQ z Ê ,

namely

U Æ ÝßÞ U Æ � Í êÇêÇê Í U Æ z . We set @ @ U Æ�@ @ �� ÝßÞ � l{Ë � Ä°ÆÇÄ z ®® ®®U Æ Æ ®® ®® �� .

With a slight change of notation, instead of ì Æ È � Ê , we denote by ì Æ ÝßÞ��³² % ²£ ² Æ È � ´ { Ê��Ì the value at the origin of the Poisson extension ì Æ . Equiv-

alently, ì Æ Þ óÆ is the mean value of Æ on ÏPÿ . Here is a summary of the mostuseful properties of

U.

Lemma 3.3. The

Q z -valued Hilbert transform

Uis a bounded linear endomor-

phism of D �� ÈsÏPÿ ÍRQ z Ê with�xc ½ �� · � £ � º � @ @ @ U @ @ @ �� ½ �� · � £ � º satisfying

U ÈÍ�a�Re Ê Þ �and

U È U Æ ÊJÞíè Æ û ì Æ êIn the sequel, we shall rather use the mild modification

U � of

Udefined by:U � Æ È � ´ { ÊoÝßÞ U Æ È � ´ { ÊÒè U Æ È³ÉeÊ ê

In fact,

U � is uniquely determined by the normalizing condition

U � Æ È³ÉeÊ Þ �.

Then

U � is also bounded:�xc ½ �� · � £ � º � @ @ @ U � @ @ @ �� ½ �� · � £ � º , also annihilates con-

stants:

U � ÈÍ�a�ReÅÊJÞ �and U � È U � Æ Ê!Þíè Æ û Æ È³ÉeÊ ê

Furthermore, most importantly:


Lemma 3.4. If Æ -ED �� ÈsÏ©ÿ ÍRQ z Ê , then Æ È � ´ { Ê û m U � Æ È � ´ { Ê extends as a holo-morphic map ÿÎL � z which is D¤�� in the closed disc ÿ .

To check that the extension is D �� in ÿ , one may introduce the Poisson integralformula and apply Lemma 2.21 (ii).

If ½ÕÞ È`¿ ÍwÀ Ê is an analytic disc attached to ² , we set Æ ÝßÞ Æ È³ÉeÊ and È ÝßÞÈ È³ÉeÊ . Since

Àis holomorphic, necessarily È È � ´ { Ê!Þ U � Æ È � ´ { Ê û È . Applying

U � to both sides, we get

U � È È � ´ { ÊúÞ§è Æ È � ´ { Ê û Æ (the left and the right handsides vanish at � ´ { Þ§É ). Applying

U � to È È � ´ { ÊüÞÉ´7ÏÄçÈ � ´ { Ê Í Å È � ´ { Ê Í Æ È � ´ { Ê above and reorganizing, we obtain that Æ satisfies a functional equation7 involvingthe Hilbert transform:

(3.5) Æ È � ´ { ÊJÞíè U � /ÐćÈ�Ä�ÈtÑßÊ Í Å ÈtÑßÊ Í Æ ÈtÑßÊ'Ê 1 È � ´ { Ê û Æ êHere, the map Æ ÝüÏ©ÿÒL Q z is the unknown, whereas the holomorphic map¿�Þ�Ä û m�Å Ý�Ï©ÿÓL � Á and the constant vector Æ are given data.

Conversely, given Ä û m�Å and Æ , assume that Æ -ÔD �� satisfies theabove functional equation. Set È È � ´ { Ê ÝßÞ U � Æ È � ´ { Ê û È , where È ÝßÞćÈ�ÄçÈ³ÉeÊ Í Å È³ÉeÊ Í Æ È³ÉeÊ'Ê . Then Æ È � ´ { Ê û m È È � ´ { Ê extends as a D �� map ÿ�¹4äÕ»LÀ Èsä¢Ê.- � z which is holomorphic in ÿ . If @ / Ä û m�Å�1 È � ´ { Ê @ �i¸ � , @ÇÆ È � ´ { Ê @ �J¸ �and @\È È � ´ { Ê @ �i¸ � , the disc ½ ÝßÞ È`¿ ÍwÀ Ê is attached to ² .

Bishop (1965) in the D �� classes and then Hill-Taiani (1978), Boggess-Pitts(1985) in the Holder classes Dz�� established existence and uniqueness of thesolution Æ to the fundamental functional equation (3.5).

Theorem 3.6. ([Bi1965, HiTa1978, BPi1985]) If ² is at least D � � � , shrinking¸ � if necessary, there exists ¸ � with� �o¸ � �º¸ � such that whenever the data¿Ö-�D � � È ÿ Í � Á Ê ¦7× Èsÿ Í � Á Ê and Æ - Q z satisfy @ ¿ È � ´ { Ê @ ��¸ � on Ï©ÿ and@ÇÆ @ �y¸ � , there exists a unique solution Æ -}D � Ø ÈsÏPÿ ÍRQ z Ê , � �°Ùi� I �

, to theBishop-type functional equation (3.5) above such that @ÇÆ È � ´ { Ê @ ��¸ � on ÏPÿ andsuch that in addition @\È È � ´ { Ê @ �i¸ � on Ï©ÿ , whereÈ È � ´ { ÊoÝßÞ U � Æ È � ´ { Ê û óÈ�ÄçÈ³ÉeÊ Í Å È³ÉeÊ Í Æ È³ÉeÊ'Ê êConsequently, the disc È`¿ Í Æ û m È Ê is attached to ² .

7The origin of this equation may be found in the seminal article [Bi1965] of Bishop.Since then, it has been further exploited in [Pi1974a, Pi1974b, HiTa1978, BeFo1978,We1982, BG1983, BPo1982, KW1982, R1983, HiTa1984, KW1984, BPi1985, FR1985,Trp1986, Fo1986, Tu1988, Ai1989, Tu1990, Trp1990, Bo1991, DH1992, Tu1994a,Tu1994b, BRT1994, CR1994, Gl1994, Me1994, HuKr1995, Jo1995, Trp1996, Tu1996,Jo1996, Jo1997, Me1997, Po1997, MP1998, Tu1998, Hu1998, CR1998, Jo1999a,Jo1999b, MP1999, BER1999, Po2000, MP2000, Tu2001, Da2001, DS2001, Me2002,MP2002, HM2002, Po2003, JS2004, Me2004c].


Notice the (substantial) loss of smoothness, occuring also in [BRT1994,BER1999], which is due to an application of a general implicit function theo-rem in Banach spaces. The main theorem of this chapter ([Tu1990, Tu1996])refines the preceding result with a negligible loss of smoothness, provided thegraphing map ´ belongs to the Holder space DÑ�� . In the geometric applications(Parts V and VI), it is advantageous to be able to solve a Bishop equation like (3.5)which involves supplementary parameters. Thus, instead of ´ , we shall consideran

Q z -valued D¤�� map Ú�ÞÛÚ È�ø Í � ´ { Í�Ü Ê , where

Üis a parameter. For fixed

Ü, we

shall denote by Ú @\Ý the map Þ zß � Ã Ï©ÿà¹íÈ�ø Í � ´ { Ê�» è�L Úisø Í � ´ { Í�Ü - Q z . Inaccordance with Section 1, we set:@ @ Ú � @ @ � ÝßÞ � l{Ë� Ä°ÆgÄ z áâ öã Ä�äÄ z ®® ®® Ú Æ �~å ®® ®® � aæç Íand similarly @ @ Ú { @ @ � Þ � l{Ë ã Ä°ÆÇÄ z ®® ®® Ú Æ { ®® ®® � .Theorem 3.7. ([Tu1990, Tu1996], [ � ]) Let Ú¶Þ�Ú ø Í � ´ { Í�Ü be an

Q z -valued

map of class D«è � é , F3� É , � � I �ÕÉ , defined for ø0- Q z , @ ø @ �y¸ ã , Ì�- QandÜ - Qëê

, @ Ü @ �Jì ã , where� �J¸ ã �|É and

� �Jì ã �IÉ . Assume that on its domainof definition Þ zß � Ã Ï©ÿ Ã Þ êí � , the map Ú and its derivatives with respect to ø andto Ì satisfy the inequalities (nothing is required about Ú Ý ):@ @ Ú @ @ � �iî ã Í @ @ Ú � @ @ � �Nîaï Í @ @ Ú { @ @ � �Nî ± Ífor some small positive constants

î ã , îðï andî ± such that

(3.8)î ã � | I ¸ ã Í îaïñ� | ï I ï#ò É û �,¬]jó Ý ó ÷ í � @ @ Ú @ Ý~@ @ ã � é�ô

£ ï Í î ± � ¸ ï ã îaï Íwhere

� ��|õ�MÉ is an absolute constant. Then for every fixed Æ satisfying@ÇÆ @ �ö¸ ã g¢Éð÷ and every fixed

Ü -ÖÞ êí � , the parameterized local Bishop-typefunctional equation: Æ È � ´ { Ê!Þíè U ã /ÐÚ3È Æ ÈtÑßÊ Í Ñ Í�Ü Ê 1 È � ´ { Ê û Æ has a unique solution: ÏPÿº¹ � ´ { » è�L Æ È � ´ { Í�Ü�Í Æ Ê�- Q z Íwith @ @ÇÆ�@ @ � � ¸ ã g ¼ which is of class D è � é on Ï©ÿ . Furthermore, this solution is

of class D©è � é £ Þµø Ø ÷ é D©è � Ø with respect to all the variables, including param-eters, namely the complete mapÏ©ÿ Ã Þ êí � Ã Þ zß � c ãbù ¹ n � ´ { Í�Ü�Í ÆYú p » è�L Æ È � ´ { Í�Ü�Í ÆÂú Ê.- Q zis D©è � é�û ú .


Since the assumptions involve only the D ã � é norm of Ú , we notice that thetheorem is also true with Ú°-HD«è � é�û ú , provided F�üJý , except that the solution willonly be D è � é�û ú with respect to � ´ { : it suffices to apply the theorem by consideringthat Ú°-ED©è � Ø , with ÙK� I

arbitrary, getting a solution that is DYè � Ø�û ú with respectto all variables and concluding from ø Ø ÷ é D è � Ø�û ú ÞyD è � é�û ú .

The main purpose of this section is to provide a thorough proof of the theorem.In the sequel, | , | ã , | ï , | ± and | ¤ will denote positive absolute constants. Wemay assure that they all will be üÕÉ � û�þ and

� É � þ .The smallness of @ @ Ú @ @ ú � ú , of @ @ Ú � @ @ ú � ú and of @ @ Ú { @ @ ú � ú guarantee the smallness of@ @ Ú @ Ü @ @ ã � é c ï by virtue of an elementary observation.

Lemma 3.9. ([ � ]) Let �y-µÞ ¡ß , o-�G , Îü�É , where� �:¸ ´ � > , and letËÕÞ ËÎÈd��Ê be D ú � é function with values in

Q z , ��ü É . If @ @ Ë @ @ ú � ú �Öî, for some

quantityî " �

, then: @ @ Ë @ @ ÿ ú � é c ï �Nî ã c ï�� ý û @ @ Ë @ @ ªú � é�� êWe apply this inequality to Ú � @ Ý and to Ú { @ Ý , pointing out that for any Ù with� �iÙ �JI

, by definition:@ @ Ú � @ Ýh@ @ ªú � Ø Þ��#l{Ëã Ä°ÆÇÄ z õ zö ä�� ã ®® Ú Æ �~å È�ø� � Í � ´ {� Í�Ü ÊÑè�Ú Æ �~å È�ø�� Í � ´ {� Í�Ü Ê ®®@ È�ø � � Í Ì � � ÊÒè�È�ø � Í Ì � Ê @ Ø þ Í@ @ Ú { @ Ýh@ @ ªú � Ø Þ��#l{Ëã Ä°ÆÇÄ z ®® Ú

Æ { È�ø�� Í � ´ {� Í�Ü ÊÑè�Ú Æ { È�ø�� Í � ´ {� Í�Ü Ê ®®@ È�ø � � Í Ì � � ÊÑè�È�ø � Í Ì � Ê @ Ø êLemma 3.10. ([ � ]) Independently of

Ü, we have:@ @ Ú � @ Ý{@ @ ÿ ú � é c ï �Nî ã c ïï � ý û @ @ Ú @ Ý~@ @ ã � é � Í@ @ Ú { @ Ý @ @ ÿ ú � é c ï �Nî ã c ï± � ý û @ @ Ú @ Ý @ @ ã � é � Í@ @ Ú @ Ý @ @ ã � é c ï �Nî ã û î ï û î ± û n î ã c ïï û î ã c ï± p � ý û @ @ Ú @ Ý @ @ ã � é � ê

The presence of the squares in the inequalities of Theorem 3.7 anticipates the rootsî ã c ïï andî ã c ï± above. These two lemmas and the next involve dry computations

with Holder norms. The detailed proofs are postponed to Section 4.

Lemma 3.11. ([ � ]) If Æ -HD ã � Ø ÈsÏPÿ ÍRQ z Ê with� �iÙ �yI

satisfies @ÇÆ È � ´ { Ê @ �i¸ ãon Ï©ÿ , then for every fixed

Ü -CÞ êí � , we have:@ @ Ú�È Æ ÈtÑßÊ Í Ñ Í�Ü Ê @ @ �� ·�� } º � @ @ Ú @ @ ú � ú û @ @ Ú { @ @ ú � ú û @ @ Ú { @ Ýh@ @ ªú � Ø�� É û n @ @ÇÆ�@ @ ªã � ú p Ø�� ûû @ @ Ú � @ @ ú � ú @ @ÇÆ�@ @ ã � Ø û @ @ Ú � @ Ýh@ @ ªú � Ø � @ @ÇÆ#@ @ ªã � ú û n @ @ÇÆ�@ @ ªã � ú p ã�� Ø�� ê


Remind @ @ÇÆ�@ @ ªã � ú Þ°�,¬ j { ®® Æ { È � ´ { Ê ®® . We then introduce the map:Æ » è�L��cÈ Æ ÊoÝßÞ Æ ú è U ã /ÇÚ È Æ ÈtÑßÊ Í Ñ Í�Ü Ê 1 È � ´ { Ê êTo construct the solution Æ , we endeavour a Picard iteration process, settingÆ��q@ � � ú ÝßÞ Æ ú with @ÇÆ ú @ �õ¸ ã g¢Éð÷ and Æ�� ã ÝßÞ��cÈ Æ�� Ê , for �º-ÖG , when-ever �cÈ Æ�� Ê may be defined, i.e. whenever @ @ÇÆ �]@ @ ú � ú ��¸ ã . We shall first work

in D ã � é c ï"! D©è � é .

Lemma 3.12. If we choose the absolute constant | �ÕÉ appearing in the theoremsufficiently small, then independently of

Ü, the sequence Æ#� satisfies the uniform

boundedness estimate: @ @ÇÆ��]@ @ ã � é c ï � ¸ ã g ¼ �N¸ ã Íhence it is defined for every �H-EG and each Æ$� belongs to D ã � é c ï ÈsÏ©ÿÚÊ .Proof. By Theorem 2.24, there exists an absolute constant | ã " É (not exactlythe same), such that @ @ @ U ã @ @ @ ã � é c ï � | ã g I êMajorating by means of the D ú � é c ï -norm:@ @ �cÈ Æ�� Ê @ @ ã � é c ï � @ÇÆÂú�@ û @ @ @ U ã @ @ @ ã � é c ï @ @ Ú�È Æ�� ÈtÑßÊ Í Ñ Í�Ü Ê @ @ �� %'& ¥ ·(� } º êAssume by induction that Æ � is D ã � é c ï and satisfies @ @ÇÆ��]@ @ ã � é c ï � ¸ ã g ¼ (this holds

for � Þ �). Clearly Æ�� ã Þ)�cÈ Æ�� Ê is D ã � é c ï . Thanks to Lemma 3.11, and to the

trivial majoration È @ @ÇÆ*�q@ @ ªã � ú Ê é c ï � È�¸ ã g ¼ Ê é c ï �|É :@ @ Ú�È Æ�� ÈtÑßÊ Í Ñ Í�Ü Ê @ @ �� %'& ¥ ·(� } º � @ @ Ú @ @ ú � ú û @ @ Ú { @ @ ú � ú û ý @ @ Ú { @ Ýh@ @ ÿ ú � é c ï ûû @ @ Ú � @ @ ú � ú @ @ÇÆ��q@ @ ã � é c ï û ý @ @ Ú � @ Ýh@ @ ÿ ú � é c ï @ @ÇÆ��]@ @ ã � é c ï êUsing then the assumptions (3.8) of the theorem together with Lemma 3.10:@ @ÇÆ�� ã @ @ ã � é c ï � ¸ ã g¢Éð÷ û | ã I û ã � î ã û î ± û ¼ î ã c ï± n É û @ @ Ú @ Ýh@ @ ã � é p ûû @ @ÇÆ��]@ @ ã � é c ï n î ï û ¼ î ã c ïï n É û @ @ Ú @ Ý @ @ ã � é p�p � êUsing the two trivial majorations

î ± � | I ¸ ã andîðï � | I together with the main

assumptions (3.8) to majorateî ã c ïï and

î ã c ï± , we get:@ @ÇÆ�� ã @ @ ã � é c ï � ¸ ã g¢Éð÷ û | ã ¸ ã ÷M| û @ @ÇÆ��]@ @ ã � é c ï | ã+ | êChoosing | � ããbù ½ � ù (whence | � ãï ½ � þ ), we finally get:@ @ÇÆ�� ã @ @ ã � é c ï � ¸ ã g-, û È³É~g�ý�Ê @ @TÆ��@ @ ã � é c ï êBy immediate induction, the assumption @ÇÆ ú @ �J¸ ã g¢Éð÷ and these (strict) inequal-ities entail that @ @ÇÆ*�q@ @ ã � é c ï � ¸ ã g ¼ for every �E-HG , as claimed. Þ


Lemma 3.13. ([Tu1990], [ � ]) For every Ù with� �¾Ù �õI

and every fixedÜ -�Þ êß � , if two maps Æ Æ -(D ã � ú ÈsÏ©ÿ ÍRQ z Ê with ®® ®® Æ Æ ®® ®® ú � ú ��¸ ã g-. for� Þ¶É Í ý are

given, the following inequality holds:®® ®® Ú È Æ ï ÈtÑßÊ Í Ñ Í�Ü ÊÎè�Ú È Æ ã ÈtÑßÊ Í Ñ Í�Ü Ê ®® ®® 0/ � � ·�� } º �21 ®® ®® Æ ï è Æ ã ®® ®® 0/ � � ·�� } º Íwhere 1 Þ ®® ®® Ú ó Ý ®® ®® ã � Ø ý � É û n ®® ®® Æ ã ®® ®® ªã � ú p Ø û n ®® ®® Æ ï ®® ®® ªã � ú p Ø�� êAgain, the (latexnically lengthy) proof is postponed to Section 4.

Lemma 3.14. If we choose the absolute constant | of the theorem sufficientlysmall, then independently of

Ü, the map:Æ » è�L��cÈ Æ ÊoÝßÞ ÆYú è U ã /ÇÚ È Æ ÈtÑßÊ Í Ñ Í�Ü Ê 1 È � ´ { Ê Í

restricted to the set of those Æ -HD ã � é c ï ÈsÏPÿ ÍRQ z Ê that satisfy @ @ÇÆ#@ @ ã � é c ï � ¸ ã g ¼ , isa contraction: ®® ®® �cÈ Æ ï ÊÎè3�cÈ Æ ã Ê ®® ®® ú � é c ï � Éý ®® ®® Æ ï è Æ ã ®® ®® ú � é c ï êProof. Let Æ Æ -ÎD ã � é c ï with ®® ®® Æ Æ ®® ®® ã � é c ï � ¸ ã g ¼ for

� Þ É Í ý . In particular,®® ®® Æ Æ ®® ®® ú � ú �i¸ ã g-. , so Lemma 3.13 applies. In the majorations below, to pass to thefourth line, we use the assumption ¸ ã � É , which enables us to majorate simplyby . the three terms in the brackets of the third line:®® ®® �cÈ Æ ï ÊÑè4��È Æ ã Ê ®® ®® ú � é c ï Þ ®® ®®

U ã65 Úi Æ ï ÈtÑßÊ Í Ñ Í�Ü è0ÚN Æ ã ÈtÑßÊ Í Ñ Í�Ü 07 ®® ®® ú � é c ï� @ @ @ U ã @ @ @ ú � é c ï ®® ®® Ú Æ ï ÈtÑßÊ Í Ñ Í�Ü èKÚ Æ ã ÈtÑßÊ Í Ñ Í�Ü ®® ®® ú � é c ï� | ãI ®® ®® Ú ó Ý ®® ®® ã � é c ï ý � É û n ®® ®® Æ ã ®® ®® ã � é c ï p é c ï û n ®® ®® Æ ï ®® ®® ã � é c ï p é c ï � ®® ®® Æ ï è Æ ã ®® ®® ú � é c ï� | ïI ®® ®® Ú ó Ý ®® ®® ã � é c ï ®® ®® Æ ï è Æ ã ®® ®® ú � é c ï Íwhere | ï " É is absolute. Then we apply Lemma 3.10 to majorate @ @ Ú @ Ý @ @ ã � é c ï ,we use the three trivial majorations

î ã Í î ï Í î ± � | I and we majorateî ã c ïï Í î ã c ï± by

means of (3.8), dropping ¸ ã �ÕÉ inî ã c ï± , which yields:@ @ Ú @ Ý{@ @ ã � é c ï �Nî ã û îaï û î ± û n î ã c ïï û î ã c ï± p � ý û ®® ®® Ú ó Ý ®® ®® ã � é �� .M| I û ýM| I û ¼ | I Þ98M| I ê

Then we conclude that®® ®® �cÈ Æ ï ÊÑè:�cÈ Æ ã Ê ®® ®® ú � é c ï � |�| ± ®® ®® Æ ï è Æ ã ®® ®® ú � é c ï êChoosing the absolute constant | of the theorem� ãï ½<; yields the desired con-

tracting factorãï . Þ


The fixed point theorem then entails that our sequence Æ=� converges in D ú � é c ï -norm towards some map Æ -�D ú � é c ï ÈsÏ©ÿ ÍRQ z Ê . More is true:

Lemma 3.15. For every fixed parameters È Ü�Í Æ ú Ê , this solutionÆ Þ Æ � ´ { Í�Ü�Í Æ ú Þ '*) � � + Ù Æ�� belongs in fact to D ã � é c ï ÈsÏPÿ ÍRQ z Ê andsatisfies @ @ÇÆ�@ @ ã � é c ï � ¸ ã g ¼ .Proof. Indeed, since @ @ Æ ��@ @ ã � é c ï � ¸ ã g ¼ is bounded, it is possible thanks to the

Arzela-Ascoli lemma to extract some subsequence converging in D ã � ú ÈsÏ©ÿÚÊ to amap, still denoted by Æ Þ Æ � ´ { Í�Ü�Í Æ ú , which is D ã � ú on ÏPÿ . Still denoting byÆ�� such a subsequence, we observe that the uniform convergence @ @ Æ#� è Æ#@ @ ã � ú L�

plus the boundedness @ @ÇÆ �q@ @ ÿ ã � é c ï � ¸ ã g ¼ entail immediately that the following

majoration holds:®® Æ { È � ´ { ÊÑè Æ { È � ´ { Ê ®®@ Ì � � è&Ì � @ é c ï Þ '*) �� + Ù ®® Æ�� { È � ´ { ÊÒè Æ�� { È � ´ { Ê ®®@ Ì � � è0Ì � @ é c ï � ¸ ã¼ Ífor arbitrary

� � @ Ì�� ¢è0Ì>� @ � � . Consequently, Æ belongs to D ã � é c ï . Passing tothe limit in @ @ Æ��q@ @ ã � é c ï � ¸ ã g ¼ , we also deduce @ @ÇÆ�@ @ ã � é c ï � ¸ ã g ¼ . Þ

The next crucial step is to study the regularity of the solutionÆ Þ Æ � ´ { Í�Ü�Í Æ ú with respect to È Ü�Í Æ ú Ê .Lemma 3.16. The solution Æ Þ Æ È � ´ { Í�Ü�Í ÆYú Ê satisfies a Lipschitz condition withrespect to the parameters

Üand Æ ú .

Proof. Consider two parameters

Ü ã Í�Ü ï -�Þ êí � and define Æ Æ ÝßÞ Æ È � ´ { Í�Ü Æ Í Æ ú Êfor

� Þ�É Í ý . Then substract the two corresponding Bishop equations, insert twoinnocuous opposite terms and majorate:®® ®® Æ ï è Æ ã ®® ®® ú � é c ï � @ @ @ V ã @ @ @ ú � é c ï � ®® ®® Ú Æ ï ÈtÑßÊ Í Ñ Í�Ü ï è�Ú Æ ï ÈtÑßÊ Í Ñ Í�Ü ã ®® ®® ú � é c ï ûû ®® ®® Úi Æ ï ÈtÑßÊ Í Ñ Í�Ü ã è0Úi Æ ã ÈtÑßÊ Í Ñ Í�Ü ã ®® ®® ú � é c ï � êTo majorate the difference in the second line, we again apply Lemma 3.13. Tomajorate the difference in the first line, we apply:

Lemma 3.17. ([ � ]) Let Ù with� ��Ù ��I

, let Æ -7D ã � ú ÈsÏ©ÿ ÍRQ z Ê with @ @ Æ#@ @ ú � ú �¸ ã and let two parameters

Ü ã Í�Ü ï -CÞ êí � . Then

®® ®® Ú Æ ÈtÑßÊ Í Ñ Í�Ü ï è�Ú Æ ÈtÑßÊ Í Ñ Í�Ü ã ®® ®® ú � Ø � ®® Ü ï è Ü ã ®® å @ @ Ú @ @ ã � ú û @ @ Ú @ @ ã � Ø � É û n @ @ Æ#@ @ ªã � ú p Ø�� é ê


With Ù�ÝßÞ é ï , we thus obtain:®® ®® Æ ï è Æ ã ®® ®® ú � é c ï � | ãI � ®® Ü ï è Ü ã ®® å @ @ Ú @ @ ã � ú û @ @ Ú @ @ ã � é c ï � É û n ®® ®® Æ ï ®® ®® ã � ú p % ¥ � é ûû �,¬ jó Ý ó ÷ í � ®® ®® Ú ó Ý ®® ®® ã � é c ï ý � É û n ®® ®® Æ ã ®® ®® ªã � ú p % ¥ û n ®® ®® Æ ï ®® ®® ªã � ú p % ¥ � ô ®® ®® Æ ï è Æ ã ®® ®® ú � é c ï êThen we apply the majoration of Lemma 3.10 to ®® ®® Ú ó Ý ®® ®® ã � é c ï and we use ¸ ã �|É to

majorate by É the terms ®® ®® Æ Æ ®® ®® ªã � ú � ¸ ã g ¼ :®® ®® Æ ï è Æ ã ®® ®® ú � é c ï � | ã I û ã ®® Ü ï è Ü ã ®® . @ @ Ú @ @ ã � é c ï û |�| ï ®® ®® Æ ï è Æ ã ®® ®® ú � é c ï êSetting ?�ÝßÞP| ã I û ã . @ @ Ú @ @ ã � é c ï , requiring | � ãï ½ ¥ and reorganizing we obtain

that Æ È � ´ { Í�Ü�Í Æ ú Ê is Lipschitzian with respect to

Ü:®® ®® Æ ï è Æ ã ®® ®® ú � ú � ®® ®® Æ ï è Æ ã ®® ®® ú � é c ï � ý@? ®® Ü ï è Ü ã ®® ê

The proof that Æ È � ´ { Í�Ü�Í Æ ú Ê is Lipschitzian with respect to Æ ú is similar and evensimpler. Þ

In summary, the solution Æ Þ Æ È � ´ { Í�Ü�Í Æ ú Ê is D ã � é c ï with respect to � ´ { andLipschitzian with respect to all the variables � ´ { Í�Ü�Í Æ ú .

Consequently, according to a theorem due to Rademacher ([Ra1919, Fe1969]),the partial derivatives ÆMÝBA , � ÞíÉ Í êÇêÇê ÍDC and Æ*EGF/ , H ÞíÉ Í êÇêÇê Í � exist in 9 Ù . Wethen have to differentiate the Bishop-type equation of Theorem 3.7 with respectto Ì , to

Ü � and to Æ Áú . However, the linear operator

U ã is not bounded in 9 Ù ; infact, according to (2.27), @ @ @ U @ @ @ á�B £ <

as< Lõ> . So we need more information.

Lemma 3.18. There exists a null-measure subset I ! Þ êí � Ã Þ zß � c ãbù and thereexists a quantity ? " �

such that at every È Ü�Í Æ ú Ê"J-KI , for every � ÞíÉ Í êÇêÇê ÍDC andfor every H ÞíÉ Í êÇêÇê Í � :

(i) the partial derivatives Æ§Ý A È � ´ { Í�Ü�Í Æ ú Ê and Æ E F/ È � ´ { Í�Ü�Í Æ ú Ê exist for every� ´ { -�ÏPÿ ;

(ii) the maps � ´ { »L ÆÂÝ A È � ´ { Í�Ü�Í Æ ú Ê and � ´ { »L Æ E F/ È � ´ { Í�Ü�Í Æ ú Ê are D ú � é c ïon Ï©ÿ and satisfy the uniform inequality@ @ÇÆ ÝBA ÈtÑ Í�Ü�Í ÆYú Ê @ @ / � %'& ¥ ·(� } º � ? L-MN ®® ®® Æ EGF/ ÈtÑ Í�Ü�Í ÆYú Ê ®® ®® / � %'& ¥ ·(� } º � ? ê

Proof. Since Æ is almost everywhere differentiable with respect to all its argu-ments, there exist a subset � ! Þ êí � Ã Þ zß � c ãbù Ã Ï©ÿ having full measure, namely

its complement has null measure, such that for every È � ´ { Í�Ü�Í ÆÂú Ê -4� , all partial


derivatives Æ { Í ÆëÝ A Í Æ*E F/ exist at È � ´ { Í�Ü�Í Æ ú Ê . Since � has full measure, there ex-ists a null measure subset I ! Þ êí � Ã Þ zß � c ãbù such that for every È Ü�Í ÆMú ÊOJ-KI , theslice � Ý � E / ÝßÞ Ï©ÿ Ã 4 Ü 5 Ã 4 Æ ú 5 ¦P�is a subset of Ï©ÿ having full measure, so that Æ { Í ÆëÝ A Í Æ*E F/ exist at È � ´ { Í�Ü�Í Æ ú Êwith � ´ { -Q� Ý � E / .

Fix È Ü�Í Æ ú ÊRJ-SI . We will treat only the partial derivatives with respect to theÜ � , arguments being similar for the Æ=E F/ . In the end of the proof of Lemma 3.17,we have shown: ®® ®® Æ ï è Æ ã ®® ®® ú � é c ï � ? ®® Ü ï è Ü ã ®® Ífor some (not the same) quantity ? " �

. Fix �N- 4 É Í ý Í êÇêÇê ÍDC 5 , take

Ü ïand

Ü ãwith

Ü ï� JÞ Ü ã� but

Ü ïä Þ Ü ãäfor TRJÞ � . The inequality above says that for every� ´ { Í � ´ { Í � ´ { -wÏPÿ with� � @ Ì�� 8è&Ì>� @ � � , we have®®®® Æ È � ´

{ Í�Ü ï Í Æ ú ÊÎè Æ È � ´ { Í�Ü ã Í Æ ú ÊÜ ï� è Ü ã� ®®®® ûû ®®®®® Æ È �´ { Í�Ü ï Í ÆYú ÊÑè Æ È � ´ { Í�Ü ã Í ÆYú ÊÜ ï� è Ü ã� è

è Æ È � ´ { Í�Ü ï Í Æ ú ÊÒè Æ È � ´ { Í�Ü ã Í Æ ú ÊÜ ï� è Ü ã� ®®®®® U @ Ì � � è&Ì � @ é c ï � ? êAssume � ´ { Í � ´ {� Í � ´ {� -V� Ý � � E / , let

Ü ï� L Ü ã� (the limits of the quotients aboveexist) and rename

Ü ãby

Ü:®® ÆëÝ A È � ´ { Í�Ü�Í Æ ú Ê ®® û ®® ÆÂÝ A È � ´ {� Í�Ü�Í Æ ú ÊÒè ÆÂÝ A È � ´ {� Í�Ü�Í Æ ú Ê ®®@ Ì � � è&Ì � @ é c ï � ? ê

This inequality says that Æ ÝBA ÈtÑ Í�Ü�Í ÆYú Ê is D ú � é c ï almost everywhere on ÏPÿ . Thenext extension lemma concludes the proof. ÞLemma 3.19. Let Óü É , let �A- Q ¡

, let H üMÉ , let �N- Q Á , let ¸ " �, letì " �

, and let Ë/Þ ËÎÈd� Í ��Ê be a function defined (only) in a full-measure subset� ! Þ ¡ß Ã Þ Á í , so that there exists a null-measure subset I ! Þ Á í with theproperty that for every �4J-3I , the slice ��WòÝßÞ�tÞ ¡ß Ã 4 � 5 Ã � has full measurein Þ ¡ß . Assume that for every �XJ-QI , every � Í � � Í � � � -K� W , we have@ ËÎÈd� Í ��Ê @ û ®® ËÎÈd�-� � Í ��ÊÒè/ËÎÈd�Y� Í ��Ê ®®@ � � � è7� � @ Ø � ? Ífor some Ù with

� �ÉÙ[Z�É and some quantity ? " �. Then for every �\J-\I ,

the function �K»L ËÎÈd� Í ��Ê admits a unique continuous prolongation to Þ ¡ß , still


denoted by ËÎÈtÑ Í ��Ê , that is D ú � Ø in Þ ¡ß with@ @ ËÎÈtÑ Í ��Ê @ @ / � � ·�]_^� º � ? êThus, for every È Ü�Í Æëú Ê`J-�I , the partial derivatives Æ Ý�A , Æ EGF/ belong toD ú � é c ï ÈsÏ©ÿ ÍRQ z Ê . Since the operator

U ã is linear and bounded in D ú � é c ï , we maydifferentiate the � scalar Bishop-type equations of Theorem 3.7 with respect to Ì ,to

Ü � , � ÞíÉ Í êÇêÇê ÍDC and to Æ Áú , H ÞíÉ Í êÇêÇê Í � , which yields, for� ÞíÉ Í êÇêÇê Í � :

(3.20)aÁÁÁÁÁÁÁÁÁÁÁÁÁb ÁÁÁÁÁÁÁÁÁÁÁÁÁc

Æ Æ{ È � ´ { ÊÎÞíè U ãPde öã Ä�äÄ z Ú Æ �~å È Æ ÈtÑßÊ Í Ñ Í�Ü Ê Æ ä{ ÈtÑßÊ û Ú Æ { È Æ ÈtÑßÊ Í Ñ Í�Ü Êgfh È � ´ { Ê ÍÆ ÆÝ A È � ´ { ÊÎÞíè U ã de öã Ä�äÄ z Ú Æ � å È Æ ÈtÑßÊ Í Ñ Í�Ü Ê Æ äÝ A ÈtÑßÊ û Ú Æ Ý A È Æ ÈtÑßÊ Í Ñ Í�Ü Ê fh È � ´ { Ê ÍÆ ÆE F/ È � ´ { ÊÎÞ�6 ÆÁ è U ã de öã Ä�ä.Ä z Ú Æ �~å È Æ ÈtÑßÊ Í Ñ Í�Ü Ê Æ äEGF/ ÈtÑßÊ fh4È � ´ { Ê Í

for every � ´ { - ÏPÿ , provided È Ü�Í Æ ú ÊVJ-`I . In the first line, we noticed thatÈ U ã È Ê { Þ U È È { Ê . We observe that as Æ is Lipschitzian, as Úö-�DYè � é and asFÊü É , the composite functions Ú Æ � å Í Ú Æ { Í Ú Æ Ý s Æ È � ´ { Í�Ü�Í Æ ú Ê Í � ´ { Í�Ü are of classD ú � é with respect to all variables.We notice that in each of the three linear systems of Bishop-type equa-

tions (3.20) above, �üÝßÞ§È Ü�Í Æ ú Ê is a parameter and there appears the same matrixcoefficients: i Æ ä È � ´ { Í �%ÊoÝßÞ�Ú Æ � å n Æ È � ´ { Í�Ü�Í Æ ú Ê Í � ´ { Í�Ü p Ífor

� Í TcÞ�É Í êÇêÇê Í � . For any Ù with� �ÛÙ �ÛI

, in order to be coherent with thedefinition of @ @ Ú � @ Ý @ @ ªú � Ø given after Lemma 3.9, we set:@ @ i @ � @ @ ªú � Ø ÝßÞj�#l{Ëã Ä°ÆÇÄ z áâ öã Ä�ä.Ä z ®® ®® i Æ ä @ � ®® ®® ªú � Ø æç êWe also set: @ @ i @ @ ú � ú ÝßÞj�Õl{Ëã Ä°ÆÇÄ z áâ öã Ä�äÄ z ®® ®® i Æ ä ®® ®® ú � ú æç Þ @ @ Ú � @ @ ú � ú êWith these definitions, it is easy to check the inequality:@ @ i @ � @ @ ú � Ø � @ @ Ú � @ Ýh@ @ ú � Ø � É û n @ @ÇÆ�@ @ ã � ú p Ø�� êAs @ @ Æ#@ @ ã � ú � ¸ ã g ¼ �|É , with Ù/ÝßÞ I

, we deduce:@ @ i @ � @ @ ú � é � ý @ @ Ú � @ Ý{@ @ ú � é � ý @ @ Ú @ Ý{@ @ ã � é ê


Taking ��¬]j Ý and then �,¬ j E / , adding É , squaring and inverting:

� É û �,¬]jÝ @ @ Ú @ Ý~@ @ ã � é � û ï � ¼ ò É û �,¬]jÝ � E / @ @ i @ � @ @ ú � é ô û ï êConsequently, the main assumption of the next Proposition 3.21, according towhich: @ @ i @ @ ú � ú � | ï I ï � É û �,¬ j� @ @ i @ � @ @ ú � é � û ï Ífor some positive absolute constant |=� É , is superseded by one of the mainassumptions, made in the theorem, according to which:@ @ Ú � @ @ ú � ú � | ï I ï � É û �,¬]jÝ @ @ Ú @ Ý~@ @ ú � é � û ï Ífor some (a priori distinct) positive absolute constant |É�ÕÉ .

The following proposition applies to the three systems (3.20) and suffices toconclude the proof of Theorem 3.7 in the case F�Þ É . The case FNüÉý shall bediscussed afterwards.

Proposition 3.21. ([Tu1996], [ � ]) Let k - G with k ü É , letl ã Þ È l ã � ã Í êÇêÇê Í l ã � m Êy- Q m with� � l ã � ´ � > , m Þ É Í êÇêÇê Í k , and denote

by Þ mn � the polycube 4 �Ê- Q m Ý @ � ´ @ � l ã � ´ 5 . Consider vector-valued andmatrix-valued Holder data:¬ Þ n ¬ Æ È � ´ { Í �'Ê p ã Ä°ÆgÄ z -ÎD ú � é n ÏPÿ Ã Þ mn � ÍRQ z p Íi Þ n i Æ ä È � ´ { Í �%Ê p ã Ä°ÆÇÄ zã Ä�äÄ z -ÎD ú � é n Ï©ÿ Ã Þ mn � ÍRQ z0o�z p Íwith

� � I �ÕÉ . Suppose that a bounded measurable map:ø Þ n ø Æ È � ´ { Í �'Ê p ã Ä°ÆgÄ z -P9 Ù ÈsÏPÿ Ã Þ mn � ÍRQ z Ê Íis D ú � é c ï on Ï©ÿ for every fixed � not belonging to some null-measure subset Iof Þ mn � and suppose that it satisfies the system of linear Bishop-type equations inD ú � é c ï ÈsÏ©ÿ ÍRQ z Ê :(3.22) ø Æ Þ U ª áâ öã Ä�äÄ z i Æ ä ø ä æç û ¬ Æ Ífor

� Þ É Í êÇêÇê Í � , where

U ª Þ Uor

U ª Þ U ã . Assume that the norm of the matrixisatisfies: @ @ i @ @ ú � ú Þj�#l{Ëã Ä°ÆgÄ z áâ öã Ä�ä.Ä z ®® ®® i Æ ä ®® ®® ú � ú æç �Nî ¤ Í


for some small positive constantî ¤ �|É~g¢Éð÷ such that

(3.23)î ¤ � | ï I ï ò É û �,¬ jó � ó ÷ n � @ @ i @ � @ @ ú � é ô û ï Í

where |É�ÕÉ is a positive absolute constant. Then, after a correction of ø on I :

(i) on its full domain of definition Ï©ÿ Ã Þ mn � , the corrected map øÑÈ � ´ { Í �%Ê isD ú � é�û ú Þ ø ú ÷ Ø ÷ é D ú � Ø and furthermore:

(ii) for every fixed � , the map � ´ { »L~øÒÈ � ´ { Í �%Ê is D ú � é on ÏPÿ .

In general, the Hilbert transform

Udoes not preserve D ú � é smoothness with

respect to parameters, so that the above solution ø�Þ øÒÈ � ´ { Í �'Ê is not better thanD ú � é�û ú .Example 3.24. ([Tu1996]) If

Ü - Qwith @ Ü @ �ÕÉ is a parameter, the function:øÑÈ � ´ { Í�Ü Ê�ÝßÞ @ Ü @ é )qp è � � Ì � è @ Ü @ ã c ï ÍÝßÞ°Ì ï é )qp è @ Ü @ ã c ï � Ì �N� ÍÝßÞ°Ì é )qp �#� Ì � @ Ü @ ÍÝßÞ @ Ü @ é )qp @ Ü @ � Ì � � Í

is ý � -periodic with respect to Ì and D ú � é with respect to È � ´ { Í�Ü Ê . As the function��fe�È��Rg�ý�ÊÛè}ý�gh� is D ú � ú on /¹è � Í �21 , the regularity properties of the singular integralU øÑÈ � ´ { ÊJÞyj ê\k¡ê ã² % ²û ² � · r s �\usr � � º�� [ · �dc ï º �� are the same as those of:tU øÒÈ � ´ { ÊcÝßÞ�j ê\k©ê É� � ²û ² øÑÈ � ´ · { û � º Ê� �� êHowever

tU øÑÈ³ÉeÊ involves the term @ Ü @ é ' �f¦ @ Ü @ which is D ú � é�û ú but not D ú � é :tU øÑÈ³ÉeÊJÞ É� õ � û ó Ý óû ó Ý ó �u& ¥ @ Ü @ é� �� û � úû ó Ý ó È³è¯�'Êté� �� û � ó Ý ó �u& ¥ú � ï é� �� þÞ Éý � å @ Ü @ é ' �f¦ @ Ü @ è @ Ü @ éI é ê

Proof of the proposition. We shall drop the indices, writing ø ,i

and ¬ , withoutarguments. Assume that �XJ-vI . For future majorations, it is necessary to haveì ú ø�Þ �

. If this is not the case, we set øw�!ÝßÞ øwè{ì ú ø in order that ì ú ø�ÎÞ �.

Since ø satisfies either ø3Þ U È i øÛÊ û ¬ or ø Þ U È i ø¡ÊÎè U / i ø 1 È³ÉeÊ û ¬ , it followsthat ø� satisfies similar equations: either øw� Þ U È i ø��YÊ û ¬ � , with ¬ �©ÝßÞ ¬ è�ì ú ø orø�ÑÞ U È i ø��YÊ û ¬ � , with ¬ �ÎÝßÞ ¬ èôì ú øØè U È i ø¡Ê_È³ÉeÊ . Notice that

iis unchanged.

It then suffices to establish the improvements of smoothness (i) and (ii) for ø�� .Equivalently, we may assume that óø ú Þíì ú ø Þ �

in the proposition.


For �xJ-:I , the function Ö is D ú � é c ï on Ï©ÿ . Applying

Ueither to the equationø�Þ U È i ø¡Ê û ¬ or to the equation ø�Þ U È i ø¡Ê!è U / i ø 1 È³ÉeÊ û ¬ , we get the same

equation for both:U ø Þ è i ø û ì ú È i øÛÊ û U ¬ ê

As ó ø ú Þ �, we may write øÒÈ � ´ { Ê Þzy � ÷ ú óø � � ´ � { û y � ý ú óø � � ´ � { Þ Ý Ö û Ö ,

where Ö extends holomorphically to ÿ . In fact, Ö is determined up to a imaginaryconstant m ½ and we choose ½IÝßÞ è�ì ú È i øÛÊRg�ý , so that:

(3.25) Ö Þ ø û m U øBè m ì ú È i ø¡Êý êEquivalently: { øØÞÕÖ û Ö ÍU ø Þíì ú È i øÛÊÒè m È ÖBè Ö©Ê êSubstituting, we rewrite (3.25) as:è m È ÖBè ÖÛÊÎÞíè i È Ö û Ö©Ê û U ¬ Íor under the more convenient form:Ö Þ Ö û � Ö û\| Íwhere the � Ã � -matrix � ÝßÞ~è ý m i È~} û m i Ê û ã and the � -vector

| ÝßÞ m È~} ûm i Ê�û ã U ¬ both belong to D ú � é .First of all, we establish (ii) before any correction of ø .

Lemma 3.26. For �OJ-XI , the map � ´ { »L�øÑÈ � ´ { Í �%Ê is D ú � é on Ï©ÿ .

Proof. By assumption, the map � ´ { »L ÖÑÈ � ´ { Í �'Ê is D ú � é c ï on ÏPÿ . Since1 �

isbounded in D ú � é c ï , we may apply

1 �to the vectorial equation Ö Þ Ö û � Ö ûQ| Ö ,

noticing that1 � È Ö¡Ê�Þ Ö and that

1 � È Ö¡ÊwÞ ì ú È ÖÛÊ , since, by construction, Öextends holomorphically to ÿ . We thus get:

(3.27) Ö Þíì ú Ö û 1 � 5 � Ö û\| 7 êRemind that ì ú_� Þ ãï ² % ²û ² � È � ´ { Ê��Ì , so that @ @ ì ú_� @ @ ú � ú � @ @ � @ @ ú � ú . Notice8 that@ @ ÖDÈtÑ Í �%Ê @ @ ú � ú �y> for �OJ-QI . Taking the D ú � é norm to (3.27) and applying crucially

Theorem 2.18 in its

Q z -valued version, as in [Tu1996], we get:@ @ Ö @ @ ú � é � ®® ®® ì ú Ö ®® ®® ú � ú û ®® ®® 1 � 5 � Ö û\| 7 ®® ®® ú � é� @ @ Ö @ @ ú � ú û | ãI È³Éúè I Ê n @ @ � @ @ ú � é @ @ Ö @ @ ú � ú û @ @ | @ @ ú � é p�y> Í8In Lemma 3.15 above, for ��Q� , the map ��s��G�u��s�'�g�g� was shown to be ��D� �Y�B�

on �� in order to insure that ��G�g��g� and �G� �(�g�g� are both bounded on �� , so that Theo-rem 2.24 may be applied in the next phrase. In [Tu1996], �G�g��g� is only shown to be in¡�¢ �£�� , but then �¤�_� �(�g�g� and � are not necessarily bounded.


whence Ö ÞÕÖÑÈtÑ Í �'Ê is D ú � é on Ï©ÿ , as claimed. ÞNext, we shall establish (i) before any correction of ø . To this aim, let � ã Í � ï J-I and simply denote by ø ã Í ø ï , by Ö ã Í Ö ï , by � ã Í � ï and by

| ã Í | ïthe functions

on Ï©ÿ with these two values of � . In the theorem, to establish that ø is D ú � é�û ú , itis natural to show that ø is D ú � Ø for every Ù0� I

arbitrarily close toI

.

Lemma 3.28. For every Ù with é ï ��Ù:� I, every two � ã Í � ï J-[I , we have®® ®® ø ï è3ø ã ®® ®® ú � ú � ¼ ? é¢� Ø ®® � ï è&� ã ®® Ø , for some positive quantity ? é¢� Ø �y> .

We shall obtain ? é¢� Ø involving a nonremovable factor É~g8È I èiÙÑÊ . This willconfirm the optimality of D ú � é�û ú -smoothness of ø with respect to the parameter � .Proof. Since ì ú øØÞ ì¥-È U øÛÊoÞ �

, applying ì ú to the conjugate of (3.25), we getì ú Ö Þ ´ï ì ú È i ø¡Ê , so that we may rewrite (3.27) under the form:Ö Þ mý ì ú È i øÛÊ û 1 � 5 � Ö û\| 7 êWe may then organize the difference Ö ï è�Ö ã as follows:Ö ï è�Ö ã Þ mý ì ú i ï È�ø ï è3ø ã Ê û mý ì ú %È i ï è i ã Êqø ã ûû 1 � n È � ï è � ã Ê Ö ï û\| ï è | ã p û 1 � n � ã È Ö ï è Ö ã Ê pÞ Ý�¦ ã û ¦ ï û ¦ ± û ¦ ¤ êTo majorate these four ¦ ´ ’s, we shall need various preliminaries.

Firstly, to majorate ¦ ã , we first observe the elementary inequality:

(3.29) ®® ®® ø ï è3ø ã ®® ®® ú � ú � ý ®® ®® Ö ï è�Ö ã ®® ®® ú � ú êAlso, we notice that:®® ®® i ï ®® ®® ú � ú Þ ®® ®® i ÈtÑ Í � ï Ê ®® ®® 0/ � / ·�� } º � @ @ i @ @ ú � ú �Jî ¤ êThe majoration of ¦ ã is then easy:@ @ ¦ ã @ @ ú � ú � Éý ®® ®® i ï ®® ®® ú � ú ®® ®® ø ï è ø ã ®® ®® ú � ú � ¼ î ¤ ®® ®® Ö ï è�Ö ã ®® ®® ú � ú ê

Secondly, to majorate ¦ ï , let again Ù with é ï ��Ù�� I. Using the inequality@ @ i @ @ ú � Ø � . @ @ i @ @ ú � é proved in the beginning of Section 1, we may majorate ¦ ï :@ @ ¦ ï @ @ ú � ú ��Éý ®® ®® ø ã ®® ®® ú � ú ®® ®® i ï è i ã ®® ®® ú � ú �`.ý ®® ®® ø ã ®® ®® ú � ú @ @ i @ @ ú � é ®® � ï è(� ã ®® Ø ê

Thirdly, to majorate ¦ ± , we need:


Lemma 3.30. Let ËØÞ{ËÎÈd� Í ��Ê be a D ú � é function, defined in the product open cubeÞ ¡ß Ã Þ Áß , where� � I �IÉ and ¸ " �

. Let Ù with� �yÙA� I

. For �� Í �Y� � -(Þ ¡ßarbitrary, the D ú � é�û Ø -norm of the function Þ Áß ¹(�Õ» è L ËÎÈd� � � Í ��ÊoèçËÎÈd� � Í ��Êñ- Qsatisfies: ®® ®® ËÎÈd� � � Í ÑßÊÒè/ËÎÈd� � Í ÑßÊ ®® ®® ú � é�û Ø � ¼ @ @ Ë @ @ ú � é ®® � � � è7� � ®® Ø êAgain, assume é ï �iÙ0� I

. Thanks to Theorem 2.18 and to the lemma above, wemay majorate ¦ ± :@ @ ¦ ± @ @ ú � ú � @ @ ¦ ± @ @ ú � é�û Ø � | ãI è(Ù n ®® ®® � ï è � ã ®® ®® ú � é�û Ø ®® ®® Ö ï ®® ®® ú � ú û ®® ®® | ï è | ã ®® ®® ú � é�û Ø p� | ïI è(Ù n @ @ � @ @ ú � é ®® ®® Ö ï ®® ®® ú � ú û @ @ | @ @ ú � é p ®® � ï è&� ã ®® Ø ê

Fourthly, to majorate ¦ ¤ , we need a statement whose proof is similar to that ofLemma 3.10.

Lemma 3.31. As @ @ i @ @ ú � ú �Nî ¤ , then independently of ��J-KI , we have:@ @ i ó � @ @ ú � é c ï �Nî ¤ û î ã c ï¤ � ý û �,¬ j� @ @ i @ � @ @ ú � é � êReminding the main assumption

î ¤ � | ï I ï � É û �,¬ j � @ @ i @ � @ @ ú � é � û ï , whenceî ¤ �| I , we deduce: @ @ i ó � @ @ ú � é c ï � .M| I ê

Choosing then |É�ÕÉ~g�÷ , we may insure that @ @ i ó � @ @ ú � é c ï � É~g�ý for every fixed � . Itfollows in particular that we may develope � Þíè ý m i y �0§ È³è m i Ê � and deducethe norm inequality:®® ®® � ó � ®® ®® ú � é c ï � ý @ @ i ó � @ @ ú � é c ïÉüè @ @ i ó � @ @ ú � é c ï � ¼ @ @ i ó � @ @ ú � é c ï Ívalid for every fixed � . Then again thanks to Theorem 2.18 and thanks to theprevious inequalities:@ @ ¦ ¤ @ @ ú � ú � @ @ ¦ ¤ @ @ ú � é c ï � | ã I û ã ®® ®® � ã ®® ®® ú � é c ï ®® ®® Ö ï è�Ö ã ®® ®® ú � ú� | ïaI û ã @ @ i @ � � @ @ ú � é c ï ®® ®® Ö ï è/Ö ã ®® ®® ú � ú� | ± | ®® ®® Ö ï è�Ö ã ®® ®® ú � ú� ¼ û ã ®® ®® Ö ï è�Ö ã ®® ®® ú � ú Íprovided | � ã¤ ½<; . We then insert these four majorations in the inequality®® ®® Ö ï è�Ö ã ®® ®® ú � ú � @ @ ¦ ã @ @ ú � ú û @ @ ¦ ï @ @ ú � ú û @ @ ¦ ± @ @ ú � ú û @ @ ¦ ¤ @ @ ú � ú Íand we get after reorganization:®® ®® Ö ï è�Ö ã ®® ®® ú � ú *Éðè ¼ î ¤ è ¼ û ã � ? é¢� Ø ®® � ï è(� ã ®® Ø Í


for some quantity ? é¢� Ø ��> whose precise expression is:¨ é¢� Ø ÝßÞ | ïI è(Ù n @ @ � @ @ ú � é ®® ®® Ö ï ®® ®® ú � ú û @ @ | @ @ ú � é p û . ý ®® ®® ø ã ®® ®® ú � ú @ @ i @ @ ú � é êIt suffices then to remind that

î ¤ � É~g¢Éð÷ in the assumptions of the theorem toinsure the uniform Holder condition:®® ®® Ö ï è�Ö ã ®® ®® ú � ú � ý@? é¢� Ø ®® � ï è(� ã ®® Ø Ívalid for � ã Í � ï J- I . From (3.29), we conclude that ®® ®® ø ï è ø ã ®® ®® ú � ú �¼ ? é¢� Ø ®® � ï è(� ã ®® Ø . The proof of Lemma 3.28 is complete. Þ

Then the correction of ø is provided by the following statement.

Lemma 3.32. ([ � ]) Let Ë Þ ËÎÈd� Í ��ÊðÝ Þ ¡ß Ã È`Þ Áß 3 IBÊ L Qbe a measurable 9 Ù

map defined only for � not belonging to some null-measure subset I ! Þ Áß andlet Ù with

� �NÙ0� I. If the map �¼»LÐËÎÈd� Í ��Ê is D ú � Ø for every �XJ-KI and if there

exists ?A�y> such that:��¬]j© ®® ËÎÈd� Í � ï ÊÑè/ËÎÈd� Í � ã Ê ®® � ? ®® � ï è�� ã ®® Ø Ífor every two � ã Í � ï J-KI , then Ë may be extended as a D ú � Ø map in the full domainÞ ¡ß Ã Þ Áß .

In sum, still denoting by ø the D ú � é�û ú extension of ø through I , property (i) ofthe proposition is proved. To obtain that ø is D ú � é with respect to � ´ { , we re-applythe reasoning of Lemma 3.26 to this extension.

The proof of Proposition 3.21 is complete. ÞIn conclusion, the functions Æ Æ{ , Æ ÆÝ A and Æ ÆEGF/ are D ú � é�û ú with respect toÈ � ´ { Í�Ü�Í Æ ú Ê and D é¢� ú with respect to � ´ { . Thus, the theorem is achieved if FàÞíÉ .If FëÞ ý , the composite functions Ú Æ � å Í Ú Æ { Í Ú Æ Ý s Æ È � ´ { Í�Ü�Í Æ ú Ê Í � ´ { Í�Ü are then

of class D ã � é�û ú with respect to È � ´ { Í�Ü�Í Æ ú Ê and of class D ã � é with respect to � ´ { .We then apply the next lemma to the three families of Bishop-type vector equa-tions (3.20).

Lemma 3.33. Let ��- Þ"mn � be a parameter with kà- G ,� � l ã � ´ �> , m Þ É Í êÇêÇê Í k , and consider vector-valued and matrix-valued Holder data¬ Þ ¬ Æ È � ´ { Í �%Ê ã Ä°ÆgÄ z and

i Þ n i Æ ä È � ´ { Í �'Ê p ã Ä°ÆgÄ zã Ä�ä.Ä z that are D ã � é�û ú with re-

spect to È � ´ { Í �%Ê and D ã � é with respect to � ´ { . Suppose that a given map ø Þ ø Æ È � ´ { Í �%Ê ã Ä°ÆgÄ z which is D ú � é�û ú with respect to È � ´ { Í �%Ê and D ú � é with respect to� ´ { satisfies the linear Bishop-type equation øIÞ U ª È i øÛÊ û ¬ , where

U ª Þ Uor

U ª Þ U ã . Assume that the norm of the matrixi

satisfies the same in-equality as in the proposition: @ @ i @ @ ú � ú �àî ¤ , for some small positive constant

HOLOMORPHIC EXTENSIONS AND REMOVABLE SINGULARITIES 145î ¤ � | ï I ï � É û �,¬]j � @ @ i @ � @ @ ú � é � û ï , where� �Û|º� É is some absolute constant.

Then ø is D ã � é with respect to � ´ { and satisfies the Lipschitz condition®® ®® øÑÈtÑ Í � ï ÊÑè3øÒÈtÑ Í � ã Ê ®® ®® ú � é c ï � ? ®® � ï è(� ã ®® Ífor some quantity ?��> . Furthermore, there exists a null-measure subset I !Þ mn � Ã Þ zß � c ãbù such that at every �OJ-XI , for every TzÞíÉ Í êÇêÇê Í k :(i) the partial derivative ø � å È � ´ { Í �'Ê exists for every � ´ { -ØÏ©ÿ ;

(ii) the map � ´ { »L~ø � å È � ´ { Í �%Ê is D ú � é c ï on ÏPÿ .

Proof. The fact that ø is D ã � é with respect to � ´ { is proved as in Lemma 3.26.For the Lipschitz condition, the reasoning is simpler than Lemma 3.17, due to thelinearity of øëÞ U ª È i ø¡Ê û ¬ . Indeed, for two parameters � ã Í � ï -0Þ mn � , if we takethe D ú � é c ï -norm of the difference:ø ï è3ø ã Þ U ª i ï È�ø ï è3ø ã Ê û U ª %È i ï è i ã Êqø ã û ¬ ï è ¬ ã Íwe get:®® ®® ø ï è3ø ã ®® ®® ú � é c ï � | ã I û ã ®® ®® i ï ®® ®® ú � é c ï ®® ®® ø ï è3ø ã ®® ®® ú � é c ï û ? ®® � ï è(� ã ®® Íand after substraction, taking account of Lemma 3.31:®® ®® ø ï è3ø ã ®® ®® ú � é c ï � ý@? ®® � ï è(� ã ®® êThen the sequel of the reasoning is already known. Þ

So for TÛÞ É Í êÇêÇê Í k , the partial derivatives ø � å exist almost everywhere and theysatisfy: ø � å Þ U ª È i ø � å Ê û ¬ ä Íwith the same matrix

i, where ¬ ä ÝßÞ U ª È i � å øÛÊ û ¬ � å is D ú � é�û ú with respect toÈ � ´ { Í �%Ê and D ú � é with respect to � ´ { .

Lemma 3.34. Proposition 3.21 holds true if more generally, ¬ is only assumedto be D ú � é�û ú with respect to È � ´ { Í �%Ê and D ú � é with respect to � ´ { , with the sameconclusion.

(It suffices only to inspect the majoration of ¦ ± .) Consequently, with ø ÞÆ { Í ÆÂÝ A Í Æ E F/ in the three equations (3.20), we have verified that the partial deriva-tives ø { , ø Ý A and ø E F/ exist everywhere, are D ú � é�û ú with respect to È � ´ { Í�Ü�Í Æ ú Ê andare D ú � é with respect to � ´ { . In summary, the theorem is achieved if FBÞ�ý .

Needless to say, we have clarified how to cover the case of general FNüÉý bypure logical induction. In conclusion, the proof of Theorem 3.7 is complete.

Open problem 3.35. Solve parametrized Bishop-type equations in Sobolevspaces.

146 JOEL MERKER AND EGMONT PORTEN� 4. APPENDIX: PROOFS OF SOME LEMMAS

4.1. Proofs of Lemmas 3.9 and 3.10. Let �C- Q ¡, �ü É , with @ � @ �Û¸ , where� �i¸ � > . Assuming @ @ Ë @ @ ú � ú �Nî

, we estimate:@ @ Ë @ @ ÿ ú � é c ï � �,¬]j© �ª� © @ ËÎÈd�-� �ÊÒè�ËÎÈd�Y�.Ê @@ � � � è�� @ é c ï Þ«�#l{ËBÈ~¬ Í� Ê � ¬ û ÎÍwhere ¬ ÝßÞ°�,¬ j ú ÷ ó © û © ó ÷�® �u&�% and

ÝßÞ°��¬]j ó © û © ó ¯ ® �u&�% satisfy:¬�Þ �,¬]jú ÷ ó © û © ó ÷�® �u&�% å @ ËÎÈd� � � ÊÎè4ËÎÈd� � Ê @@ � � � è7� � @ é ®® � � � è7� � ®® é c ï é � @ @ Ë @ @ ªú � é î ã c ï Í Þ �,¬]jó © û © ó ¯ ® �u&�% @ ËÎÈd� � � ÊÒè/ËÎÈd� � Ê @@ � � � è7� � @ é c ï � ý @ @ Ë @ @ ú � úî ã c ï � ý î ã c ï êLemma 3.9 is proved. Þ

Applying this to ��Þ È�ø Í Ì°Ê , we deduce:@ @ Ú � @ Ýh@ @ ÿ ú � é c ï �Nî ã c ïï � ý û @ @ Ú � @ Ýh@ @ ªú � é�� Nî ã c ïï � ý û @ @ Ú @ Ý~@ @ ã � é � Í@ @ Ú { @ Ýh@ @ ÿ ú � é c ï �Nî ã c ï± � ý û @ @ Ú { @ Ý{@ @ ªú � é � �Jî ã c ï± � ý û @ @ Ú @ Ý~@ @ ã � é � êConsequently:@ @ Ú @ Ý{@ @ ã � é c ï Þ @ @ Ú @ Ý{@ @ ú � ú û @ @ Ú � @ Ýh@ @ ú � ú û @ @ Ú { @ Ýh@ @ ú � ú û @ @ Ú � @ Ý{@ @ ÿ ú � é c ï û @ @ Ú { @ Ýh@ @ ÿú � é c ï�Nî ã û îaï û î ± û n î ã c ïï û î ã c ï± p � ý û @ @ Ú @ @ ã � é � êLemma 3.10 is proved. Þ4.2. Proof of Lemma 3.11. We shall abbreviate ��¬]j ú ÷ ó { û { ó Ä ² by �,¬]j { ~ª� { . Bydefinition:®® ®® Ú Æ ÈtÑßÊ Í Ñ Í�Ü ®® ®® �� ·�� } º ÞÞ°�,¬]j{ ®®® Ú Æ È � ´ { Ê Í � ´ { Í�Ü ®®® ûû ��¬]j{ ®®® öã Ä�äÄ z Ú � å Æ È � ´ { Ê Í � ´ { Í�Ü Æ ä{ È � ´ { Ê û Ú { Æ È � ´ { Ê Í � ´ { Í�Ü ®®® ûû �,¬]j{ ª� { ®®® öã Ä�äÄ z Ú � å Æ È � ´ { Ê Í � ´ { Í�Ü Æ ä{ È � ´ { Ê û Ú { Æ È � ´ { Ê Í � ´ { Í�Ü è

è öã Ä�ä.Ä z Ú �hå Æ È � ´ { Ê Í � ´ { Í�Ü Æ ä{ È � ´ { ÊÒè0Ú { Æ È � ´ { Ê Í � ´ { Í�Ü ®®® ®® Ì � � è&Ì � ®® û Ø


Majorating and inserting some appropriate new terms whose sum is zero:� @ @ Ú @ @ ú � ú û @ @ Ú � @ @ ú � ú @ @ Æ#@ @ ªã � ú û @ @ Ú { @ @ ú � ú ûû �,¬]j{ ª� { ®®® öã Ä�äÄ z Ú � å Æ È � ´ {� Ê Í � ´ {� Í�Ü 5 Æ ä{ È � ´ {� ÊÒè Æ ä{ È � ´ {� Ê 7 ®®® ®® Ì � � è0Ì � ®® û Ø ûû �,¬]j{ ª� { ®®® öã Ä�äÄ z � Ú � å Æ È � ´ {� Ê Í � ´ {� Í�Ü è�Ú � å Æ È � ´ {� Ê Í � ´ {� Í�Ü � Æ ä{ È � ´ {� Ê ®®® ®® Ì � � è&Ì � ®® û Ø ûû �,¬]j{ ª� { ®®® öã Ä�äÄ z � Ú � å Æ È � ´ { Ê Í � ´ { Í�Ü è�Ú � å Æ È � ´ { Ê Í � ´ { Í�Ü � Æ ä{ È � ´ { Ê ®®® ®® Ì � � è&Ì � ®® û Ø ûû �,¬]j{ ª� { ®®® Ú { Æ È � ´ { Ê Í � ´ { Í�Ü èKÚ { Æ È � ´ { Ê Í � ´ { Í�Ü ®®® ®® Ì � � è&Ì � ®® û Ø ûû �,¬]j{ ª� { ®®® Ú { Æ È � ´ { Ê Í � ´ { Í�Ü è�Ú { Æ È � ´ { Ê Í � ´ { Í�Ü ®®® ®® Ì � � è0Ì � ®® û Ø êMajorating:� @ @ Ú @ @ ú � ú û @ @ Ú � @ @ ú � ú @ @ÇÆ�@ @ ªã � ú û @ @ Ú { @ @ ú � ú ûû @ @ Ú � @ @ ú � ú @ @TÆ�@ @ ªã � Ø û @ @ Ú � @ Ýh@ @ ªú � Ø @ @ÇÆ�@ @ ªã � ú û @ @ Ú � @ Ýh@ @ ªú � Ø @ @ Æ�@ @ ªã � ú Ø @ @ÇÆ#@ @ ªã � ú ûû @ @ Ú { @ Ý{@ @ ªú � Ø û @ @ Ú { @ Ý{@ @ ªú � Ø @ @ÇÆ�@ @ ªã � ú Ø Íwhich yields the lemma, noticing that @ @ Ú � @ @ ú � ú @ @ÇÆ#@ @ ªã � ú û @ @TÆ�@ @ ªã � Ø �@ @ Ú � @ @ ú � ú @ @ Æ�@ @ ã � Ø . Þ4.3. Proof of Lemma 3.13. We need two preparatory lemmas.

Lemma 4.4. Let �ü�É , let �(- Q ¡, let H=ü�É , let �(- Q Á , let ¸ " �

and letËÕÞ ËÎÈd� Í ��Ê be a -°D ã � é map, with� � IÓ� É , defined in the strip 4 Èd� Í ��ÊH-Q Á Ã Q ¡ Ý @ � û � @ �i¸ 5 and valued in

Q z , ��üÕÉ . If four vertices Èd�>� Í �Y�YÊ , Èd�-� � Í �-�.Ê ,Èd�-� Í �Y� �YÊ and Èd�-� � Í �-� �.Ê of a rectangle belong to the strip, then:®® ËÎÈd� � � Í � � � ÊÑè�ËÎÈd� � Í � � � ÊÒè/ËÎÈd� � � Í � � Ê û ËÎÈd� � Í � � Ê ®® � @ @ Ë @ @ ªã � é ®® � � � è7� � ®® ®® � � � è�� ®® é êA similar inequality holds by reversing the roles of the variables � and � .

Proof. We apply twice the Taylor integral formula (1.2) with respect to the vari-able � and we majorate:®® ËÎÈd� � � Í � � � ÊÑè/ËÎÈd� � Í � � � Ê è ËÎÈd� � � Í � � ÊÑè/ËÎÈd� � Í � � Ê ®® �� ãú öã Ä ´ Ä ¡ ®®®® ÏPËÏ]� ´ �� ûV° Èd� � � è7� � Ê Í � � � è ÏPËÏ]� ´ �� û±° Èd� � � è7� � Ê Í � � ®®®® ®® � � �´ è7� �´ ®® � °� @ @ Ë @ @ ªã � é ®® � � � è�� ®® é ®® � � � è7� � ®® ê Þ


Lemma 4.5. Let KüíÉ , let ��- Q ¡, let ¸ " �

and let × Þ × È³²%Ê be a D ã � é map,with

� � IN� É , defined in the open cube Þ ¡ß Þ 4 @ ² @ �J¸ 5 and valued in

Q z . Let� , � , ´ with @ � @ Í @ � @ Í @ ´ @ � ¸]g-. , so that the four points � , � , ´ and � û �!èP´ constitutethe vertices of a parallelogram which is contained in Þ ¡ß . Then:@ × Èd� û � èµéÊÑè × Èd��ÊÎè × È��Ê û × È~éÊ @ � ¼ @ @ × @ @ ªã � é @ ��èµ´ @a@ �FèS´ @ é êA similar inequality holds after exchanging � and � .

Proof. To estimate the second difference of × , we introduce a new mapËÎÈd� Í ��ÊcÝßÞ × Èd� û ��Ê Íof Èd� Í ��Ê - Q ¡ Ã Q ¡

, whose domain is the strip 4 @ � û � @ �Û¸ 5 . Let � , � , ´ with@ � @ Í @ � @ Í @ ´ @ � ¸qg-. . Fixing � � - Q ¡arbitrary, there exist unique � � , � � � and � � �

solving the linear system:{ � � û � � Þ9´ Í � � � û � � � ÞJ� û �Fèµ´ Í� � û � � � Þy� Í � � � û � � Þ�� êIn fact, �>�ÔÞ9óè}�Y� , �Y� �Þy�#è4´ û �-� and �-� � ÞJ�#èH�-� � . Taking the norms @ Ñ @ of thefour equations above, we see that the rectangle Èd� � Í � � Ê , Èd� � � Í � � Ê , Èd� � Í � � � Ê , Èd� � � Í � � � Êis contained in the strip 4 @ � û � @ �y¸ 5 . Applying then Lemma 4.4 (with H¾Þ� ),we get: @ × Èd� û ��èµéÊÑè × Èd��ÊÑè × È��Ê û × È~éÊ @ ÞÞ ®® ËÎÈd� � � Í � � � ÊÑè/ËÎÈd� � Í � � � ÊÎè/ËÎÈd� � � Í � � Ê û ËÎÈd� � Í � � Ê ®®� @ @ Ë @ @ ªã � é ®® � � � è7� � ®® ®® � � � è7� � ®® éÞ @ @ Ë @ @ ªã � é @ ��èµ´ @�@ �Fèµ´ @ é êWe claim that @ @ Ë @ @ ªã � é � ¼ @ @ × @ @ ªã � é , which will conclude. Carefulness and rigor arerequired. In fact, to estimate:@ @ Ë @ @ ªã � é Þ ¡ö ´ � ã ��¬]j· © � W º ª� · © � W º @ Ë © s Èd� � �

Í � � � ÊÒè/Ë © s Èd� � Í � � Ê @ û @ Ë W s Èd� � � Í � � � ÊÎè/Ë W s Èd� � Í � � Ê @@ Èd� � � Í � � � ÊÑè Èd� � Í � � Ê @ é Íwe shall first transform the denominator. By definition:®® Èd� � � Í � � � ÊÎè�Èd� � Í � � Ê ®® Þ9�#l{Ë� ®® � � � è7� � ®® Í ®® � � � è(� � ®® êIf we set � ÝßÞ @ � � � è7� � @ and

C ÝßÞ @ � � � è�� @ and if we invert the inequality È � ûC Ê é � ýx�#l{ËòÈ � é ÍDC é Ê , noticing ý é � ý , we obtain:É@ Èd� � � Í � � � ÊÒè�Èd� � Í � � Ê @ é Þ É�#l{ËòÈ � é ÍDC é Ê � ýÈ � û C Ê é Þ ýÈ @ � � � è7� � @ û @ � � � è�� @ Ê é� ý@ � � � û � � � è7� � è�� @ é ê


Replacing the denominator above, we then transform and majorate the numerator:@ @ Ë @ @ ªã � é � ý ¡ö ´ � ã �,¬]j· © � W º ª� · © � W ºÑå @ ×x¶ s Èd� � � û � � � ÊÑè ×x¶ s Èd� � û � � Ê @ û @ ×x¶ s Èd� � � û � � � ÊÑè ×x¶ s Èd� � û � � Ê @@ � � � û � � � è7� � è�� @ é é� ¼ ¡ö ´ � ã ��¬]j¶ �ª� ¶ å @ ×·¶ s È³² � � ÊÑè ×x¶ s È³² � Ê @@ ² � � è:² � @ é éÞ ¼ @ @ × @ @ ªã � é ê

This completes the proof of Lemma 4.5. ÞWe can now state a slightly simplified version of Lemma 3.13.

Lemma 4.6. ([Tu1990], [ � ]) Let ø�- Q z , ��ü|É , let ¸ ã " �and let ¸IÞ)¸ È�øÛÊ be

a D ã � é map, with� � Ii� É , ø&- Q z , defined in the cube 4 @ ø @ �J¸ ã 5 and valued

in

Q z . Let Æ ã Í Æ ï -(D ã � ú ÈsÏ©ÿ ÍRQ z Ê with ®® Æ Æ È � ´ { Ê ®® ��¸ ã g-. on Ï©ÿ , for� Þ�É Í ý .

For every Ù with� �NÙ �JI

the following inequality holds:®® ®® ¸ È Æ ï ÈtÑßÊ'ÊÎè¹¸ È Æ ã ÈtÑßÊ'Ê ®® ®® 0/ � � ·(� } º �)º ®® ®® Æ ï è Æ ã ®® ®® ú � Ø Íwith º Þ @ @ ¸ @ @ ã � Ø � É û ý n ®® ®® Æ ã ®® ®® ªã � ú p Ø û ý n ®® ®® Æ ï ®® ®® ªã � ú p Ø�� êProof. Firstly and obviously:®® ®® ¸ È Æ ï ÊÒèS¸ È Æ ã Ê ®® ®® ú � ú � @ @ ¸ @ @ ªã � ú ®® ®® Æ ï è Æ ã ®® ®® ú � ú êSecondly, we have ®® ®® ¸ È Æ ï ÊÒè¹¸ È Æ ã Ê ®® ®® ªú � Ø Þ �,¬ j ú ÷ ó { û { ó Ä ² �» g @ Ì � � è&Ì � @ Ø ,where:»�ÝßÞ ®® ¸ Æ ï È � ´ { Ê è¹¸ Æ ã È � ´ { Ê è¹¸ Æ ï È � ´ { Ê û ¸ Æ ã È � ´ { Ê ®® êTo majorate » , we start by inserting the term ¸ 5 Æ ã È � ´ { Ê û Æ ï È � ´ { Ê�è Æ ã È � ´ { Ê 7 ,well-defined, thanks to the assumption ®® ®® Æ Æ ®® ®® ú � ú �i¸ ã g-. :» � ®® ¸ Æ ï È � ´ { Ê è¹¸ 5 Æ ã È � ´ { Ê û Æ ï È � ´ { ÊÑè Æ ã È � ´ { Ê 7 ®® ûû ®® ¸ 5 Æ ã È � ´ { Ê û Æ ï È � ´ { ÊÑè Æ ã È � ´ { Ê 7 èV¸H Æ ã È � ´ { Ê èè¹¸H Æ ï È � ´ { Ê û ¸H Æ ã È � ´ { Ê ®® êTo estimate the second absolute value, we apply Lemma 4.5 with � Þ Æ ã È � ´ { Ê ,with �òÞ Æ ï È � ´ { Ê and with ´úÞ Æ ã È � ´ { Ê :» � @ @ ¸ @ @ ªã � ú ®® / Æ ï è Æ ã 1 È � ´ {� ÊÑèJ/ Æ ï è Æ ã 1 È � ´ {� Ê ®® ûû ¼ @ @ ¸ @ @ ªã � Ø ®® Æ ï È � ´ { ÊÑè Æ ã È � ´ { Ê ®® ®® Æ ã È � ´ { ÊÒè Æ ã È � ´ { Ê ®® Ø ê


We then achieve the remaining majorations:» � @ @ ¸ @ @ ªã � ú ®® ®® Æ ï è Æ ã ®® ®® ªú � Ø ®® Ì � � è&Ì � ®® Ø ûû ¼ @ @ ¸ @ @ ªã � Ø ®® ®® Æ ï è Æ ã ®® ®® ú � ú n ®® ®® Æ ã ®® ®® ªã � ú p Ø ®® Ì � � è0Ì � ®® Ø êReminding that ®® ®® Æ ï è Æ ã ®® ®® ú � ú û ®® ®® Æ ï è Æ ã ®® ®® ªú � Ø Þ ®® ®® Æ ï è Æ ã ®® ®® ú � Ø and summing,we obtain:®® ®® ¸ È Æ ï ÊÒèS¸ È Æ ã Ê ®® ®® ú � Ø � @ @ ¸ @ @ ã � Ø � É û ¼ n ®® ®® Æ ã ®® ®® ªã � ú p Ø�� ®® ®® Æ ï è Æ ã ®® ®® ú � Ø êA similar inequality holds with ®® ®® Æ ï ®® ®® ªã � ú Ø instead of ®® ®® Æ ã ®® ®® ªã � ú Ø . Taking thearithmetic mean, we find the symmetric quantity

ºof the lemma. The proof is

complete. ÞProof of Lemma 3.13. By definition:¼ ÝßÞ ®® ®® Úi Æ ï ÈtÑßÊ Í Ñ Í�Ü è�Úi Æ ã ÈtÑßÊ Í Ñ Í�Ü ®® ®® Y/ � � ·(� } ºÞ7�,¬]j{ ®®® Ú Æ ï È � ´ { Ê Í � ´ { Í�Ü è�Ú Æ ã È � ´ { Ê Í � ´ { Í�Ü ®®® ûû �,¬ j{ �ª� { ®®® Ú Æ ï È � ´ { Ê Í � ´ { Í�Ü è�Ú Æ ã È � ´ { Ê Í � ´ { Í�Ü èè�Ú Æ ï È � ´ { Ê Í � ´ { Í�Ü û Ú Æ ã È � ´ { Ê Í � ´ { Í�Ü ®®® ®® Ì � � è&Ì � ®® û Ø êIn the numerator, we insert èOÚS Æ ï È � ´ { Ê Í � ´ { Í�Ü û Ú� Æ ã È � ´ { Ê Í � ´ { Í�Ü plus itsopposite:¼ � @ @ Ú @ @ ªã � ú ®® ®® Æ ï è Æ ã ®® ®® ú � ú ûû ��¬]j{ ~ª� { ®®® Ú� Æ ï È � ´ { Ê Í � ´ { Í�Ü è�Ú� Æ ã È � ´ { Ê Í � ´ { Í�Ü èè�Ú Æ ï È � ´ { Ê Í � ´ { Í�Ü û Ú Æ ã È � ´ { Ê Í � ´ { Í�Ü ®®® ®® Ì � � è&Ì � ®® û Ø ûû ��¬]j{ ~ª� { ®®® Ú� Æ ï È � ´ { Ê Í � ´ { Í�Ü èKÚ� Æ ã È � ´ { Ê Í � ´ { Í�Ü èè�Ú Æ ï È � ´ { Ê Í � ´ { Í�Ü û Ú Æ ã È � ´ { Ê Í � ´ { Í�Ü ®®® ®® Ì � � è&Ì � ®® û Ø êTo majorate the first �,¬ j { ~ª� { , we apply Lemma 4.4 with �>��Þ Æ ã È � ´ { Ê , �-��Þ� ´ { , � � � Þ Æ ï È � ´ { Ê and � � � Þ � ´ { , where

Üis considered as a dumb parameter.

To majorate the second �,¬ j { �ª� { , we apply Lemma 4.6 to ¸ È�øÛÊüÝßÞ Ú ø Í � ´ { Í�Ü ,


where È � ´ { Í�Ü Ê are considered as dumb parameters. We get:¼ � @ @ Ú @ @ ªã � ú ®® ®® Æ ï è Æ ã ®® ®® ú � ú ûû @ @ Ú @ @ ªã � Ø �,¬]j{ ®®® Æ ï È � ´ { ÊÒè Æ ã È � ´ { Ê ®®® ûû @ @ Ú @ @ ã � Ø � É û ý n ®® ®® Æ ã ®® ®® ªã � ú p Ø û ý n ®® ®® Æ ï ®® ®® ªã � ú p Ø�� ®® ®® Æ ï è Æ ã ®® ®® ú � Ø êTo conclude, we use @ @ Ú @ @ ªã � ú û @ @ Ú @ @ ªã � Ø � @ @ Ú @ @ ã � Ø and we get the term

1of

Lemma 3.13. ÞWith these techniques, the proofs of Lemmas 3.17, 3.19, 3.30, 3.31 and 3.32

are easily guessed and even simpler.


V: Holomorphic extension of CR functions

Table of contents

1. Hartogs theorem, jump formula and domains having the holomorphic extensionproperty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2. Trepreau’s theorem, deformations of Bishop discs and propagation on hyper-surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.3. Tumanov’s theorem, deformations of Bishop discs and propagation on genericsubmanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.4. Holomorphic extension on globally minimal generic submanifolds . . . . . . . . 192.

[19 diagrams]

The method of analytic discs is rooted in the very birth of the theory of func-tions of several complex variables. The discovery by Hurwitz and Hartogs ofthe compulsory extension of holomorphic functions relied upon an application ofCauchy’s integral formula along a family of analytic discs surrounding an illusorysingularity. Since H. Cartan, Thullen, Behnke and Sommer, various versions ofthis argument were coined “Kontinuitattsatz” or “Continuity principle”.

The removal of compact singularities culminated in the so-called Hartogs-Bochner theorem, usually proved by means of integral formulas or thanks to theresolution of a Ï problem with compact support. Contradicting all expectations,a subtle example due to Fornaess (1998), shows that on a non-pseudoconvex do-main, the disc method may fail to fill in the domain, if the discs are required tostay inside the domain.

Nevertheless, it is of the highest prize to build constructive methods in orderto describe significant parts of the envelope of holomorphy of a domain, of aCR manifold, as well as the polynomial hull of certain compact sets. In suchproblems, analytic discs with boundary in the domain, the CR manifold or thecompact set remain the most adequate tools.

The precise existence Theorem 3.7(IV) for the solutions of Bishop’s equationthat was established in the previous Part IV may now be applied systematically to avariety of geometric situations. In this respect, we just followed Bishop’s genuinephilosophy that required to ensure an explicit control of the size of solutions interms of the size of data, instead of appealing to some general, imprecise versionof the implicit function theorem.

Thanks to the jump theorem, holomorphic extension of CR functions definedon a hypersurface ² is equivalent to the extension of the functions that are holo-morphic in one of the two sides to the other side. Trepreau’s original theorem(1986) states that such an extension holds at a point

iif and only if there does

not exist a local complex hypersurface ½ of ¾ ¡ withi -¿½ ! ² . A deeper


phenomenon of propagation (Trepreau, 1990) holds: if CR functions extend holo-morphically to one side at a point

i, a similar extension holds at every point of

the CR orbit ofi

in ² . By means of deformations of attached Bishop discs, thereis an elementary (and folklore) proof that contains both the local and the globalextension theorems on hypersurfaces, yielding a satisfactory understanding of thephenomenon.

On a generic submanifold ² of ¾ ¡ of higher codimension, the celebrated Tu-manov extension theorem (1988) states that CR functions defined on ² extendholomorphically to a local wedge of edge ² at a point

iif the local CR orbit of

icontains a neighborhood of

iin ² . A globalization of this statement, obtained in-

dependently by Joricke and the first author in 1994, states that the same extensionphenomenon holds if ² consists of a single CR orbit, i.e. is globally minimal.Both proofs heavily relied on the local Tumanov theorem and on a precise controlof the propagation of directions of extension.

A clever proof that treats both locally minimal and globally minimal genericsubmanifolds on the same footing constitutes the main Theorem 4.12 of thepresent Part V: if ² is a globally minimal D ï � éàÈ � � I �|ÉeÊ generic submanifoldof ¾ ¡ of codimension ü~É and of CR dimension üMÉ , there exists a wedgelikedomain À attached to ² such that every continuous CR function ËA-�D ú½wÁ ÈÏ²íÊpossesses a holomorphic extension ÂÉ-7× È³ÀíÊ�¦ D ú ÈÏ²zÃ#À Ê with Â @ÅÄ Þ{Ë . Thisbasic statement as well as the techniques underlying its proof will be the very start-ing point of the study of removable singularities in Parts VI and in [MP2006a].� 1. HARTOGS THEOREM, JUMP FORMULA

AND DOMAINS HAVING THE EXTENSION PROPERTY

1.1. Hartogs extension theorem: brief history. 9 In 1897, Hurwitz showed thata function holomorphic in ¾ ï 3�4 ��5 extends holomorphically through the origin.In his thesis (1906), Hartogs generalized this discovery, emphasizing the com-pulsory holomorphic extendability of functions that are defined on the nowadayscelebrated Hartogs skeleton (diagram below). The main argument is to applyCauchy’s integral formula along families of analytic discs having their boundaryinside the domain and whose interior goes outside the domain. In fact, the thin-ness of an embedded circle in ¾ ¡ ( &üJý ) offers much freedom to include illusorysingularities inside a disc.

In 1924, Osgood stated the ultimate generalization of the discovery of Hurwitzand Hartogs: if Æ ! ¾ ¡ È� Aü�ý�Ê is a domain and if

¨ ! Æ is any compact suchthat Æ 3 ¨ connected, then × È�Æ 3 ¨ Ê�ÞÖ× È�Æ#Ê @ÅÇ�ÈsÉ . This statement is nowadayscalled the Hartogs-Bochner theorem. Although the proof of Osgood was correctfor geometrically simple complements Æ 3 ¨ , as for instance spherical shells, itwas incomplete for general Æ 3 ¨ . In fact, unpleasant topological and monodromy

9Further historical information may be found in [Ra1986, Str1988, Fi1991, Ra2002,HM2002].


obstructions occur for general Æ 3 ¨ when pushing analytic discs. In 1998, For-naess exhibited certain domains in which discs are forced to first leave some in-termediate domain Æ 3 ¨ ã , ¨ ã ! ¨ , before Æ may be filled in.

In the late 1930’s, a rigorous proof of Osgood’s general statement was obtainedby Fueter, by means of a generalization of the classical Cauchy-Green-Pompeiuintegral formula to several variables, in the context of complex and quaternionicfunctions (Moisil 1931, Fueter 1935). In 1943, Martinelli simplified the formaltreatment of Fueter. Then Bochner observed that the same result holds more gen-erally if one assumes given on ÏGÆ just a CR function.ó � ¥ ó

ú � r�Ê ãË ÊË ÊË Ê

Ì � Í�

ó � ¥ óúó � � ó

Î�Ï³ÐÒÑ(Ó�ÔwÕÎ Ï ÐÒÑ�ÔwÕ

ÖFilling in the Hartogs skeleton by means of analytic discs

1.2. Hartogs domain. Consider the × -Hartogs skeleton (pot-looking) domain:Ø Ö ÝßÞ 4 È æ ã Í æ ï Ê.-K¾ ï Ý @ æ ã @ �ÕÉ Í @ æ ï @ �Ù× 5xÚ 4 Éðèµ× � @ æ ã @ �ÕÉ Í @ æ ï @ �|É 5 êWe draw two diagrams: in È @ æ ã @ Í @ æ ï @ Ê and in È�¶ ã ÍR· ã Í @ æ ï @ Ê coordinates.

Lemma 1.3. Every holomorphic function Ë�-°× È Ø Ö Ê extends holomorphicallyto the bidisc ÿ Ã ÿ , the convex hull of

Ø Ö .Proof. Letting 6 with

� �J6 �\× , for every æ ï -X¾ with @ æ ï @ �ÕÉ , the analytic discÿº¹wä¼» è2L¾½ � ¥ Èsä°ÊoÝßÞ ÈR/¹Éðè06 1 ä Í æ ï Ê¯-X¾ ïhas its boundary ½ � ¥ ÈsÏ©ÿÚÊ contained inØ Ö , the domain where the function Ë is

defined. Thus, we may compute the Cauchy integralÂÚÈ æ ã Í æ ï ÊoÝßÞ Éý ��m � � } ËÎÈÍ½ � ¥ Èsä°Ê'Êä�è æ ã �¢ä êDifferentiating under the sum, this extension Â is seen to be holomorphic. Inaddition, for @ æ ï @ �«× , it coincides with Ë . Obviously, the discs ½ � ¥ ÈsÿÚÊ fill in thehole of the domain

Ø Ö . Þ


1.4. Bounded domains in ¾ ¡ and Hartogs-Bochner extension phenomenon.Let Æ be a connected open subset of ¾ ¡ , a domain. We assume it to be bounded,i.e. Æ is compact and that its boundary ÏGÆ¶ÝßÞ Æ 3 Æ is a hypersurface of ¾ ¡ ofclass at least D ã . By means of a partition of unity, one can construct a real-valuedfunction Û defined on ¾ ¡ such that Æ Þ 4 æ Ý>ÛÈ æ Ê.� ��5

and ÏGÆ�Þ 4 æ Ý�ÛÈ æ ÊJÞ ��5,

with ��ÛÈ æ Ê"JÞ �for every æ -wÏÆ . Then ÏGÆ is orientable.

Extensions of the above disc argument led to the most general10 form of theHartogs theorem: if Æ is a bounded domain in ¾ ¡ ( Öü ý ) having connectedboundary ÏGÆ , then every function holomorphic in a neighborhood of ÏÆ uniquelyextends as a function holomorphic in Æ . There are three classical methods ofproof: ! using the Bochner-Martinelli kernel;! using solutions of Ï ø Þ ³ having compact support;! pushing analytic discs, in successive Hartogs skeletons.

The first two are rigorously established and we shall review the first in a while.For almost one hundred years, it has been a folklore belief that the third methodcould be accomplished somehow. Let us be precise.

1.5. Fornaess’ counterexample and a disc theorem. Thus, let Æ be a boundeddomain of ¾ ï having connected D ã boundary. For 6 " �

small, consider theone-sided neighborhood of ÏGÆ defined by:tÆÝÜúÝßÞ 4 æ -XÆ{Ý � ) �Re¤È æ Í ÏGÆúÊ �J6 5 êThe complement Æ 3 tÆÝÜ is a compact hole. Remind that the bidisc ÿ ï is the convexhull of the Hartogs skeleton

Ø Ö . Following [F1998], we say that Æ can be filledin by analytic discs if for every 6 " �

, there exist a finite sequence of subdomainsof Æ having D ã boundary,

tÆÝÜ�ÞÞÆ ã ! Æ ï ! ÑaÑaÑ ! Æ � ÞßÆ and for each� Þ É Í êÇêÇê Í � èÕÉ , an × Æ " �and a univalent holomorphic map Ú Æ defined in a

neighborhood of ÿ ï such that:

(1) Æ Æ � ã ! Æ Æ ÃHÚ Æ Èsÿ ï Ê ! Æ ;

(2) Ú Æ È Ø Ö Ê ! Æ Æ ;(3) Æ Æ ¦}Ú Æ Èsÿ ï Ê is connected;

(4) Æ Æ � ã ¦}Ú Æ Èsÿ ï Ê is connected.

For such domains, by pushing analytic discs in the embedded Hartogs figure,taking account of connectedness, we have × È�Æ Æ � ã Ê ®® Ç�à Þ�× È�Æ Æ Ê . Then by induc-tion, uniquely determined holomorphic extension holds from Æ ã up to Æ . Impor-tantly, the intermediate domains are required to be all contained in Æ .

10Often, some authors consider instead a compact á�âvã with ã äåá connected andstate that æ��uã ä�áP�Oçèæé�³ã*�åêê ë�ìBí ; a technical check shows that the two statements are

equivalent.


In 1998, Fornaess [F1998] constructed a topologically strange domain Æ ! ¾ ïthat cannot be filled in this way. This example shows that the requirement thatÆ$î ! Æ î � ã ! Æ is too stringent.

Nevertheless, taking care of monodromy and working in the envelope of holo-morphy of Æ , one may push analytic discs by allowing them to wander in theoutside, in order to get the general Hartogs theorem stated above. As a pre-liminary, one perturbs and smoothes out the boundary. Denote by @ @ æ @ @ ÝßÞ @ æ ã @ ï û ÑaÑaÑ û @ æ ¡ @ ï ã c ï the Euclidean norm of æ Þ�ï æ ã�ð êÇêÇê ð æ ¡�ñ -2¾ ¡ and byò ¡ ï i ð 6 ñ ÝßÞ¿ó @ @ æ è i @ @ �J6<ô the open ball of radius 6 " �

centered at a pointi

.

Theorem 1.6. ([MP2006c]) Let ² õ9¾ ¡ ( iü�ý ) be a connected D Ù hypersur-face bounding a domain Æ Ä õ\¾ ¡ . Suppose to fix ideas that ý � ��) �Re � ð Æ Ä �+ and assume that the restriction Û Ä ÝßÞ)Û @ Ä of the distance function Û�ï æ ñ Þ @ @ æ @ @to ² is a Morse function having only a finite number F of critical points ói�ö -�² ,÷¼�vø�� F , located on different sphere levels:ý � ó Û ã ÝßÞ«Û�ïsói ã ñ ��ÑaÑaÑ2��ó Û è ÝßÞ«Û�ïsói è ñ � + û diam Æ Ä êThen there exists 6 ã " �

such that for every 6 with� �:6K�:6 ã , the (tubular)

neighborhood ù Ü0ïÏ² ñ ÝßÞ)Ã_ú § Ä ò ¡ ï i ð 6 ñenjoys the global Hartogs extension property into Æ Ä :× ù Ü0ïÏ² ñ Þ�× Æ Ä Ã ù Ü-ïÏ² ñ ®®Åû-üDý Ä·þ ðby � � pushing � � analytic discs inside a finite number of Hartogs figures, withoutusing neither the Bochner-Martinelli kernel, nor solutions of some auxiliary ÿproblem.

1.7. Hartogs-Bochner theorem via the Bochner-Martinelli kernel. By ×Qï`| ñ ,where | ! ¾ ¡ is closed, we mean ×Qï ù ï`| ñsñ for some open neighborhood

ù ï`| ñof | . Here is the general statement.

Theorem 1.8. ([HeLe1984, He1985, Ra1986]) Let Æ be a bounded domain in ¾ ¡having connected boundary. Then for every neighborhood Æ of ÿGÆ in ¾ ¡ andevery holomorphic function ��-°×Qï Æ ñ , there exists a function Â�-�×Qï Æ ñ withÂ @ � Ç Þ�� @ � Ç .

In the thin neighborhood Æ of the not necessarily smooth boundary ÿÆ , bymeans of a partion of unity, one may construct a connected boundary ÿÆ ã !Æ close to ÿGÆ which is D ã , or D Ù , or even D�� , using Whitney approximation([Hi1976]; in addition, one may assure that Û�ï æ ñ ®® � Ç � is as in Theorem 1.6, whenceboth statements are equivalent). Then the restriction Â @ � Ç � is CR on ÿGÆ ã and theprevious theorem is a consequence of the next.


Theorem 1.9. ([Ra1986, He1985]) Let Æ be a bounded domain in ¾ ¡ ï� �üÓý ñhaving connected D«è � é boundary, with

÷Õ� F � > ,�}�ÊI°�`÷

. Then for everyCR function � Ý6ÿÆ�L ¾ of class D è � é , there exists a function Â¨-É×Qï�Æ ñ ¦D è � éwï Æ ñ with Â @ � Ç Þ�� .

Some words about the proof. With ä ð æ -2¾ ¡ , consider the Bochner-Martinellikernel:�� ïsä ð æ ñ ÝßÞ ï� Øè ÷ ñ��ïÏý ��m ñ ¡ @ ä�è æ @ û ï ¡

¡öî � ã ùäsîðè ùæ î �¢äsî� ª� î � ùä � °�¢ä � êThis is a ï� ð �è ÷ ñ

-form which is D�� off the diagonal 4 ä Þ æ 5 . For Þ�ý ,it coincides with the Cauchy kernel

ãï �� ã� û � . If � and ÿÆ are D ã , the integralformula: ÂRï æ ñ��

� � Ç �*ï�� ñ �� ï�� ð æ ñprovides the holomorphic extension Â .

1.10. Hypersurfaces of ¾ ¡ and jump theorem for CR functions. Let ² be areal hypersurface of ¾ ¡ without boundary. In the sequel, we shall mainly dealwith three geometric situations.! Local: ² is defined in a small open polydisc centered at one point

i -² .! Global: ² is a connected orientable embedded submanifold of ¾ ¡ .! Boundary: ¾ ¡ 3 ² consists of two open sets Æ � , bounded and Æ û , un-bounded.

Then there exists some appropriate neighborhood � of ² in ¾ ¡ in which ²is relatively closed, in the sense that ² ¦�� ² .

More generally, let � be an arbitrary complex manifold of dimension yü ÷and let ² ! � be a hypersurface of class at least D ã which is relatively closedin � and oriented. The complement � 3 ² then consists of two connected com-ponents Æ � and Æ û , where Æ � is located on the positive side to ² . Also, let� � ² L ¾ be a CR function of class at least D ú . By definition, � is CR if thecurrent of integration on ² of bidegree ï � ð ÷ ñ defined by11:� Ä ï�� ñ�� Ä �� ð �N-�� ¡ � ¡ û ã ðsatisfies % Ä � ÿ��

for every � -�� ¡ � ¡ û ï . Equivalently, ÿ�� Ä � �in the

sense of currents, where � Ä is interpreted as a ï � ð ÷ ñ -form having measure coef-ficients.

11Here, "! � # is the space of � ¢ forms of bidegree �%$��'&�� having compact support;fundamental notions about currents may be found in [Ch1989].


To formulate the jump theorem in arbitrary complex manifolds, we shall mainlyassume that the Dolbeault ÿ -complex on � is exact in bidegree ï � ð ÷ ñ , namely( ú � ã� ï)� ñ��

. This assumption holds for instance when � � ÿ ¡ , ¾ ¡ or � ¡ ï~¾ ñ .It means that the equation ÿø � ³ , where ³ is a ÿ -closed ï � ð ÷ ñ -form on � havingD+* , 9 ï or distributional coefficients has a D,* , 9 ï or distributional solution ø on� .

Consequently, there exists a distribution Â on � with ÿGÂ � � Ä . As��¬]j j-� Ä ! ² , such a function Â is holomorphic in � 3 ² . The differenceÂ ï èVÂ ã of two solutions to ÿ�Â � � Ä is holomorphic in � . In the case where� � ¾ ¡ , a solution to ÿÂ � � Ä may be represented ([Ch1975, Ra1986]) bymeans of the Bochner-Martinelli kernel as ÂRï æ ñ.�� % Ä �*ï�� ñ �� ï�� ð æ ñ . In com-plex dimension � ÷

, such a solution coincides with the classical Cauchy trans-form.

In 1975, after previous work of Andreotti-Hill ([AH1972b]), Chirka obtained aseveral complex variables version of the Sokhotski /ı-Plemelj Theorem 2.7(IV).

Theorem 1.11. ([Ch1975]) Assume that( ú � ã� ï)� ñ0� �

and that the hypersurface² ! � is orientable and relatively closed, i.e. ²u¦�� ² . Assume � ) �1� � �ü ÷and let ïÏF ð I ñ with

�µ� F � > ,� � I � ÷

. If ² is D è � ã � é andif the current � Ä associated to a D«è � é function � � ² L ¾ is CR, then everydistributional solution ÂÉ-7×Qï)� 3 ² ñ

to ÿGÂ � � Ä extends to be D è � é in the twoclosures Æ-2 � Æ 2 Ã�² , yielding two functions Â 2 -(×Qï�Æ 2 ñ ¦�D è � éwï�Æ 2 Ã�² ñwhose jump across ² equals � :Â � ï æ ñ èSÂ û ï æ ñ�� *ï æ ñ ð 3 æ -}² êA similar jump formula holds for �o-É9 ;ä%4 5 � 6 Á ïÏ² ñ

, with ² at least D ã (or aLipschitz graph) and for �C-7� �6 Á ïÏ² ñ

, with ² -ED * .

When � � ¾ , the conditions that � is CR and that( ú � ã� ï)� ñ0� �

are automat-ically satisfied and we recover the Sokhotski /ı-Plemelj jump formula. However,we mention that in several complex variables ( �üÓý ), there is no analog of thesecond formula

ãï /ÅÂ � ï�� ú ñ û Â û ï�� ú ñ 1 � j ê\k¡ê ãï �� %98;: ý � þ� û � / �<� . The reason is the

inexistence of a universal integral formula solving ÿ�Â � � Ä . Nevertheless, thereshould exist generalized principal value integrals which depend on the kernel.

If ² is only D ã and � is only D ú , it is in general untrue that Â û and Â �extend continuously to ² . Fortunately, there is a useful substitute result, analogto Theorem 2.9(IV). Consider a open subset ²[� ! ² having compact closure² � not meeting ÿ ² � ² 3 ² . We may embedd ² � in a one-parameter familyïb² �Ö ñ ó Ö ó ÷ Ö / , × ú " �

, of hypersurfaces that foliates a strip thickening of ² � .Theorem 1.12. ([Ch1975]) If � is CR and D è on a D©è � ã hypersurface ² , then'*) �Ö + ú ®® ®® Â @ Ä Ê èµÂ @ Ä r>Ê è=� ®® ®® è � � ê


1.13. CR extension in the projective space. Unlike in ¾ ¡ , there is no privileged“interior” side of an orientable connected hypersurface in the projective space� ¡ ï~¾ ñ , Êüºý . Nevertheless, a version of the Hartogs-Bochner theorem holds.The proof is an illustration of the use of the jump theorem.

Theorem 1.14. ( Pü . : [HL1975]; � ý : [Sa1999, DM2002]) Let ² be acompact orientable connected D ï real hypersurface of � ¡ ï~¾ ñ that divides the pro-jective space into two domains Æ û and Æ � . Then:

(i) there exists a side, Æñû or Æ � , to which every function holomorphic insome neighborhood of ² extends holomorphically12;

(ii) every function that is holomorphic in the union of the other side of ²together with a neighborhood of ² must be constant.

Let us summarize the proof. Let � be holomorphic in some neighborhoodù ïÏ² ñof ² in � ¡ ï~¾ ñ . As the Dolbeault cohomology group

( ú � ã ï � ¡ ï~¾ ñ ñ van-ishes for �üÎý ([HeLe1984, He1985]), thanks to Theorem 1.11 above, the CRfunction � @ÅÄ on ² can be decomposed as the jump � � Â � è«Â$û betweentwo functions Â 2 holomorphic in Æ 2 which are (at least) continuous up to ² . Itsuffices then to show that either Â � or Â�û is constant, since clearly, if Â � (resp.Â�û ) is constant equal to k � (resp. k{û ), then � extends holomorphically to Æ û(resp. to Æ � ) as k � èµÂ$û (resp. as Â � èµkhû ).

By contradiction, assume that both Â � and Â û are nonconstant. We choosetwo domains Æ � and Æ û with

ù ïÏ² ñ ÃQÆ 2?> Æ 2?> ²�ÃQÆ 2 . By a preliminary(technical) deformation argument, we may assume that Â 2 is holomorphic inÆ 2 . According to a theorem due to Takeuchi [Ta1964], holomorphic functionsin an arbitrary domain of � ¡ ï~¾ ñ ( �üÉý ), either are constant or separate points.Since Â û is nonconstant, ×Qï Æ û ñ separates points. Conjugating with elementsof the group @�A.B#ï� ð ¾ ñ of projective automorphisms of � ¡ ï~¾ ñ , shrinking

ù ïÏ² ñand Æ û slightly if necessary, we may verify ([DM2002]) that ×Qï Æ û ñ separatespoints and provides local system of holomorphic coordinates at every point. Bystandard techniques of Stein theory ([Ho1973]), it follows that Æ û is embeddablein some ¾�C , with D large. The image of ² under such an embedding Ú is acompact CR submanifold of ¾EC that is filled by the relatively compact complexmanifold ½ û � Ú"ï Æ û ñ with boundary Ú"ïÏ² ñ

. Two applications of the maximumprinciple to the nonconstant holomorphic function Â �1F Ú û ã say that it mustdecrease inside ½û , since ½û is interior to ÚéïÏ² ñ

in ¾ C , and that it must increase,since the one-sided neighborhood Æ û ¦ ù ïÏ² ñ

is exterior to Æ � . This is the desiredcontradiction.

12Using propagation techniques of Section 3, the theorem holds assuming that G isglobally minimal and considering continuous CR functions on G .


1.15. Levi extension theorem. A D ï hypersurface ² ! ¾ ¡ may always be rep-resented as ² � 4 æ - Æ � Û�ï æ ñ.� ��5

, where Æ is some open neighborhood of² in ¾ ¡ , and where Û � Æ L Qis a D ï implicit defining function that satisfies��Û�ï)H ñ J� �

at every point H of ² . Two defining functions Û ã ð Û ï are nonzero mul-tiple of each other in some neighborhood È ! Æ of ² : there exists

ø � È L Qnowhere vanishing with Û ï � ø Û ã .

At a pointi -}² , the Levi form of Û :I ú�Û�ïÍ9�ú ð 9�ú ñ�� öãKJ î � � J ¡ ÿ ï Ûÿ æ î ÿ ùæ � ï

i ñ 9 îú 9 �ú ð 9�ú�- V ã � úú ² ðis a Hermitian form that may be diagonalized. Its signature at

i:ï � ú ð C ú ñE�� ïML positive eigenvalues ð L negative eigenvalues

ñis the same for Û ã and Û ï if they are positive multiples of each other. It is alsoinvariant through local biholomorphic changes of coordinates æ »L tæ ï æ ñ that donot reverse the orientation of ² . Reversing the orientation or taking a negativefactor

øcorresponds to the transposition ï � ú ð C ú ñ »L ï C ú ð � ú ñ .

The Levi form may be read off a graphed equation ³ � ´=ï�¶ ð · ð ø ñ for ² .

Lemma 1.16. There exist local holomorphic coordinates ï æ ð ø û m ³ ñ - ¾ ¡ û ã Ã ¾centered at

iin which ² is represented as a graph of the form:³ � ´#ï æ ð ø ñE� öãKJ � J�N OÒ� ê O × � æ � ùæ � û ��ï @ æ @ ï ñ ûQP ï @ æ @a@ ø @ ñ ûQP ï @ ø @ ï ñ ð

where × � � û ÷for

÷ � � � � ú , × � � è ÷ for � ú û ÷ñ� � � � ú û C ú . If � ú � Fè ÷ ,the open set 4 ³ " ´ 5 is strongly convex (in the real sense) in a neighborhood ofi

.

Assuming that ² is orientable, it is surrounded by two open sides. By an openside of ² , we mean a connected component of

ù 3 ² for a (thin) neighborhoodùof ² which is divided by ² in two components. As germs of open sets along² , there exist two open sides (if ² were not orientable, there would exist only

one).Assuming that ² is represented either by an implicit equation Û � �

or as agraph è ³ û ´=ï�¶ ð · ð ø ñ0� �

, we adopt the convention of denoting:Æ � �� 4 Û$� ��5 �SR Æ � �� 4 ³ " ´=ï�¶ ð · ð ø ñ 5 ðÆ û �� 4 Û " ��5 �SR Æ û �� 4 ³ ��´=ï�¶ ð · ð ø ñ 5 êOnce a local side Æ of ² has been fixed, ² is oriented and the indeterminationÛUT èÝÛ disappears. By convention, we will always represent Æ � 4 ÛJ� ��5

.Then the number of positive and of negative eigenvalues of the Levi-form of Û ata point

i - ÿÆ is an invariant. By common abuse of language, we speak of theLevi form of ÿGÆ .


At one of its pointsi

, a boundary ÿGÆ is called strongly pseudoconvex (resp.strongly pseudoconcave) if its Levi form has all its eigenvalues " �

(resp. � �)

ati

. It is called weakly pseudoconvex (resp. weakly pseudoconcave) ati

if alleigenvalues are ü �

(resp.�J�

). Often, the term “weakly” is dropped in commonuse.

Definition 1.17. If Æ is an open side of ² , we say that Æ is holomorphicallyextendable at

iif for every open13 polydisc Æ ú centered at

i, there exists an open

polydisc È ú centered ati

such that for every �°-y×Qï�ÆQV Æ ú ñ , there exists Âö-×Qï È ú ñ with Â @ Ç�WYX O � � @ Ç�W<X O .In 1910, Levi localized the Hartogs extension phenomenon.

Theorem 1.18. ([Bo1991, Trp1996, Tu1998, BER1999]) If the Levi form of ² !ÿGÆ has one negative eigenvalue at a pointi

, then Æ is holomorphically extendableati

.

Proof. As Æ is given by 4 ³ " è æ ã ùæ ã û ÑaÑaÑ 5 , for × " �small, the disc ½ Ö ï�� ñZ��ï~×,� ð � ð êÇêÇê ð � ñ has its boundary ½ Ö ï�ÿ©ÿ ñ contained in Æ near

i.

Lemma 1.19. Assume ² is D ã , leti -�² and Æ be an open side of ² at

i.

Suppose that for every open polydisc Æ ú centered ati

, there exists an analyticdisc ½ � ÿ8L Æ ú continuous in ÿ with ½xï � ñ.� i

and ½xï�ÿPÿ ñ ! Æ . Then Æ isholomorphically extendable at

i.

To draw ½ ïsÿ ñ , decreasing by÷

its dimension, we represent it as a curve. Thecusp illustrates the fact that ½xïsÿ ñ is not assumed to be embedded.

ÄÎGÑ�Ó ÔwÕÎGÑ�ÔwÕÄ [�\ X OE O E O] ^ Ä! Ä] ^

Pushing (translating) an analytic disc

Ç ÇTo prove the lemma, we may assume that

i � �. Since ½xï�ÿPÿ ñ is contained

in Æ , for æ -Ù¾ ¡ very small, say @ æ @ �Ó6 , the translates æ û ½ ï�ÿ©ÿ ñ of the discboundary are also contained in Æ .

13Although the shape of polydiscs is not invariant by local biholomorphisms, theirtopology is. To avoid dealing implicitly with possibly wild open sets, we prefer to speakof neighborhoods ^ ! , _ ! , ` ! of $ that are polydiscs.


Consequently, the Cauchy integral:ÂRï æ ñ��÷ý ��m � � } �*ï æ û ½ ï�� ñsñ �<��

is meaningful and it defines a holomorphic function of æ in the polydisc È ú ��4 @ æ @ �J6 5 .Does it coincide with � in È ú�VRÆ ? The assumption that ² is D ã yields a local

real segment asú transversal to ² ati

. If Æ ú is sufficiently small and if æ -baDúEVPÆgoes sufficiently deep in Æ , the disc æ û ½ ï ÿ ñ is contained in Æ , so that the Cauchyintegral ÂRï æ ñ coincides with �*ï æ ñ for those æ . Þ1.20. Contact of weakly pseudoconvex domains with complex hypersurfaces.The domain Æ is said to admit a support complex hypersurface at

i - ÿGÆ if thereexists a local (possibly singular) complex hypersurface ½ passing through

ithat

does not intersect Æ . In this situation, if ½ � 4�c ï æ ñd� ��5with c holomorphic,

the function÷ g c does not extend holomorphically at

i, being unbounded. WithI " �

not integer, one may define branches c�é which are uniform in Æ andcontinuous up to ÿÆ , but whose extension through

iwould be ramified around ½ .

Consequently, the existence of a support complex hypersurface prevents ×Qï�Æ ñ tobe holomorphically extendable at

i. Is it the right obstruction ? For instance, at a

strongly pseudoconvex boundary point, the complex tangent plane is support.Nevertheless, in 1973, Kohn-Nirenberg constructed a special pseudoconvex do-

main Æ � egf in ¾ ï showing that:! not every weakly pseudoconvex smoothly bounded domain is locally bi-holomorphically equivalent to a domain which is weakly convex in thereal sense;! the holomorphic non-extendability of ×Qï�Æ ñ at

iis totally independent

from the existence of a local supporting complex hypersurface ati

.

The boundary of this domain² egf �� óï æ ðKh ñ -X¾ ï � ñ�� h � @ æ h @ ï û æ ¤ ùæ ¤ û ÷ + ï æ i ùæ û ùæ i æ ñ g ÷kj ômay be checked to be strongly pseudoconvex at every point except the origin,where it is weakly pseudoconvex. Hence ×Qï�Æ � elf ñ is not holomorphically ex-tendable at the origin. However, ² elf

has the striking property that every lo-cal (possibly singular) complex hypersurface ½ passing through the origin meetsboth sides of ² elf

in every neighborhood of the origin. By means of a Puiseuxparametrization ([JaPf2000]), such a complex curve ½ is always the image of acertain holomorphic disc

ø � ÿÓL ¾ ï withø ï � ñ0� �

.

Theorem 1.21. ([KN1973]) Whatever the holomorphic discø

, for every × " �,

there are � û and � � in ÿ with @ � 2 @ �Ù× such thatø ï�� 2 ñ - Æ 2 elf .

Clearly, Æ � egf is not locally convexifiable at the origin (otherwise, the biholo-morphic image of the complex tangent line would be support).


Often for technical reasons, certain results in several complex variables requireboundaries to be convex in the real sense. Although this condition is not biholo-morphically invariant, it is certainly meaningful to characterize the class of con-vexifiable domains, at least locally: does there exist an analytico-geometric crite-rion enabling to recognize local convexifiability by reading a defining equation ?

1.22. Holomorphic extendability across finite type hypersurfaces. Let ² bea D � hypersurface and let

i -Ê² . Then ² is of type H ati

, in the sense ofDefinition 4.22(III), if and only if there exists a local graphed equation centeredati

of the form:³ � ´ Á ï æ ð ùæ ñ ûmP ï @ æ @ Á � ã ñ ûQP ï @ æ @a@ ø @ ñ ûQP ï @ ø @ ï ñ ðwhere ´ Á -X¾¯/ æ ð ùæ 1 is a nonzero homogeneous real-valued polynomial of degreeH having no pluriharmonic term, namely� v ´ Á ï � ð ùæ ñ v ´ Á ï æ ð � ñ . The restric-

tion ´ Á ïna>ï�� ñ ð a>ï�� ñ ñ of ´ Á to a complex line ¾Û¹o�$»Lpa�ï�� ñ -�¾ ¡ û ã , a>ï � ñ�� , is

a polynomial in ¾ /q� ð ù� 1 . For almost every choice of a , this polynomial is nonzero,homogeneous of the same degree H and contains no harmonic term. After a rota-tion, such a line is the complex æ ã -axis. Denoting æ � � ï æ ï ð êÇêÇê ð æ ¡ û ã ñ , we obtain:

(1.23) ³ � ´ Á ï æ ã�ð ùæ ã ñ ûmP ï @ æ ã @ Á � ã ñ ûQr ï @ æ � @ ñ ûQP ï @ æ @a@ ø @ ñ ûQP ï @ ø @ ï ñ êTheorem 1.24. ([BeFo1978, R1983, BT1984]) If H is even, at least one sideÆ � or Æ û is holomorphically extendable ati

. If H is odd, both sides have thisproperty.

Proof. Let × " �arbitrarily small, let � -S¾ with @ � @ � ÷

, let � be in the closedunit disc ÿ , and introduce a ¾ ¡ -valued analytic disc:½ Ö ï�� ñE�� ï~× ï � û � ñ ð � ð êÇêÇê ð � ð ×ðÁ À ï�� ñsñhaving zero æ � -component and æ ã -component being a disc of radius × centered atè � . We assume its h -component

À ï�� ñ be defined by requiring that the ¾ ï -valueddisc s Ö ï�� ñ�� ï~×�ï � û � ñ ð × Á À ï�� ñsñhas its boundary

s Ö ï�ÿPÿ ñ attached to ³ � ´ Á ï æ ã'ð ùæ ã ñ . By homogeneity, × Á dropsand it is equivalent to require that

s ã is attached to ³ � ´ Á ï æ ã�ð ùæ ã ñ . Equivalently,the imaginary part È ï�� ñ of

À � Æ û m È should satisfy:È ï � � { ñ�� ´ Á ï � û � � { ð ù� û � û � { ñ ðfor all � � { -9ÿPÿ . To obtain a harmonic extension to ÿ of the function È thusdefined on ÿ©ÿ , no Bishop equation is needed. It suffices to take the harmonicprolongation by means of Poisson’s formula, as in � 2.20(IV):È ï�t ñE� ì È ï�t ñ�� ÷ý ��m � � } ´ Á ï � û � ð ù� û ù� ñ ÷ è @ t @ ï@ �òèut @ ï �<�� ê


Since ´ Á has no harmonic term, it may be factored as ´ Á � æ ã ùæ ã � ã ï æ ã�ð ùæ ã ñ ,with � ã -X¾ / æ ã ð ùæ ã 1 homogeneous of degree ï³H è(ý ñ and nonzero. In the integralabove, we put t �� è � and we replace ´ Á � æ ã ùæ ã � ã to get the value of È at è � :È ï³è � ñ�� ÷ý ��m � � } ´ Á ï � û � ð ù� û ù� ñ ÷ è @ � @ ï@ � û � @ ï �g�� ÷ è @ � @ ïý � � �û � � ã ï � û � � { ð ù� û � û � { ñ ��Ì êAs a function of � - ÿ , the last integral is identically zero if and only if thepolynomial � ã is zero. Thus, there exists � such that È ï³è � ñ J� �

. (However, wehave no information about the possible signs of È ï³è � ñ in terms of ´ Á .) Thenwe define Æ ï�� ñ to be the harmonic prolongation of è U È that vanishes at è � ands Ö ï�� ñ�� ï~× ï � û � ñ ð × Á À ï�� ñsñ .

The positivity (resp. negativity) of the sign of È ï³è � ñ means thats Ö ï³è � ñv�ï � ð m È ï³è � ñsñ is in Æ � (resp. Æ û ). Then after translating slightly

s Ö in the rightdirection along the ³ -axis, Lemma 1.19 applies to deduce that Æ ûã ! ¾ ï (resp.Æ � ã ! ¾ ï ) is holomorphically extendable at the origin. Since ´ Á ï³è æ ãÒð è ùæ ã ñw�ï³è ÷ ñ Á ´ Á ï æ ã�ð ùæ ã ñ , in the case where H is odd, the disc è s Ö will also be attachedto ² ã and will provide extendability of the other side.

Thanks to basic majorations of the “P

” remainders in the equation (1.23) of² , if × " �is sufficiently small, then Æ û ! ¾ ¡ (resp. Æ � ! ¾ ¡ ) has the same

extendability property. ÞIf ² is a real analytic hypersurface, it is easily seen, by inspecting the Taylor

series of its graphing function, that ² is not of finite type at a pointi

if and onlyif it may be represented by ³ � ø t´@ï æ ð ø ñ , with

t´J-7D � . Then the local complexhypersurface 4 ³ � ø � ��5

is contained in ² .

Corollary 1.25. ([BeFo1978, R1983, BT1984]) If ² is Dx� and ifi -Û² , the

following properties are equivalent:! ² has finite type ati

;! ² does not contain any local complex analytic hypersurface passingthrough

i;! Æ � or Æ û is holomorphically extendable at

i.

1.26. Which side is holomorphically extendable ? We claim that it suffices tostudy osculating domains in ¾ ï of the form:Æ �y F �� 4 è ³ û ´ Á ï æ ð ùæ ñ � ��5 ð æ -X¾ ðzh � ø û m ³ -X¾ ðwhere ´ Á J� �

is real, homogeneous of degree H�üJý and has no harmonic term.Indeed, extendability properties of such domains transfer to perturbations (1.23).Also, extendability properties of Æ ûy F are just the same, via ´ Á T è´ Á . For thisreason, if H is odd, both Æ �y F and Æ ûy F are holomorphically extendable at

i.


The local complex line 4 ï æ ð � ñ 5 intersects the closure Æ �y F in regions that areclosed angular sectors (cones), due to homogeneity. We call these regions interior.The complement ¾ ï 3 Æ �y F intersects 4 ï æ ð � ñ 5 in open, exterior sectors.

Theorem 1.27. ([R1983, BT1984, FR1985]) If there exists an interior sector ofangular width " �Á , then Æ �y F is holomorphically extendable at

i.

The proof consists in choosing an appropriate truncated angular sector as the

æ -component of a disc attached to ÿGÆ �y F , instead of the round disc � »L ×�ï � û � ñ .Example 1.28. Every homogeneous quartic ³ � ´ ¤ ï æ ð ùæ ñ in ¾ ï is biholomorphi-cally equivalent to a model� � Û N �� è ³ û æ ï ùæ ï û � æ ùæ ï æ ï û ùæ

ï ñ ðfor some � - Q

. Such a hypersurface bounds two open sides Æ �N � 4 Û N � ��5andÆ ûN � 4 Û N " ��5

which enjoy the following properties ([R1983, BT1984]):! Æ ûN is holomorphically extendable ati

, for every � ;! @ � @ �Jý�g-. if and only if Æ �N is everywhere strongly pseudoconvex;! @ � @ � ÷|{~} ý if and only if Æ �N is not holomorphically extendable ati

;! @ � @ " ÷|{ } ý if and only if Æ �N is holomorphically extendable ati

;! the above extendability property holds true for any perturbation of ÿGÆ byhigher order terms.

Finer results, strictly more general than the above theorem that apply to sixtics,were obtained in [FR1985]. If we remove all exterior sectors of angular widthü �Á , the rest of the complex line 4 ï æ ð � ñ 5 is formed by disjoint closed sectors,which are called supersectors of order H of Æ �y F at

i. A supersector is proper if

it contains points of Æ �y F .

Theorem 1.29. ([FR1985])

(i) If Æ �y F has a proper supersector of angular width " �Á , then Æ �y F isholomorphically extendable at

i.

(ii) If all supersectors of Æ �y F have angular width � �Á , then there exists�7-C×Qï�Æ �y F ñ VSD ú ï Æ �y F ñ that does not extend holomorphically ati

.

Even in the case H � ÷ , some cases in this theorem are left open. Examplesmay be found in [FR1985].

Open problem 1.30. In the case where H is even, find a necessary and sufficientcondition for Æ �y F � 4 ³ " ´ Á ï æ ð ùæ ñ 5 to be holomorphically extendable at

i, or

show that the problem is undecidable.

One could generalize this (already wide open) question to a not necessarilyfinite type boundary, D � , D+* , D ï or even D ú graph.

166 JOEL MERKER AND EGMONT PORTEN� 2. TREPREAU’S THEOREM, DEFORMATIONS OF BISHOP DISCS

AND PROPAGATION ON HYPERSURFACES

2.1. Holomorphic extension of CR functions via jump. Let ² be a hypersur-face in ¾ ¡ of class at least D ã � é with

� � I � ÷and let � be a continuous CR

function on ² . At each pointi

of ² , we may restrict � to a small open ball (orpolydisc) Æ$ú centered at

i. Applying the jump Theorem 1.11, we may represent� � Â � èVÂ û , with Â 2 -0×Qï�Æ 2ú ñ V�D ú ï Æ 2ú ñ . If Æ �ú (resp. Æ ûú ) is holomorphi-

cally extendable ati

, then Â � (resp. Â�û ) extends to a neighborhood �wú ofi

in¾ ¡ as ��-y×Qï��Gú ñ (resp.( -J×Qï��Gú ñ ). Then � extends holomorphically to the

small one-sided neighborhood � ûú (resp. to � �ú ) as �ÕèµÂ û (resp. as Â � è ( ).

Lemma 2.2. On hypersurfaces, at a given point, local holomorphic extendabilityof CR functions to one side is equivalent to holomorphic extendability to the sameside of the holomorphic functions defined in the opposite side.

Consequently, the theorems of � 1.22 yield gratuitously extension results aboutCR functions. For instance:

Corollary 2.3. ([BeFo1978, R1983, BT1984]) On a real analytic hypersurface² , at a given pointi

, continuous CR functions extend holomorphically to oneside if and only if ² does not contain any local complex hypersurface passingthrough

i.

The assumption of real analyticity, or the assumption of finite typeness in case² is D+* , both consume much smoothness. The removal of these assumptionswas accomplished by Trepreau in 1986.

Theorem 2.4. ([Trp1986]) Let ² be a D ï hypersurface of ¾ ¡ , Îü�ý and leti -}² . The following two conditions are equivalent:! ² does not contain any local complex hypersurface passing throughi

.! for every open subset Æ ú ! ² containingi

, there exists a one-sided

neighborhood � 2ú of ² ati

with � 2ú V0² õ Æ ú such that for every�7-ED ú6 Á ï Æ ú ñ , there exists ÂÉ-7×Qï�� 2ú ñ VSD ú ï�� 2 Ã Æ ú ñ with Â @ E O � � .

We have seen that characterizing the side of extension is an open question, evenfor rigid polynomial hypersurfaces ³ � ´ Á ï æ ð ùæ ñ and even for H � ÷ . Althoughthe above theorem constitutes a neat answer for holomorphic extension to someimprecise side, it does not provide any control of the side of extension.

Let ² be a D ï orientable connected hypersurface and let Æ �Ä be an open sideof ² . One could hope to characterize holomorphic extension to the other sideat every point of ² , since weak pseudoconvexity characterizes holomorphic non-extendability at every point of ² , by Oka’s theorem.


Example 2.5. ([Trp1992]) In ¾ ± , let Æ �Ä be ó ³ " ´ Á ï æ ã�ð ùæ ã ñ è @ æ ï @ ï @ æ ã @ ï C ôwhere ´ Á Jv �, of degree H with . � Hõ��D is as in Open problem 1.30. One

verifies that holomorphic extension at every point of ² entails a characterizationof holomorphic extension at the origin for the domain ó ³ " ´ Á ï æ ð ùæ ñ è�× @ æ @ ï C ô .

In the sequel, we shall abandon definitely the difficult, still open question ofhow to control sides of holomorphic extension.

Although Theorem 2.4 is well known in Several Complex Variables, there is amore general formulation with a simpler proof than the original one. The remain-der of this section will expose such a proof.

By a global one-sided neighborhood of a connected (not necessarily orientable)hypersurface ² ! ¾ ¡ , we mean a domain Æ Ä with Æ Ä > ² such that for everypoint HÕ-�² , at least one open side � 2� of ² at H is contained in Æ Ä . In fact, toinsure connectedness, Æ Ä is the interior of the closure of the union Ã � § Ä � 2� ofall (possibly shrunk) one-sided neighborhoods.

��

�� ¡�¡�¡�¡�¡¡�¡�¡�¡�¡¢�¢�¢�¢¢�¢�¢�¢£�£�£�£�£�£¤�¤�¤�¤�¤�¤¥�¥�¥�¥�¥�¥�¥¥�¥�¥�¥�¥�¥�¥¦�¦�¦�¦�¦�¦�¦¦�¦�¦�¦�¦�¦�¦§�§�§¨�¨©�©�©�©�©�©©�©�©�©�©�©ª�ª�ª�ª�ªª�ª�ª�ª�ª«�«�«�«�«�««�«�«�«�«�«¬�¬�¬�¬�¬�¬¬�¬�¬�¬�¬�¬��®�®�®�®�®®�®�®�®�®¯�¯�¯�¯�¯¯�¯�¯�¯�¯°�°�°�°°�°�°�°±�±�±�±�±�±�±�±�±�±�±�±�±�±�±�±�±�±±�±�±�±�±�±�±�±�±�±�±�±�±�±�±�±�±�±²�²�²�²�²�²�²�²�²�²�²�²�²�²�²�²�²²�²�²�²�²�²�²�²�²�²�²�²�²�²�²�²�²³�³�³�³�³�³�³�³�³�³�³�³�³�³�³�³�³�³�³�³�³�³³�³�³�³�³�³�³�³�³�³�³�³�³�³�³�³�³�³�³�³�³�³´�´�´�´�´�´�´�´�´�´�´�´�´�´�´�´�´�´�´�´�´´�´�´�´�´�´�´�´�´�´�´�´�´�´�´�´�´�´�´�´�´

µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µµ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µµ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ�µ¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶�¶

Global one-sided neighborhood of ·Then Æ Ä contains a neighborhood in ¾ ¡ of every point Û�-}² which belongs

to at least two one-sided neighborhoods that are opposite. The classical Moreratheorem insures holomorphicity in a neighborhood of such points Û .

Remind that ² is called globally minimal if it consists of a single CR orbit.The assumption that ² does not contain any complex hypersurface at any pointmeans that for every

i -Ö² , every open Æ ú ¹ i, the CR orbit × 6 Á ï Æ ú ð i ñ

contains a neighborhood ofi

in ² . This implies that ² is globally minimal andhence, Theorem 2.4 is less general than the following.

Theorem 2.6. ([Trp1990, Tu1994a]) Let ² be14 a connected D ï � é ï � � I � ÷ ñhypersurface of ¾ ¡ ï� Aü�ý ñ . If ² is globally minimal, then there exists a globalone-sided neighborhood Æ Ä of ² such that for every continuous CR function��-HD ú6 Á ïÏ² ñ

, there exists ÂÉ-C×Qï�Æ Ä ñ VSD ú ï�Æ Ä ÃH² ñwith Â @ Ä � � .

14This theorem also holds true with G � � � and even with G � �~¸ � � ( ¹»ºU¼zº¾½ ),provided one redefines the notion of CR orbit in terms of boundaries of small attachedanalytic discs.


It will appear that Æ Ä is contructed by gluing discs to ² and to subsequentopen sets ÆÝ� ! Æ Ä which are all contained in the polynomial hull of ² :�² ��À¿~Á - ¾ ¡ � @ � ï Á�ñ @ � �,¬]jÂ § Ä @ � ï h ñ @ ð,3 � -X¾ / Á 1KÃEÄ

Let us summarize the proof. Although the assumption of global minimality isso weak that ² may incorporate large open Levi-flat regions, there exists at leastone point

i -}² in a neighborhood of whichV � ² � V 5� ² û / V 5� ² ð V 5� ² 1 ð H$- Æ úlÄOtherwise, the distribution

i »L V 5ú ² would be Frobenius-integrable and all CRorbits would be complex hypersurfaces ! At such a point

i, the classical Lewy

extension theorem ( � 2.10 below) insures that D ú6 Á ïÏ² ñextends holomorphically

to (at least) one side ati

.

Theorem 2.7. ([Trp1990, Tu1994a]) Let ² be a connected D ï � é hypersurface,not necessarily globally minimal. If D ú6 Á ïÏ² ñ

extends holomorphically to a one-sided neighborhood at some point

i -�² , then holomorphic extension to one side� 2� holds at every point H$-C× 6 Á ïÏ² ð i ñ .When × 6 Á ïÏ² ð i ñ�� ² as in Theorem 2.6, the global one-sided neighborhoodÆ Ä will be the interior of the closure of the union Ã � § Ä � 2� of all (possibly

shrunk) one-sided neighborhoods.

The next paragraphs are devoted to expose a detailed proof of both the Lewytheorem and of the above propagation theorem.

2.8. Approximation theorem and maximum principle. According to the ap-proximation Theorem 5.2(III), for every

i -�² , there exist a neighborhoodÆ ú ofi

in ² and a sequence ï �ÆÅ ï Á�ñsñ Å § of holomorphic polynomials with'd) � Å + * @ @ �ÇÅ�È � @ @ 0/ ý E O þ � �.

Lemma 2.9. For every analytic disc ½ -µ×Qïsÿ ñ V�D ú ï ÿ ñ with ½xï�ÿ©ÿ ñ ! Æ ú ,the sequence �ÆÅ also converges uniformly on the closed disc ½xï ÿ ñ , even if ½xï ÿ ñgoes outside Æ ú .Proof. By assumption, '*) � Å � É + * @ @ �ÇÅ.È&� É @ @ 0/ ý E O þ � �

. Let tÛ- ÿ arbitrary.Thanks to the maximum principle and to ½ ï�ÿ©ÿ ñ ! Æ ú :@ @ �,Å ïÍ½xï�t ñsñ È&� É ïÍ½xï�t ñsñ @ @ � �Õl{Ë� § � } @ @ �ÇÅ ïÍ½xï�� ñsñ È&� É ïÍ½xï�� ñsñ @ @� �,¬]j� § E O @ @ �,Å ï Á�ñ È&� É ï Á ñ @ @ È L � ÄThe same argument shows that �ÊÅ converges uniformly in the polynomial hull ofÆ ú (we shall not need this). Þ


Next, suppose that we have some family of analytic discs ½ Ý , with

Üa small

parameter, such that Ã Ý ½ Ý ïsÿ ñ contains an open set in ¾ ¡ , for instance a one-sided neighborhood at

i -(² . Then ï �ÊÅ ñ Å § converges uniformly on Ã Ý ½ Ý ïsÿ ñand a theorem due to Cauchy assures that the limit is holomorphic in the interiorof Ã Ý ½ Ý ïsÿ ñ . It then follows that � extends holomorphically to the interior ofÃ Ý ½ Ý ïsÿ ñ .

Remarkably, this short argument based on an application of the approximationTheorem 5.2(III) shows that15:

In order to establish local holomorphic extension of CRfunctions, it suffices to glue appropriate families of analyticdiscs to CR manifolds.

In the sequel, the geometry of glued discs will be studied for itself; thus, it willbe understood that statements about holomorphic or CR extension follow imme-diately; elementary details about continuity of the obtained extensions will beskipped.

2.10. Lewy extension. Since ² is globally minimal, there exists a pointi

atwhich V ú ² � V 5ú ² û / V 5ú ² ð V 5ú ² 1 . Granted Lemma 2.2, holomorphic ex-tension to one side at such a point

ihas already been proved in Theorem 1.18.

Nevertheless, we want to present a geometrically different proof that will producepreliminaries and motivations for the sequel.

Since V 5 ² � ËZÌ V ã � ú ² � ËZÌ V ú � ã ² , we have equivalently5 V ã � ú ² ð V ú � ã ² 7 ï i ñ J! ¾ÎÍ V 5ú ² , namely the intrinsic Levi form of ²ati

is nonzero. In other words, there exists a local section 9 of V ã � ú ² with9Oï i ñ J� �and 5 9 ð 9 7 ï i ñ J-)¾¾Í V 5ú ² . After a complex linear transformation

of V 5ú ² , we may assume that 9�ï i ñ7� �� . After removing the second orderpluriharmonic terms, there exist local coordinates ï Á ã�ð Á � ðKh ñ vanishing at

isuch

that ² is represented by³ � È Á ùÁ ã ûQP ï @ Á ã @ ï � é ñ ûQP ï @ Á � @ ñ ûQP ï @ Á @ @ ø @ ñ ûmP ï @ ø @ ï ñ ÄThe minus sign is set for clarity in the diagram of � 2.12 below. We denote by´#ï Á ã'ð Á � ð ø ñ the right hand side. Let × ã " �

be small. For × satisfying� �\× � × ã ,

we introduce the analytic disc½ Ö ï�� ñE�� ×�ï ÷ È � ñ ð � � ð Æ Ö ï�� ñ û m È Ö ï�� ñ with zero

Á � -component, withÁ ã -component equal to a (reverse) round disc of

radius × centered at÷ -V¾ and with ø -component Æ Ö satisfying the Bishop-type

equation: Æ Ö ï � � { ñ�� ÈU ã 5 ´=ï~× ï ÷ È Ñ ñ ð � � ð Æ Ö ïtÑ ñsñ 7 ï � � { ñ Ä

15This is the so-called Method of analytic discs ; � techniques are also powerful.


Acoording to Theorem 3.7(IV), a unique solution Æ Ö ï � � { ñ exists and is D ï � é�û úwith respect to ï � � { ð × ñ . Since

U ã ï � ñ ï ÷ ñv� �by definition, we have Æ Ö ï ÷ ñw� �

and then È Ö �� U ã ï Æ Ö ñ also satisfies È Ö ï ÷ ñw� �. Consequently, ½ Ö ï ÷ ñw� �

. Byapplying

U ã to both sides of the above equation, we see that the disc is attachedto ² : È Ö ï � � { ñ�� ´ n ×�ï ÷ È � � { ñ ð � � ð Æ Ö ï � � { ñ p ÄWe shall prove that for × ã sufficiently small, every disc ½ Ö ïsÿ ñ with

� �)× � × ãis not tangent to ² at

i. We draw two diagrams: a ý -dimensional and a . -

dimensional view. In both, the ³ -axis is vertical, oriented down.

Ï�ÐÒÑÏ u.Ó)ÔÖÕ!+× ÎGÑ ¸ Õ!Ø× ÎGÑ ¸ ÕÎ Ñ Ñ�ÔwÕû ÏÙÐÒÑÏÙÚ ý ã þ

Ä Ä ÄÄÎ Ñ Ñ(Ó�ÔwÕ û ÏÙÐ ÑÏÙÚ ý ã þ Î Ñ Ñ�ÔwÕ

Nontangency of a small disc to the paraboloid Û�ÜÞÝxßa�Ùàß��Just now, we need a geometrical remark. Let ½µ-C×Qïsÿ ñ VÂD ã ï ÿ ñ be an arbitrary

but small analytic disc attached to ² with ½ ï ÷ ñ�� . We use polar coordinates to

denote � � Û � � { .

G

á �âÊã

â ÓÓåä æ¹��

½â ½ã

ç ý ã þ Ð ��r � �ÏÙÐÏ u ý r sÇu þ ó u'è /ÎGÑ � ÕÓÓ � û ÏÙÐÏÙÚ ý ã þexit vector

Direction of exit of an attached analytic disc

The holomorphicity of ½ yields the following identities between vectors in V ú ¾ ¡ :m ÿ ½ÿ Ì ï � � { ñ ®®®® { � ú � È ÿ�½ÿ�Û ï³Û ñ ®®®®qé � ã � È ÿ ½ÿ�� ï�� ñ ®®®® � � ã ÄThe multiplication by m (or equivalently the complex structure ê ) provides anisomorphism V ú>¾ ¡ g V ú�² L V ú�²°g V 5ú ² ; in coordinates, V ú>¾ ¡ g V ú�² ë QÊì

,V ú ²°g V 5ú ² ë Q � and ê sends

Q � to

Q ì. It follows that

� ç� é ï ÷ ñ is not tangent to² ati

if and only if� ç� { ï ÷ ñ is not complex tangent to ² at

i.


Coming back to ½ Ö , we call the vector

È ÿ ½ Öÿ�Û ï ÷ ñ �� V ú ² � È ÿÀ Öÿ�Û ï ÷ ñ �� V ú ²

the exit vector of ½ Ö . By differentiating È Ö � ´ at Ì � �, taking account of�q´#ï � ñ��

, we get� X Ê� { ï ÷ ñ��

. So only the real part� E Ê� { ï ÷ ñ of

�|í Ê� { ï ÷ ñ may benonzero.

Lemma 2.11. Shrinking × ã if necessary, the exit vector of every disc ½ Ö with� �\× � × ã is nonzero:

È ÿÀ Öÿ�Û ï ÷ ñ0� m ÿ À Öÿ�Ì ï ÷ ñ�� m ÿ Æ Öÿ Ì ï ÷ ñ J� � Ä

Proof. The principal term of ´ is È Á ã ùÁ ã . We compute first:U ã 5 È ¿ ã ï�� ñ ¿ ã ï�� ñ 7 � U ã � × ï ï � û � { È ý û � � { ñ �� ÷ m × ï ï È � û � { û � � { ñ ÄProceeding as carefully as in Section 3(IV), we may verify thatÆ Ö ï � � { ñ�� È

U ã 5 È ¿ ã ï�� ñ ¿ ã ï�� ñ û ¼�îðï�ñYò�óõôYîðö 7 ï � � { ñ� È ý�× ï � )dW Ì û tÆ Ö ï � � { ñ ðwith a D ï � é�û ú remainder satisfying ®® ®® tÆ Ö ®® ®® ã � ú � ? × ï � é , for some quantity ? " �

.

So� E Ê� { ï ÷ ñ0� È ý�× ï ûQP ï~× ï � é ñ J� �

. Þ2.12. Translations of a nontangent analytic disc. We now fix × with

� �\× � × ãand we denote simply by ½ the disc ½ Ö . So the vectorÿ�½ÿ Ì ï ÷ ñ�� È m × ð � � ð È ý�× ï ûmP ï~× ï � é ñ is not tangent to V 5ú ² � 4 ³ � ø � ��5

at the origin. Furthermore, it is not tangentto the ïÏý{ È ý ñ -dimensional sub-plane 4 · ã � ³ � ��5

of V ú�² � 4 ³ � ��5.

We now introduce parameters of translation ¶ ú ã - Q,Á �ú -±¾ ¡ û ï and ø ú - Q

with @ ¶ ú ã @ ð @ Á �ú @ ð @ ø ú @ �J6 ã , where� �J6 ã �¼�\× . The points in ² of coordinates ¶ ú ã ð Á �ú ð ø ú û m ´#ï�¶ ú ã ð Á �ú ð ø ú ñ

cover a small ïÏý{ È ý ñ -dimensional submanifold¨ ú with V ú ¨ ú � 4 · ã � ³ � ��5

transverse to the disc boundary ½ Ö ï�ÿ©ÿ ñ ati

that we draw below.


View inside · ß ø÷

û Öù û ï Ö ¥ The interiorç ý } þlies outside Ä Í�É O

Ì �ç ý � } þûúéÄú

Nontangency at ü of the disc boundary ý Ê Óÿþ��Õ to � ÷ Ü7ü��To conclude the proof of one-sided holomorphic extension at the Levi nonde-

generate pointi

, it suffices to deform the disc ½ Ì / � � � / � � / so that its distinguishedpoint ½ Ì / � � � / � � / ï ÷ ñ covers the submanifold

¨ ú , namely

(2.13) ½ Ì / � � � / � � / ï ÷ ñ0� Í¶ ú ã ð Á �ú ð ø ú û m ´#ï�¶ ú ã ð Á �ú ð ø ú ñ ÄThis may be achieved easily by defining ¿ ã � Ì / � ï�� ñ ð ¿ �� / ï�� ñ �� × ã ï ÷ È � ñ û ¶ ú ã ð Á �ú and by solving the Bishop-type equation:

(2.14) Æ Ì / � � � / � � / ï � � { ñ0� ø ú È U ã � ´ n ¿ ã�� Ì / � ïtÑ ñ ð ¿ �� / ïtÑ ñ ð Æ Ì / � � � / � � / ïtÑ ñ p � ï � � { ñfor the ø -component of the sought disc ½ Ì / � � � / � � / . Thanks to Theorem 3.7(IV), the

solution exists and is D ï � é�û ú with respect to all the variables. We finally define the³ -component of ½ Ì / � � � / � � / :(2.15) È Ì / � � � / � � / ï � � { ñ�� U ã � Æ Ì / � � � / � � / ïtÑ ñ � ï � � { ñ û ´#ï�¶ ú ã ð Á �ú ð ø ú ñ ÄApplying

U ã to (2.14), we see that this disc is attached to ² ; also, putting � � { �� ÷in (2.14) and in (2.15), we see that (2.13) holds. Geometrically, the ïÏý{ È ý ñ addedparameters ï�¶ ú ã ð Á �ú ð ø ú ñ correspond to translations in ² of the original disc ½ Ö � .

� � � � � �� ÎGÑ ¸ Õ��

ç ý } þTranslates of the disc

exit vectorÝ ÏÙÐÏÙÚ Ó)ÔKÕ not tangent to ·� Ð � � �ç ý � } þ

Translations of an attached analytic disc


Define the open circular region ÿ ã �� 4 �Õ-wÿ � @ � È ÷ @ �J6 ã 5 around÷

in theunit disc. Then we claim that shrinking 6 ã " �

if necessary, the set¿ ½ Ì / � � � / � � / ï�� ñE� �#-wÿ ãåð @ ¶ ú ã @ �J6 ã�ð @ Á �ú @ �y6 ã�ð @ ø ú @ �J6 ã Ãcontains a one-sided neighborhood of ² at

i � ½ ú � ú � ú ï ÷ ñ . Indeed, by computa-tion, one may check that the ý{ vectors of V ú>¾ ¡ÿ ½ ú � ú � úÿ�¶ ã ï ÷ ñ ð ÿ ½ ú � ú � úÿ�Ì ï ÷ ñ ð ÿ ½ ú � ú � úÿ ¶ � � ï ÷ ñ ð ÿ ½ ú � ú � úÿ · �� ï ÷ ñ ð ÿ ½ ú � ú � úÿ ø ï ÷ ñ ð È ÿ�½ ú � ú � úÿ�Û ï ÷ ñ ðare linearly independent; geometrically and by construction, the first ïÏý{ È ÷ ñ

ofthese vectors span V ú�² and the last one is linearly independent, since by con-struction the exit vector of ½ Ö � is nontangent to ² at

i. Þ

Incidentally, we have proved an elementary but crucial statement: by “translat-ing” (through a suitable Bishop-type equation) any small attached disc whose exitvector is nonzero, we may always cover a one-sided neighborhood.

Lemma 2.16. If a small disc ½ attached to a hypersurface ² satisfies� ç� { ï ÷ ñ J-V 5ç ý ã þ ² , or equivalently È � ç� é ï ÷ ñ J- V ç ý ã þ ² , then continuous CR functions on ²

extend holomorphically at ½ ï ÷ ñ to the side in which points the nonzero exit vectorm � ç� { ï ÷ ñ�� È � ç� é ï ÷ ñ .Of course, the choice of the point

÷ -Sÿ©ÿ is no restriction at all, since after aMobius reparametrization, any given point � ú -�ÿPÿ becomes

÷ - ÿPÿ .

2.17. Propagation of holomorphic extension. The Levi form assumptionV ú�² � V 5ú ² û / V 5ú ² ð V 5ú ² 1 was strongly used to insure the existence of a dischaving a nonzero exit vector at

i. But if a disc ½ is attached to a highly degenerate

part of ² , for instance to a region where the Levi form is nearly flat, the disc ½might well satisfy

� ç� { ï�� ú ñ - V 5ç ý � / þ ² (or equivalently, È � ç� é ï�� ú ñ - V ç ý � / þ ² ),for every � ú -�ÿPÿ . Then we are stuck.

To go through, two strategies are known in the literature.! Devise refined pointwise “finite type” assumptions insuring the existenceof small discs having nonzero exit vector at a given central point.! Devise deformation arguments that propagate holomorphic extensionfrom Levi nondegenerate regions up to highly degenerate regions.

Unfortunately, the first, more developed strategy is unable to provide any proofof Theorem 2.6. Indeed, a smooth globally minimal hypersurface may well con-tain large Levi-flat regions, as for instance 4 ï Á ðKh ñ - ¾ ï � ³ � �4ï�¶ ñ 5 with a D+*function � satisfying �4ï�¶ ñ v �

for ¶ �Ó�and �3ï�¶ ñ " �

for ¶ " �(to check

global minimality, proceed as in Example 3.10); Theorem 4.8(III) shows that a


Levi-flat portion ²�� of a hypersurface ² is locally foliated by complex ï� È ÷ ñ -dimensional submanifolds; the uniqueness in Bishop’s equation16 then entails thatevery small analytic disc ½Ö-i×Qïsÿ ñ V�D ã ï ÿ ñ with ½xï�ÿ©ÿ ñ ! ²�� must satisfy½xï�ÿ©ÿ ñ ! ½ , where ½ ! ²�� is the unique local complex connected ï� È ÷ ñ

-dimensional submanifold of the foliation that contains ½xï ÷ ñ ; then the uniquenessprinciple for holomorphic maps between complex manifolds yields ½ ï ÿ ñ ! ½ ;finally, È � ç� é ï�� ú ñ - V ç ý � / þ ½ � V 5ç ý � / þ ² has exit vector tangential to ² at every� ú -�ÿPÿ .

For this reason, we will focus our attention only on the second, most powerfulstrategy, starting with a review.

After works of Sjostrand ([HS1982, Sj1982a, Sj1982b]) on propagation ofsingularities for certain classes of partial differential operators, of Baouendi-Chang-Treves [BCT1983], and of Hanges-Treves [HT1983], Trepreau [Trp1990]showed that the hypoanalytic wave-front set of a CR function or distributionpropagates along complex-tangential curves. The microlocal technique involvesdeforming V ª ² inside conic sets and controlling a certain oscillatory integralcalled Fourier-Bros-Iagolnitzer (FBI) transform. In 1994, Baouendi-Rothschild-Trepreau [BRT1994] showed how to deform analytic discs attached to a hyper-surface in order to get some propagation results (however, Theorem 2.7 whichappears in [Trp1990] is not formulated in [BRT1994]). Then Tumanov [Tu1994a]showed how to deformation discs attached to generic submanifolds of arbitrarycodimension and provided extension results that cannot be obtained by means ofthe usual microlocal analysis.

Until the end of Section 4, our goal will be to describe and to exploit thistechnique of propagation. The geometric idea is as follows.

As in Theorem 2.7, assume that holomorphic extension is already known tohold in a one-sided neighborhood � 2� at some point H�-õ² . Referring tothe diagram after the main Proposition 2.21 below, we may pick a disc ½ with½xï È ÷ ñE� H . Then a small part of its boundary, namely for � � { near È ÷ , lies in � 2� .If the vector

� ç� { ï ÷ ñ is not complex tangential at the opposite pointi � ½xï ÷ ñ , it

suffices to apply Lemma 2.16 just above to get holomorphic extension ati

, almostgratuitously. On the contrary, if

� ç� { ï ÷ ñ is complex tangential ati

, we may wellhope that by slightly deforming ² as a hypersurface ²�� which goes inside � 2�a bit, there exists a deformed disc ½ � attached to ² � with again ½ � ï ÷ ñZ� i

thatwill be not tangential: È � ç�� é ï ÷ ñ J- V ç � ý ã þx² . Then a translation of the disc ½ � asin Lemma 2.16 will provide holomorphic extension at

i.

16A more general property holds true (see [Trp1990, Tu1994a, MP2006b]): everysmall attached disc is necessarily attached to a single (local or global) CR orbit; here,�

is a local orbit, whence æ �£��â � .


2.18. Approximation theorem and chains of analytic discs. To prove Theo-rem 2.7, we first formulate a version of the approximation theorem which is app-propriate for our purposes.

Lemma 2.19. ([Tu1994a]) For everyi -}² , there exists a neighborhood Æ ú of

iin ² such that for every HE- Æ ú , for every one-sided neighborhood Æ 2� of Æ ú atH , there exists a smaller one-sided neighborhood � 2� ! Æ 2� of Æ ú at H such thatthe following approximation property holds:! for every Âo-}D ú ïÏ² ÃXÆ 2� ñ which is CR on ² and holomorphic in Æ 2� ,

there exists a sequence of holomorphic polynomials ï �0Å ï Á�ñsñ Å § such that� � '*) � Å + * @ @ �ÇÅ�È � @ @ / ý E O�� þ .The proof is an adaptation of Theorem 5.2(III). It suffices to allow the maxi-

mally real submanifolds � � ! ² be slightly deformed in Æ 2� . With a control ofthe smallness of their D ã norm, one may insure that they cover not only Æ ú butalso � 2� . Further details will not be provided.

To establish local holomorphic extension of CR functions, itis allowed to glue discs not only to ² but also to previouslyconstructed one-sided neighborhoods.

Pursuing, we formulate a lemma and a main proposition.

Lemma 2.20. ([Tu1994a]) Leti -0² and let Æ ú be a neighborhood of

iin ² ,

arbitrarily small. For every HC-J× 6 Á ïÏ² ð i ñ and every small × " �, there exista -EG with a � P ï ÷ gY× ñ and a chain of D ï � é�û ú analytic discs ½ ã ð ½ ï ð Ä Ä Ä ð ½��Rû ã ð ½��

attached to ² with the properties:! ½ ã ï È ÷ ñ - Æ ú , i.e. this point is arbitrarily close toi

;! ½ ã ï ÷ ñE� ½ ï ï È ÷ ñ , ½ ï ï ÷ ñE� ½ ± ï È ÷ ñ , . . . , ½��Rû ã ï ÷ ñ0� ½��'ï È ÷ ñ ;! ½ � ï ÷ ñ�� H ;! @ @ ½ � @ @ �� / ý } þ � × , for � � ÷ ð ý ð Ä Ä Ä ð a ;! each ½ � is an embedding ÿÓL ¾ ¡ .

�� "! � � ý � } þì �

� �$#&%(' ý ) þ ý ú þ� ç�* ý ã þç � ç ¥

E O úç � ý û ã þ ç�* r � ç *

String of analytic discs approximating a CR curve


Such a chain of analytic discs will be constructed by approximating a complex-tangential curve that goes from H to

i, using families of discs

s � � ì � � � ï�� ñ to beintroduced in a while. The above lemma is essentially obvious, whereas the nextproposition constitutes the very core of the argument.

Proposition 2.21. ([BRT1994, Tu1994a]) (Propagation along a disc) Let ½ bea small D ï � é�û ú analytic disc attached to ² which is an embedding ÿÒL ¾ ¡ .If D ú6 Á ïÏ² ñ

extends holomorphically to a one-sided neighborhood � 2ç ý û ã þ at the

point ½ ï È ÷ ñ , then it also extends holomorphically to a one-sided neighborhoodat ½xï ÷ ñ . With more precisions:! if the exit vector È � ç� é ï ÷ ñ is not tangent to ² at ½xï ÷ ñ , extension holds to

the side in which points È � ç� é ï ÷ ñ : this is already known, by Lemma 2.16;! if the exit vector È � ç� é ï ÷ ñ is tangent to ² at ½xï ÷ ñ , there exists an arbi-trarily small deformation ½+� of ½ with ½,��ï ÷ ñ-� ½ ï ÷ ñ having boundary½,��ï�ÿPÿ ñ contained in ² Ã � 2ç ý û ã þ such that the new exit vector È � ç-�� é ï ÷ ñis not tangent to ² at ½ � ï ÷ ñ ; then by translating ½ � as in Lemma 2.16,holomorphic extension holds at ½ ï ÷ ñ .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . ./ / / / / / // / / / / / // / / / / / // / / / / / // / / / / / // / / / / / /

0 0 00 0 00 0 01 1 11 1 11 1 12 2 22 2 22 2 22 2 22 2 2

3 33 33 33 33 3 û ÏÙÐ54ÏÙÚ ý ã þ not tangent to Äç ý ã þÄ ç ý û ã þ ç ý } þç-� ý } þ û ÏÙÐÏ�Ú ý ã þ tangent

to ÄTranslation after perturbation

ç-� ý û ã þ � Ð ��r � �Perturbation of the exit vector

Indeed, thanks to the flexibility of the solutions to the parametrized Bishopequation provided by Theorem 3.7(IV), we can easily, as in Lemma 2.16, addtranslation parameters ï�¶ ú ã ð Á �ú ð ø ú ñ to a slightly deformed disc ½ � attached to ²�Ã� 2ç ý û ã þ and then ½ �Ì / � � � / � � / ïsÿ ã ñ covers a small one-sided neighborhood of ² at½xï ÷ ñ-� ½ � ï ÷ ñ , thanks to the crucial condition È � ç �� é ï ÷ ñ J� �

. We shall not copythe details.

We claim that the proposition ends the proof of Theorem 2.7. By assumption,D ú6 Á ïÏ² ñextends holomorphically to a one-sided neighborhood � 2ú at

i. The

closure � 2ú contains an open neighborhood Æ ú ofi

. Let HJ- × 6 Á ïÏ² ð i ñ andconstruct a chain of analytic discs from H up to a point

i � - Æ ú . The endpointi � � ½ ã ï È ÷ ñ of the chain of analytic discs being arbitrarily close toi

, hencein Æ ú , holomorphic extension holds at ½ ã ï È ÷ ñ . We then apply the propositionsuccessively to the discs ½ ã ð ½ ï ð Ä Ä Ä ð ½ � and deduce holomorphic extension at H .


We now explain Lemma 2.20. To approximate a complex-tangential curve, itsuffices to construct families of analytic discs that are essentially directed alonggiven vectors ³ � - V 5� ² .

Lemma 2.22. For every point H�-�² and every nonzero complex tangent vec-tor ³ � - V 5� ² 3�4 ��5 , there exists a family of D ï � é�û ú analytic discs

s � � ì � � � ï�� ñparametrized by ��- Q

with @ � @ �N� ã , for some � ã " �, that satisfies:! s � � ì � � � ï�ÿ©ÿ ñ ! ² ;! H � s � � ì � � � ï ÷ ñ ;!�³ � � � � � � � � /� � ï È ÷ ñ ;! ®® ®® s � � ì � � � ®® ®® �� / ý } þ � ?�� , for some ? " �

.

Proof. In coordinates centered at H , represent ² by ³ � ´#ï Á ð ø ñ with ´#ï � ñ � �and ��´#ï � ñ � �

. The vector ³ � - V 5ú ² � 4 h � ��5has coordinates ï76Á � ð � ñ for

some nonzero 6Á � -X¾ ¡ û ã . Introduce the family of analytic discss � � ì � � � ï�� ñ�� ï��+6Á � ï ÷ È � ñ g�ý ð À � ï�� ñsñ ðwhere the real part Æ � of

À � is the unique D ï � é�û ú solution of the Bishop-typeequation: Æ � ï � � { ñ0� È

U ã�5 ´ ïÍ�,6Á � ï ÷ È Ñ ñ g�ý ð Æ � ïtÑ ñsñ 7 ï � � { ñ ÄProceeding as carefully as in Section 3(IV), we may verify that the assumption�q´#ï � ñE� �

implies that @ @ À � @ @ ã � ú � P ï @ � @ ï ñ . Then it is obvious that ³ � � ï86Á � ð � ñ0�� /� � ï È ÷ ñ . ÞWe now complete the proof of Lemma 2.20. Any point Hy-µ× 6 Á ïÏ² ð i ñ is

the endpoint of a finite concatenation of integral curves of sections 9 of V 5 ² . Itsuffices to construct the chain of discs for a single such curve

Ì Ë�j ï�� 9 ñ ï i ñ . Af-ter multiplying 9 by a suitable function, we may assume that H is the time-oneendpoint H �¾Ì Ë�j�ïÍ9 ñ ï i ñ .

Moving backwards, we start from H � �� H , we define ½ � ï�� ñ ��s � * �\û ) ý � * þ � ã c � ï�� ñ and we set H �Rû ã �� s � * �\û ) ý � * þ � ã c � ï È ÷ ñ . Clearly,H � û ã � H � È ã� 9Oï)H � ñ ûÎP ï ã� ¥ ñ . Starting again from H �Rû ã , we again movebackwards and so on, i.e. we define by descending induction:! ½ � ï�� ñ�� s � A �\û ) ý � A þ � ã c � ï�� ñ ;! H � û ã �� s � A �\û ) ý � A þ � ã c � ï È ÷ ñ ,until � � ÷

. Since H � û ã � H � È ã� 9Oï)H � ñ û7P ï ã� ¥ ñ for � � ÷ ð Ä Ä Ä ð a , the sequence ofpoints H � is a discrete approximation of the integral curve of 9 , hence the endpointH ú � ½ ã ï È ÷ ñ is arbitrarily close to

i, provided a is large enough. Finally, by

construction @ @ ½ � @ @ ã � ú � P ï ã� ñ . Þ


The proof of the main Proposition 2.21 does not use special features of hy-persurfaces. For this reason, we will directly deal with generic submanifolds ofarbitrary codimension, passing to a new section.

� 3. TUMANOV’S THEOREM, DEFORMATIONS OF BISHOP DISCS

AND PROPAGATION ON GENERIC MANIFOLDS

3.1. Wedges and CR-wedges. Assume now that ² is a connected generic sub-manifold in ¾ ¡ of codimension ��ü ÷

and of CR dimension H � � È &ü ÷. The

case � � ÷corresponds to a hypersurface. The notion of local wedge at a point

igeneralizes to codimension �Õüyý the notion of one-sided neighborhood at a pointof a hypersurface.

More briefly that was has been done in Section 4(III), a wedge may be definedas follows. Choose a � -dimensional real subspace

( ú of V ú>¾ ¡ satisfying V ú>¾ ¡ �V ú ( ú:9 V ú ² and a small convex open salient truncated cone | ú ! ( ú withvertex

i. Then a local wedge of edge ² at

iis:À ï Æ ú ð |�ú ñ�� 4 H û î � H$- Æ ú ð î -C|�ú 5 Ä

This is not yet the most effective definition. Up to shrinking open sets andparameter spaces, all definitions of local wedges will coincide. Concretely, thewedges we shall construct will always been obtained as unions of small pieces offamilies of analytic discs partly attached to ² . So we formulate all the technicalconditions that will insure that such pieces of discs cover a wedge.

Definition 3.2. A local wedge of edge ² ati

is a set of the form:Àµú �� ¿ ½ � � Ý Û � � { � @ � @ �i� ãåð @ Ü @ � Ü ãåð @ Ì @ �JÌ ãåð Û ã � Û�� ÷ Ã ðwhere, �}- Q z û ã is a rotation parameter, � ã " �

is small,

Ü - Q ï Á � z û ã is atranslation parameter,

Ü ã " �is small, Ì ã " �

is small, Û ã � ÷is close to

÷and½ � � Ý ï�� ñ , with �#- ÿ , is a parametrized family of D ï � é�û ú analytic discs satisfying:! ½ � � ú ï ÷ ñE� i for every � ;! the boundaries ½ � � Ý ï�ÿPÿ ñ are partly (sometimes completely) attached to² , namely ½ � � Ý ï � � { ñ -C² , at least for @ Ì @ � ± �ï ;! for every fixed � , the mapping ï Ü ð � � { ñ »L¨½ � � Ý ï � � { ñ is a diffeomorphism

from 4 @ Ü @ � Ü ã 5 Ã 4 @ Ì @ �NÌ ã 5 onto a neighborhood ofi

in ² ;! the exit vector È � ç / � /� é ï ÷ ñ is not tangent to ² ati

, namely it hasnonzero projection

< ö<;>=8? O ] ^ c ? O Ä ï È ÿ�½ � � ú g-ÿ�Û�ï ÷ ñsñ onto the normal spaceV úY¾ ¡ g V ú�² to ² ati

;

HOLOMORPHIC EXTENSIONS AND REMOVABLE SINGULARITIES 179! choose any linear subspace( ú of V ú>¾ ¡ satisfying V ú ( ú 9 V ú�² � V ú>¾ ¡ ,

so that( ú ë V ú ¾ ¡ g V ú ² , denote by

< ö<;>=A@ O � V ú ¾ ¡ L ( ú the projec-tion onto

( ú parallel to V ú ² , define the associated exit vector

ex ïÍ½ � � ú ñ�� < ö<;>=A@ O ï È ÿ ½ � � úÿ�Û ï ÷ ñsñ - ( úand the associated normalized exit vector n-ex ïÍ½ � � ú ñ ��ex ïÍ½ � � ú ñ g @ ex ïÍ½ � � ú ñ @ ; then the rank at � � �

of the mappingQ z û ã ¹H�ë» È L n-ex ïÍ½ � � ú ñ -CB z û ã ! Q zshould be maximal equal to � È ÷ .

A local (curved) wedge of edge G at $D �ý

· ·

EÔ

� D �

D � ü�Ý E

Ý Ôý � � F ÓHG(I sbu ÕJ

K OG �L �Ý L �G �

The last, most significant condition means that n-ex ïÍ½ � � ú ñ describes an openneighborhood of n-ex ïÍ½ ú � ú ñ in the unit sphere B z û ã ! Q z . This is of courseindependent of the choice of

( ú . Then, fixing

Ü � �and Ì � �

, as the rotationparameter �#- Q z û ã varies with @ � @ �Ó� ã , and as the radius Û with Û ã �¿Û0� ÷varies, the curves ½ � � ú ï³Û ñ generate an open truncated (curved) cone in some � -dimensional local submanifold transverse to ² at

i. Finally, as the translation

parameter

Üvaries, the points ½ � � Ý ï³Û � � { ñ describe a (curved) local wedge of edge² at

i.

Lemma 3.3. Shrinking � ã " �,

Ü ã " �, Ì ã " �

and÷ È Û ã " �

if necessary,the points of ÀSú are covered injectively: ½ � � Ý ï³Û � � { ñÊ� ½ � � Ý ï³Û � � � { ñ if and only if� � �B� , Ü � Ü � , Û � Û-� and Ì � Ì�� .

This property follows directly from all the rank conditions. It will be useful toinsure uniqueness of holomorphic extension (monodromy).

Definition 3.4. ([Tu1990, Trp1990]) A local CR-wedge of edge ² ati

of dimen-sion ýYH û � û � , with

÷7� � � � , is a set À 6 Á � rú defined similarly as a localwedge, but assuming that the rotation parameter � belongs to

Q r û ã and that therank of the normalized exit vector mappingQ r û ã ¹H�M» È L n-ex ïÍ½ � � ú ñ -CB z û ã ! Q z


is equal to � È ÷ .Then, fixing

Ü � �and Ì � �

, as the rotation parameter �.- Q r û ã with @ � @ �i� ãvaries, and as the radius Û with Û ã �vÛÕ� ÷

varies, the curves ½ � � ú ï³Û ñ describe anopen truncated (curved) cone in some � -dimensional local submanifold transverseto ² at

i. These intermediate wedges of smaller dimension will play a crucial

technical role in the sequel.The case � � ÷

deserves special attention. A CR-wedge is then just a manifoldwith boundary ² ãú with � ) �o² ãú � ÷ û � ) �:² that is attached to ² at

i, namely

there exists an open neighborhood Æ ú ofi

in ² with Æ ú ! ÿ2² ãú . If in addition² has codimension � � ÷, we recover the notion of one-sided neighborhood. It

is clear that after a possible shrinking, every D ï � é�û ú manifold with boundary ² ãúattached to ² at

imay be prolonged as a local D ï � é�û ú generic submanifold � ãú vÀ 6 Á � ãú containing a neighborhood of

iin ² (as shown in the right diagram).

M M M MM M M MM M M MM M M MM M M MM M M MM M M MN N N NN N N NN N N NN N N NN N N NN N N NN N N N

O O O O O OO O O O O OO O O O O OO O O O O OO O O O O OO O O O O OO O O O O OP P P P P PP P P P P PP P P P P PP P P P P PP P P P P PP P P P P PP P P P P P] ^

ú ·· ·] ^

úQSRUT � VO

Prolongation of a CR-wedge as a generic submanifold

W VO W VOQ RUT � VO

By elementary differential geometry, for � üÊý , it may be verified that a localCR-wedge À 6 Á � rú of edge ² at

idefined by means of a D ï � é�û ú family of discs,

namelyÀ 6 Á � rú �� ¿ ½ � � Ý Û � ��X � @ � @ �i� ã�ð @ Ü @ � Ü ãåð @ Ì @ �NÌ ãåð Û ã �\Û$� ÷ Ã ðmay also be prolonged as a local generic submanifold � rú of dimension ýYH û � û �containing a neighborhood of

iin ² . The left diagram is an illustration; in it,� � � � ý , so that ² of codimension ý is (unfortunately for intuition) collapsed

toi

.However, the smoothness of � rú can decrease to D ã � é�û ú , because as in

a standard local blowing down ï Á ã�ð Á ï ñ »L ï Á ã�ð Á ã Á ï ñ , the rank of the mapï³Û ð Ì ð Ü ð � ñ » È L ½ � � Ý Û � ��X degenerates when Û � ÷, since the discs (partial)

boundaries ó�½ � � Ý � ��X � @ Ì @ � ± �ï ô are constrained to stay in ² . For techni-cal reasons, we will need in the sequel the existence of a prolongation � rú that isD ï � é�û ú also when � ü�ý . The following modification of the definition of À 6 Á � rúinsures the existence of a D ï � é�û ú prolongation � rú . It will be applied implicitlyin the sequel without further mention.


So, assume � ü¨ý , let ½ � � Ý be a family of discs as in Definition 3.4 withex ïÍ½ ú � ú ñ J� �

in V ú ¾ ¡ g V ú ² and �(»L n-ex ïÍ½ � � ú ñ of rank � È ÷at � � �

.Fix � ��

and define firstlyÀ 6 Á � ãú �� ó ½ ú � Ý ï³Û � ��X ñ�� @ Ü @ � Ü ãåð @ Ì @ �JÌ ãåð Û ã � Û$� ÷ ô�ÄThis is a manifold with boundary attached to ² at

i. So there is a small D ï � é�û ú

prolongation � ãú > À 6 Á � ãú .Choose � J� �

small with ½ � � ú having exit vector nontangent to � ãú ati. Introduce a one-parameter family ² í

, ìÔ- Q, @ ì @ � ì ã , ì ã " �

, ofgeneric submanifolds obtained by deforming slightly ² inside � ãú near

i, with² í ! ² ÃKÀ 6 Á � ãú for ìAü �

. The ² íare “translates” of ² in � ãú near

i. To

understand the process, we draw two diagrams in different dimensions.

Y Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YZ Z Z Z Z ZZ Z Z Z Z ZZ Z Z Z Z ZZ Z Z Z Z ZZ Z Z Z Z ZZ Z Z Z Z ZZ Z Z Z Z Z

ú � ÄQSR�T � ¥O Q[R�T � �O

W �O W ¥O ÄW �O QSR�T � �O ÄÄ]\Ä \

Construction of a ^ ¥ � % r / prolongation _ VO ofK R�T � VO

ç � � F�� \ ý ` þ

Thanks to the flexibility of Bishop’s equation (Theorem 3.7(IV)), the ½ � � Ý maybe deformed as a D ï � é�û ú family ½ � � Ý � í and we define secondlyÀ 6 Á � ïú �� ó ½ � � Ý � í ï³Û � ��X ñ�� @ Ü @ � Ü ãåð � �Nì0�Nì ãåð @ Ì @ �NÌ ãÒð Û ã � Û�� ÷ ô ÄThen this set constitutes a local CR-wedge of dimension ýYH û � û ý with edge² at

i. Letting ì run in ï È ì ã�ð ì ã ñ above, we get instead a certain manifold with

boundary attached to � ãú that may be extended as a D ï � é�û ú generic submanifold� ïú of dimension ýYH û � û ý . Then À 6 Á � ïú is essentially one quarter of � ïú .We neither draw À 6 Á � ïú nor À ïú in the right diagram above, but the reader seesthem. By induction, using that � »L n-ex ïÍ½ � � ú ñ has rank � È ÷

at � � �, we get

the following.

Lemma 3.5. After a possible shrinking, a suitably constructed local D ï � é�û ú CR-wedge À 6 Á � rú of edge ² at

imay be prolonged as a local D ï � é�û ú generic sub-

manifold � rú of dimension ýYH û � û � containing a neighborhood ofi

in ² .

In the sequel, similar technical constructions will be applied to insure the exis-tence of D ï � é�û ú prolongations � rú > À 6 Á � rú without further mention.


3.6. Holomorphic extension of CR functions in higher codimension. In 1988,Tumanov [Tu1988] established a theorem that is nowadays celebrated in SeveralComplex Variables. Recall that by definition, ² is locally minimal at

iif the

local CR orbit × ä�4 56 Á ïÏ² ð i ñ contains a neighborhood ofi

in ² . Equivalently, ²does not contain any local submanifold D passing through

iwith a Ë ��) �1D �a Ë � ) �N² and � ) �UD�� ) �N² .

Theorem 3.7. ([Tu1988, BRT1994, Trp1996, Tu1998, BER1999]) Let ² be alocal D ï � é generic submanifold of ¾ ¡ and let

i -0² . If ² is locally minimal ati, then there exists a local wedge ÀVú of edge ² at

isuch that every ��-ED ú6 Á ïÏ² ñ

possesses a holomorphic extension ÂÎ-C×Qï³ÀVú ñ VSD ú ïÏ² Ã Àµú ñ with Â @ Ä � � .

Conversely, recall that according to Theorem 4.41(III), if ² is not locally min-imal at

i, there exists a local continuous CR function that is not holomorphically

extendable to any local wedge ati

.Since the literature already contains abundant restitutions17, we will focus in-

stead on propagation phenomena that are less known.

In 1994, as an answer to a conjecture formulated by Trepreau in [Trp1990],it was shown simultaneously by Joricke and by the first author that Tumanov’stheorem generalizes to globally minimal ² . The preceding statement is a directcorollary of the next. Its proof given in [Me1994, Jo1996] used techniques andideas of Tumanov [Tu1988, Tu1994a] and of Trepreau [Trp1990].

Theorem 3.8. ([Me1994, Jo1996]) Let ² be a connected D ï � é generic submani-fold of ¾ ¡ . If ² is globally minimal then at every point

i -}² , there exists a localwedge ÀSú of edge ² at

isuch that every continuous CR function ��-CD ú6 Á ïÏ² ñ

possesses a holomorphic extension ÂÎ-C×Qï³ÀVú ñ VSD ú ïÏ² Ã Àµú ñ with Â @ Ä � � .

With this statement, the extension theorem for CR function has reached a final,most general form. Philosophically, the main reason why it is true lies in thepropagation of holomorphic extendability along complex-tangential curves. Thiswas developed by Trepreau in 1990, using microlocal analysis.

Theorem 3.9. ([Trp1990]) Let ² be a connected D * generic submanifold of ¾ ¡ .If D ú6 Á ïÏ² ñ

extends holomorphically to a local wedge at some pointi -�² , then

at every point HK-�× 6 Á ïÏ² ð i ñ , there exists a local wedge À � of edge ² at Hsuch that every ��-7D ú6 Á ïÏ² ñ

possesses a holomorphic extension Â:-0×Qï³À � ñ VD ú ïÏ² Ã À � ñ with Â @ Ä � � .

17We recommend mostly the two elegant presentations [Trp1996] and [Tu1998]; otherreferences are: [BRT1994, BER1999]. Excepting a conceptual abstraction involving theimplicit function theorem in Banach spaces and the conormal bundle to G , the majorarguments: differentiation of Bishop’s equation and a crucial correspondence between anexit vector mapping and an evaluation mapping defined on the space of discs attached toG , the geometric structure of the proof is exactly the same in the original article [Tu1988]as in the restitutions.


Before surveying the original proof ([Me1994, Jo1996]) of this theorem in Sec-tion 5, we shall expose in length a substantially simpler proof of Theorem 3.8 thatwas devised by the second author in [Po2004]. This neat proof treats locallyand globally minimal generic submanifolds on the same footing. It relies partlyupon a natural deformation proposition due to Tumanov in [Tu1994a], but withoutany notion of defect of an analytic disc, without any needs to control the varia-tion of the direction of CR-extendability, and without any partial connection, asin [Trp1990, Tu1994a, Me1994]. The next paragraphs and Section 4 are devotedto the proof of this most general Theorem 3.8.

Example 3.10. A globally minimal manifold may well be not locally minimal atany point.

Indeed, let b � Q L Q �be D * with b � �

on ï È > ð ÷ 1 , with b " �onï ÷ ð û > ñ

and with second derivative b ÌåÌ " �on ï ÷ ð û > ñ

. Consider the genericmanifold ² of ¾ ± defined by the two equations³ ã � b#ï�¶ ñ ð ³ ï � b#ï È ¶ ñ ðin coordinates ï�¶ û m · ð ø ã û m ³ ã�ð ø ï û m ³ ï ñ . Then V ã � ú ² is generated by9 � ÿÿ Á û m b Ì ï�¶ ñ ÿÿ h ã È m b Ì ï È ¶ ñ ÿÿ h ï ÄIn terms of the four coordinates ï�¶ ð · ð ø ã ð ø ï ñ on ² , the two vector fields gener-ating V 5 ² are 9 ã �� ý ËZÌ 9 � ÿÿ ¶ ð9 ï �� ýdc��J9 � ÿÿ · û b Ì ï�¶ ñ ÿÿø ã È b Ì ï È ¶ ñ ÿÿ ø ï(we have dropped b Ì ï�¶ ñ �� ì � È b Ì ï È ¶ ñ �� ì ¥ in ý ËZÌ 9 ). Denote by e ú the systemof these two vector fields 4 9 ã ð 9 ï 5 on

Q ¤ ë�² and by e the D * ï Q ¤ ñ -hull of e ú .Observe that the Lie bracket5 9 ã ð 9 ï 7 � b ÌåÌ ï�¶ ñ ÿÿø ã û b ÌåÌ ï È ¶ ñ ÿÿø ïis zero at points

i � ï�¶<ú ð · ú ð ø ú ã ð ø ú ï ñ with È ÷ ��¶�ú�� ÷, has non-zero

�� ¥ -

component at pointsi

with ¶<úH� È ÷ and has non-zero�� -component at pointsi

with ¶ ú " ÷. It follows that the local e -orbit of a point

iwith ¶ ú � È ÷ is4 ø ã � ø ú ã 5 , of a point

iwith È ÷ �i¶<ú�� ÷

is 4 ø ã � ø ú ã ð ø ï � ø ú ï 5 and of a point¶ ú with ¶ ú " ÷is 4 ø ï � ø ú ï 5 . Also, observe that since the vector field 9 ã � �� Ì

belongs to e , the local e -orbit of any pointi � ï�¶�ú ð · ú ð ø ú ã ð ø ú ï ñ contains points of

coordinates ï�¶ ú û � ð · ú ð ø ú ã ð ø ú ï ñ , with � small. We deduce that the local e -orbit of


pointsi

with ¶ ú � È ÷ or ¶ ú � ÷are three-dimensional, hence in conclusion:× ä�4 5f ï Q ¤ ð i ñ�� agb gc Æ úZV 4 ø ã � ø ú ã 5 ) p ¶ ú � È ÷ ðÆ úZV 4 ø ã � ø ú ã ð ø ï � ø ú ï 5 )qp È ÷ �i¶ ú�� ÷ ðÆ úZV 4 ø ï � ø ú ï 5 ) p ¶ ú ü ÷ ð

where Æ ú is a neighborhood ofi

in ² . So e is nowhere locally minimal.

Lemma 3.11. The system e is globally minimal.

Proof. We check that any two pointsi ð HK- Q ¤ are in the same e -orbit. Using

the flow of 9 ã � �� Ì and then the flow of 9 ï on 4 ¶ � ��5, the original two

pointsi

and H may be joined to points, still denoted byi � ï � ð � ð ø ú ã ð ø ú ï ñ andH � ï � ð � ð ø � ã ð ø �ï ñ , having zero ¶ -component and zero

·-component.

We claim that the global e -orbit × f ï Q ¤ ð i ñ of every pointi � ï � ð � ð ø ú ã ð ø ú ï ñ

contains a neighborhood ofi

in

Q ¤ . Since the two-dimensional plane 4 ¶ � · ��5is connected, this will assure that any two points

i � ï � ð � ð ø ú ã ð ø ú ï ñ and H �ï � ð � ð ø � ã ð ø �ï ñ are in the same e -orbit.Indeed, by means of

�� Ì , every pointi � ï � ð � ð ø ú ã ð ø ú ï ñ is joined to the two

pointsi û �� ï È ÷ ð � ð ø ú ã ð ø ú ï ñ and

i � �� ï ÷ ð � ð ø ú ã ð ø ú ï ñ . Let Æ ú r and Æ úih be smallneighborhoods of

i û and ofi �

. Denote by( û �� 4 ø ã � ø ú ã 5 V Æ ú r and by( � �� 4 ø ï � ø ú ï 5 V Æ úih small pieces of the three-dimensional local e -orbits ofi û and of

i �. ÷ ¥j r j h

k O�lk O�m k O n÷ �

JJ r J h·poÜSqsr

Verification that t�u vxw Ó ü Õ Ü[y / q r

z|{i} Ó Ý�~ � Õz�{�} Ó ~ � Õ

The flow of 9 ã � �� Ì being a translation, we deduce:Ì Ë�j ïÍ9 ã ñ ï ( û ñ�� 4 ø ã � ø ú ã 5 V Æ ú ðÌ Ëqj ï È 9 ã ñ ï ( � ñ�� 4 ø ï � ø ú ï 5 V Æ ú ðwhere Æ ú is a small neighborhood of

iin ² ë Q ¤ . Observe that the two . -

dimensional planes are transversal in V ú Q ¤ . Lemma 1.28(III) yields:e Z\[>� ï i û ñ > V ú r × ä�4 5f ï i û ñ�� 4 ø ã � ø ú ã 5 ðe Z\[>� ï i � ñ > V ú h × ä�4 5f ï i � ñ�� 4 ø ï � ø ú ï 5 Ä


By the very definition of e Z\[�� , we necessarily have:

e Z\[�� ï i ñ > Ì Ëqj ïÍ9 ã ñ ª e Z\[�� ï i û ñ û Ì Ëqj ï È 9 ã ñ ª e Z\[>� ï i � ñ � 4 ø ã � ø ú ã 5 û 4 ø ï � ø ïh5� V ú Q ¤ ðso e Z\[�� ï i ñ � V ú Q ¤ . Consequently, × f ï Q ¤ ð i ñ contains a neighborhood ofï � ð � ð ø ú ã ð ø ú ï ñ in

Q ¤ . Þ3.12. Setup for propagation. Let ² be connected, generic and D ï � é , let HÕ-(²and let À 6 Á � r� be a CR-wedge of dimension ýYH û � û � at H , with

÷ñ� � � � . Forshort, we will say that D ú6 Á ïÏ² ñ

extends to be CR on À 6 Á � r� if for every �7-ED ú6 Á ,there exists ÂÓ-ED ú6 Á ïÏ²�ÃQÀ 6 Á � r� ñ

with Â @ÅÄ � � .

Theorem 3.13. Let � -�G with÷&� � � � . Assume that D ú6 Á ïÏ² ñ

extends tobe CR on a CR-wedge À 6 Á � rú of dimension ýYH û � û � at some point

i -N² .Then for every H�-C× 6 Á ïÏ² ð i ñ , there exists a CR-wedge À 6 Á � r� at H of the samedimension ýYH û � û � to which D ú6 Á ïÏ² ñ

extends to be CR.

In the case � � � , we recover18 Trepreau’s Theorem 3.9, since continuous CRfunctions on an open set of ¾ ¡ (here a usual wedge) are just holomorphic. If ² isglobally minimal, then extension holds at every H$-}² . Notice that this statementcovers the propagation Theorem 2.7, stated previously in the hypersurface case� � � � ÷

.

Let us start the proof. Through a chain of small analytic discs, every HÊ-× 6 Á ïÏ² ð i ñ is joined to a pointi � arbitrarily close to

i: indeed, Lemma 2.20 and

its proof remain the same in arbitrary codimension �Kü ÷. At

i � , CR extensionholds, because the edge of À 6 Á � rú contains a small open neighborhood Æ ú of

iin² . To deduce CR extension at H , it suffices therefore to propagate CR extension

along a single disc, as stated in the next main proposition.

18Classical microlocal analysis was devised to measure the analytic wave front set of adistribution in terms of the exponential decay ot the Fourier transform restricted to open,conic submanifolds of the cotangent bundle. We suspect that there might exist highergeneralizations of microlocal analysis in which one takes account of the good decay ofthe Fourier transform on submanifolds of positive codimension in the cotangent bundle.


@ Oç �� ý ` þQSRUT � VÐ ��r � �6 O ÄÄ ç ý û ã þ ç ý ` þ ú

Normal deformations of an analytic disc

Proposition 3.14. (Propagation along a disc) ([Tu1994a, MP1999], [ � ]) Let ½be a small D ï � é�û ú analytic disc attached to ² which is an embedding ÿ:L ¾ ¡ .Let � -7G with

÷ � � � � . Assume that there exists a D ï � é�û ú CR-wedge À 6 Á � rç ý û ã þat ½xï È ÷ ñ of dimension ýYH û � û � to which D ú6 Á ïÏ² ñextends to be CR. Then

there exists a D ï � é�û ú CR-wedge À 6 Á � rç ý ã þ at ½xï ÷ ñ of the same dimension ýYH û � û �to which D ú6 Á ïÏ² ñ

extends to be CR.With more precisions, the CR-wedge À 6 Á � rç ý ã þ is constructed by translating a

certain family of analytic discs ½ � having the following properties. Settingi ��½xï ÷ ñ , there exists a D ï � é�û ú family ½ � of analytic discs, � � - Q r

, @ � � @ �i� � ã , � � ã " �,

with ½ � @ � � ú � ½ , with ½ � ï ÷ ñw� i, satisfying ½ � ï � ��X ñ -i² for @ Ì @ � ± �ï and

having their boundaries ½ � ï�ÿ©ÿ ñ ! ² ÃQÀ 6 Á � rç ý û ã þ for � � belonging to some open

truncated cone1 � ! Q r

, such that the exit vector mapping:Q r ¹H� � » È L ex ïÍ½ � ñÊ� < ö<;>=�@ O � È ÿ�½ � ÿ�Û ï ÷ ñ|� - Q zis of maximal rank equal to � at �s� � �

, where( úbë Q z is any linear subspace

of V ú>¾ ¡ such that( ú 9 V úf² � V ú>¾ ¡ , and where

< ö<;>= @ O is the linear projectionparallel to V ú ² .

Geometrically, as �� varies, the exit vectors ex ïÍ½ � ñ describe an open cone |*ú !( ú , drawn in the diagram.

We claim that this statement covers the second, delicate case of Proposi-tion 2.21. Indeed assuming that � � � � ÷

and that the exit vector È � ç� é ï ÷ ñis tangent to ² at ½xï ÷ ñ , the above proposition includes ½ in a one-parameterfamily ½ � whose direction of exit in the normal bundle has nonzero derivativewith respect to �� . Hence for every nonzero �s� , the direction of exit of ½ � is nottangent to ² at

i. Thus, a non-tangential deformed disc ½�� as in Proposition 2.21

may be chosen to be any ½ � , with ��J� �.

Proof of Proposition 3.14. We first explain how to get CR extension ati

from thefamily ½ � , taking for granted its existence.


(I) Suppose firstly that the exit vector of ½ � ½ ú is non-tangential to ² ati

. Wehave to restrict the parameter space � ��- Q r

to some parameter space ��- Q r û ã soas to reach Definition 3.4.

Let us take for granted the fact that the exit vector mapping has rank � at �� .

Then the normalized exit vector mappingQ r ¹�� » È L n-ex ïÍ½ � ñ�� ex ïÍ½ � ñ g @ ex ïÍ½ � ñ @ -CB z û ãhas rank ü � È ÷ at � � � �

. So there exists a small piece of an ï � È ÷ ñ -dimensionallinear subspace � ú of

Q z , parameterized as � � �� ï�� ñ for some linear map�

, with�¯- Q r û ã small, namely @ � @ �i� ã , for some � ã " �, such that �M»L n-ex ïÍ½+� ý � þ ñ has

rank ï � È ÷ ñ at � � �.

Setting ½ � �� ½ � ý � þ , we thus reach Definition 3.4, without the translation pa-rameter

Ü.

But proceeding exactly as in the hypersurface case, it is easy to include sometranslation parameter getting a family ïÍ½ � ñ Ý � ½ � � Ý . The proof is postponedto the end � 3.24. Then the desired family ½ � � Ý of the proposition is just ½�� ý � þ � Ý ,shrinking � ã " �

and

Ü ã " �if necessary.

Lemma 3.15. There exists a deformation ½ � � Ý of ½ � , with

Ü - Q ï Á � z û ã , @ Ü @ �Ü ã , such that:! the boundaries ½ � � Ý ï�ÿPÿ ñ are contained in ² Ã À 6 Á � rç ý û ã þ and ½ � ï � ��X ñ -² for @ Ì @ � ± �ï ;! for every fixed �� , the mapping ï Ü ð � ��X ñ » È L¾½ � � Ý ï � ��X ñ is a diffeomorphismfrom 4 @ Ü @ � Ü ã 5 Ã 4 @ Ì @ �NÌ ã 5 onto a neighborhood of

iin ² .

Therefore, the final family ½ � � Ý yields a CR-wedge À 6 Á � rú ati � ½xï ÷ ñ , as

in Definition 3.4. The mild generalization of the approximation Theorem 5.2(III)stated as Lemma 2.19 above in the case � � ÷

holds in the general case �Aü ÷without modification. Consequently, D ú6 Á ïÏ² ñ

extends to be CR on À 6 Á � rú .

(II) Suppose secondly that the exit vector of ½ � ½ ú is tangential to ² ati

.Thanks to the fact that the exit vector mapping has rank � at � � � �

, for everynonzero �B� ú , the disc ½ � / is nontangential to ² at

i. In this case, we fix a small�B� ú J� �

and we proceed with ½ � � � / just as above.

In summary, it remains only to construct the family ½ � having the crucial prop-erty that the exit vector mapping has rank � at � � � �

. Þ3.16. Normal deformations of analytic discs. Thus, we now expose how to con-struct ½ � . We shall introduce a parameterized family ² � of D ï � é�û ú genericsubmanifolds by pushing ² near ½xï È ÷ ñ inside À 6 Á � rç ý û ã þ in � independent nor-

mal directions, � being the number of degrees of freedom offered by À 6 Á � rç ý û ã þ .Outside a neighborhood of ½xï È ÷ ñ , each ² � shall coincides with ² and also² � @ � � ú � ² .


We may assume that the pointi �� ½ ï ÷ ñ is the origin in coordinates ï Á ð ø ûm ³ ñ -9¾ Á Ã ¾ z in which ² is represented by ³ � ´#ï Á ð ø ñ , where ´ satisfies´#ï � ñE� �

and ��´#ï � ñE� �. Let ��2- Q r

be small, namely @ �s� @ �N�� ã , with �� ã " �.

In terms of graphing equations, the deformation ² � may be represented by³ � Ú"ï Á ð ø ð � � ñ ðwith Ú -7D ï � é�û ú defined for @ � � @ �°� � ã satisfying Ú"ï Á ð ø ð � ñ v ´#ï Á ð ø ñ . The point½xï È ÷ ñ has small coordinates ï Á û ã�ð ø û ã û m ´#ï Á û ã0ð ø û ã ñsñ . We require that the �vectors Ú � A ï Á û ã�ð ø û ã�ð � ñ ð � � ÷ ð Ä Ä Ä ð � ðare linearly independent. There exists a truncated open cone

1 � ! Q rwith the

property that ² � ! ² Ã À 6 Á � rç ý û ã þ ðwhenever �B�Â- 1 � . In fact, we implicitly assume in Proposition 3.14 that the CR-wedge based at ½xï È ÷ ñ may be extended as a D ï � é�û ú generic submanifold � rç ý û ã þof dimension ýYH û � û � passing through ½xï È ÷ ñ so that ² � is contained in² Ã7� rç ý û ã þ , for every @ � � @ �� ã . The original CR-wedge À 6 Á � rç ý û ã þ may then be

viewed as a curved real wedge of edge ² which is contained inside � 6 Á � rç ý û ã þ .The starting D ï � é disc ½xï�� ñ�� ï`¿ ï�� ñ ð À ï�� ñsñ with

À ï�� ñÊ� ï Æ ï�� ñ û m È ï�� ñsñ isattached to ² with ½ ï ÷ ñ0� �

. Equivalently:ab c È ï � ��X ñ�� ´ n ¿ ï � ��X ñ ð Æ ï � �(X ñ p ðÆ ï � ��X ñ�� ÈU ã 5 ´ ¿ ïtÑ ñ ð Æ ïtÑ ñ '7 ï � �(X ñ ð

for every � ��X - ÿPÿ . Thanks to the existence Theorem 3.7(IV), there existsa D ï � é�û ú deformation ½ � of ½ , where each ½ � ï�� ñU�� ïx¿ ï�� ñ ð À ï�� ð �B� ñsñ with½ � ï ÷ ñ�� i has the same

Á-component19 as ½ and is attached to ² � , namely:

(3.17)

ab c È ï � ��X ð � � ñ�� Ú n ¿ ï � ��X ñ ð Æ ï � �(X ð � � ñ ð � � p ðÆ ï � ��X ð � � ñ�� ÈU ãÝ5 Ú"ï`¿ ïtÑ ñ ð Æ ïtÑ ð � � ñ ð � � ñ 7 ï � �(X ñ ð

for every � ��X -vÿ©ÿ . Observe that

À ï � ��X ð � ñ v À ï � ��X ñ . We then differentiatethe first line above with respect to � � � at � � � �

, for � � ÷ ð Ä Ä Ä ð � , which yields inmatrix notation:(3.18)È � A ï � �(X ð � ñ�� Ú � n ¿ ï � ��X ñ ð Æ ï � �(X ñ ð � p Æ � A ï � ��X ð � ñ û Ú � A n ¿ ï � ��X ñ ð Æ ï � �(X ñ ð � p Ä

19Since first order partial derivatives `S� �� >��x�� , �·ç ½'�(��(� �� , will appear in a while,we do not write the parameter �"� as a lower index in ^"��>��"�(�� ã _"��g�x� � .


Also, the D ã � é�û ú discs ½ � A ï�� ð � ñ satisfy the linear Bishop-type equationÆ � A ï � ��X ð � ñ0� ÈU ã � Ú � ï`¿ ïtÑ ñ ð Æ ïtÑ ñ ð � ñ Æ � A ïtÑ ð � ñ û Ú � A ï`¿ ïtÑ ñ ð Æ ïtÑ ñ ð � ñ � ï � ��X ñ Ä

As a supplementary space to V ú�² in V ú>¾ ¡ , we choose( ú �� 4 ��5 Ã m Q z �4 h � � ð ø � ��5

. Then< ö<;>=A@ O ï È ÿ�½ � ï ÷ ñ g-ÿ�Û ñv� È ÿ È ï ÷ ð �B� ñ g-ÿ�Û , which yields

after differentiating with respect to � � � at � � � �:

(3.19)ÿÿ�� ®®®® � � ú < ö<;>= @ O � È ÿ�½ � ÿ�Û ï ÷ ñ � � È ÿ È � Aÿ�Û ï ÷ ð � ñ ð

for � � ÷ ð Ä Ä Ä ð � . We will establish that if the local deformations ² � of ²inside the CR-wedge À 6 Á � rç ý û ã þ are concentrated in a sufficiently thin neighborhoodof ½xï È ÷ ñ , then the above � vectors È ÿ È � A g-ÿ�Û�ï ÷ ð � ñ , � � ÷ ð Ä Ä Ä ð � , are linearlyindependent. This will complete the proof of the proposition.

There is a singular integral operator � which yields the interior normal deriva-tive at

÷ - ÿ©ÿ of any D ã � é�û ú mapping ³ � ÿÎL Q z which is harmonic in ÿ andvanishes at

÷ - ÿ©ÿ :

(3.20) �3ï ³ ñ�� j Ä k Ä ÷� � �û � ³ ï � ��X ñ@ � ��X È ÷ @ ï ��Ì � È ÿ ³ÿ�Û ï ÷ ñ ÄThe proof is postponed to Lemma 3.25 below. If c � ÿ?L ¾ z is D ã � é�û ú andholomorphic in ÿ , we have in addition� ï c ñ0� È ÿ cÿ�Û ï ÷ ñ�� m ÿ cÿ Ì ï ÷ ñ ÄWith the singular integral � , we may thus reformulate (3.19):< ö<;>= @ O � È ÿ ï ½ úÿ�� ÿ�Û ï ÷ ñ � � ��ï È � A ñ ÄLemma 3.21. Let ø ð ³ -HD ã � é�û ú ï ÿ ð Q z ñ be harmonic in ÿ and vanish at

÷ -Xÿ©ÿ .Then: � � � ï�ø ³ È U ã ø U ã ³ ñ ÄIn addition, ø (and also ³ ) satisfies the two equations:� ï�ø ñ�� È ÿ�ï

U ã ø ñÿ Ì ï ÷ ñ l W � � ï U ã ø ñ�� ÿ øÿ Ì ï ÷ ñ ÄProof. The holomorphic product h �� ï�ø û m U ã ø ñ ï ³ û m U ã ³ ñ vanishes to secondorder at

÷ -�ÿPÿ , so ��ï h ñ0� �, hence� �¾ËZÌ ��ï h ñE� � ï�ø ³ È U ã ø U ã ³ ñ Ä

The pair of equations satisfied by ø is obtained by identifying the real and imagi-nary parts of � ï c ñ�� m �� X ï ÷ ñ , where c �� ø û m U ã ø . Þ


Following [Tu1994a], we now introduce a � Ã � matrix � of D ã � é functions onÿ©ÿ defined by the functional equation�·ï � ��X ñ0� } û U ã / �·ïtÑ ñ Ú � ï`¿ ïtÑ ñ ð Æ ïtÑ ñ ð � ñ 1 ï � ��X ñ ÄHere Ú � � ï`Ú î �~å ñ ãKJ î J zãKJ ä J z is a � Ã � matrix. Since Ú � ï Á ð ø ð � ñ v ´ � ï Á ð ø ñ is small,the solution � exists and is unique, by an application of Proposition 3.21(IV).Notice that �·ï ÷ ñE� } . Applying

U ã to both sides, we get

U ã � � È �&Ú � û �a�Re Ä ,without writing the arguments. In fact, the constant vanishes, since Ú � ï � ð � ð � ñ��´ � ï � ð � ñE� �

. So we get:U ã � � È �&Ú � Ä

We also notice that È � A � U ã Æ � A and Æ � A � ÈU ã È � A .

Next, we rewrite (3.18) without arguments: Ú � A � È � A È Ú � Æ � A , � � ÷ ð Ä Ä Ä ð � ,we apply the matrix � to both sides, we replace �0Ú � by È

U ã � as well as Æ � A byÈU ã È � A and we let appear a term ø ³ È U ã ø U ã ³ :�0Ú � A � � È � A È �&Ú � Æ � A� � È � A È ï U ã � ñ ï U ã È � A ñ� È � A û ïM� È } ñ È � A È U ã ïM� È } ñ U ã È � A Ä

Finally20, applying the singular operator � and remembering Lemma 3.21, weobtain:

(3.22) ��ïM�0Ú � A ñ�� 3ï È � A ñ ÄWe claim that if the support of the deformation ² � is sufficiently concentratednear ½xï È ÷ ñ , the � vectors ��ï È�� A ñ�� ïM�&Ú � A ñ - Q z are linearly independent.

Indeed, since the deformations ² � are localized near ½xï È ÷ ñ , we haveÚ � A ï`¿ ï � �(X ñ ð Æ ï � ��X ñ ð � ñ v �, except for @ Ì û � @ �ÖÌ ï , with Ì ï " �

small. Wededuce:

(3.23)��ïM�&Ú � A ñÊ� ÷� � ó X � � ó ÷ X ¥ �·ï � �(X ñ Ú � A ¿ ï � ��X ñ ð Æ ï � ��X ñ ð � @ � ��X È ÷ @ ï ��Ì� ÷� �·ï È ÷ ñj � ó X � � ó ÷ X ¥ Ú � A b¿ ï � ��X ñ ð Æ ï � ��X ñ ð � ��Ì Ä

Since, by assumption, the � vectors Ú � A ï Á û ã�ð ø û ã�ð � ñ are linearly independent, thelinear independence of the above (concentrated) vector-valued integrals follows.

The proofs of Proposition 3.14 and of Theorem 3.13 are complete. Þ20We can also check that � �)^�� Oç â � �£� ¸ _�� ç �û_�� ½'� ¹0��Pç ¹ . Indeed, it

suffices to differentiate (3.18) with respect to � at �4ç ¹ , noticing that �¡ ��n¹�� ¹��'¹Y� ç¢ <� ¹��'¹Y�¤çu¹ , that ^£� �� ½'�'¹Y�wçz¹ and that �d� �� <¤��g�G�'¹Y�wçu¹ for �<¤��g�� near � ¹��'¹0� .


3.24. Proofs of two lemmas. Firstly, we check formula (3.20).

Lemma 3.25. Let ø&-CD ã � Ø ï ÿ ñ ï � ��Ùi� ÷ ñbe harmonic in ÿ , real-valued and

satisfying ø�ï ÷ ñ"� �. Then the interior normal derivative of ø at

÷ -:ÿ©ÿ is givenby:

È ÿøÿ�Û ï ÷ ñ�� j Ä k Ä ÷� � �û � ø�ï � �(X ñ@ � �(X È ÷ @ ï ��Ì � j Ä k Ä m� � � ` ø�ï�� ñï�� È ÷ ñ ï �g� ÄProof. The function c �� ø û m U ø is holomorphic in ÿ and D ã � Ø in ÿ . SinceU ø is also harmonic in ÿ , since

�i�� é ï ÷ ñ�� é ï ÷ ñ û m ��¥ �� é ï ÷ ñ , and since the kernel@ � �(X È ÷ @ û ï is real, we may prove the lemma with ø replaced by c -�×Qïsÿ ñ VD ã � Ø ï ÿ ñ .Let � � Û � �(X and denote c ã �� ï ÷ ñ � �� é ï ÷ ñ , so that c ï�� ñ�� ï�� È ÷ ñ c ã ûP ï @ � È ÷ @ ã�� Ø ñ . We remind that, for any � ú -�ÿ©ÿ , by an elementary modification

of Cauchy’s formula, we have j Ä k Ä ãï �õ� % � ` z �� û � / � ãï . We deduce that the linearterm ï�� È ÷ ñ c ã provides the main contribution:j Ä k Ä m� � � ` ï�� È ÷ ñ c ãï�� È ÷ ñ ï �<� � È ý c ã j Ä k Ä ÷ý ��m � � ` �<�� È ÷ � È c ã ÄThus, we have to prove that the remainder Û�ï�� ñE�� c ï�� ñ È ï�� È ÷ ñ c ã , which belongsto ×Qïsÿ ñ VÕD ã � Øwï ÿ ñ , gives no contribution, namely satisfies % � ` é�ý � þý � û ã þ ¥ �g� � �

.

Set

Ü ï�� ñ�� é�ý � þý � û ã þ ¥ . Then

Ü -É×Qïsÿ ñ is continuous on ÿ 3�4 ÷{5 and satisfies@ Ü ï�� ñ @ � ? @ � È ÷ @ Ø�û ã for some ? " �. We claim that by an application of

Cauchy’s theorem, the integral % � ` Ü ï�� ñ �g� , which exists without principal value,vanishes.

Indeed, let × with� � ×S�¼� ÷

and consider the open disc ÿXï ÷ ð × ñ of radius ×centered at

÷. The drawing of this disc delineates three arcs of ÿ :

(i) the open arc ÿ©ÿ 3 ÿXï ÷ ð × ñ , of length � ý � È ý�× ;(ii) the closed arc ÿPÿ V ÿXï ÷ ð × ñ , of length � ý�× ;

(iii) the closed arc ÿPÿXï ÷ ð × ñ V ÿ , of length is � � × . ú ã ÊWe then decompose the integral of

Üon ÿPÿ as integrals on the first two arcs:� � ` Ü ï�� ñ �g� � � � ` È ` ý ã � Ö þÜ ï�� ñ �<� û � � ` W ` ý ã � Ö þ

Ü ï�� ñ �g� ÄThe estimate @ Ü ï�� ñ @ � ? @ � È ÷ @ Ø�û ã insures the smallness of the second integral:®®®®®

� � ` W ` ý ã � Ö þÜ ï�� ñ �<� ®®®®® � | ã × Ø Ä


To transform the first integral, we observe that Cauchy’s theorem entails that in-tegration of

Ü ï�� ñ �<� on the closed contour 5 ÿPÿ 3 ÿ ï ÷ ð × ñ 7 Ã 5 ÿPÿXï ÷ ð × ñ V ÿ 7 van-ishes: � � � � ` È ` ý ã � Ö þ

Ü ï�� ñ �<� û � � ` ý ã � Ö þ W `Ü ï�� ñ �g� Ä

Hence the first integral % � ` È ` ý ã � Ö þ may be replaced by the integral È % � ` ý ã � Ö þ W `on the third arc. The estimate @ Ü ï�� ñ @ � ? @ � È ÷ @ Ø�û ã again insures that this secondintegral is bounded by | ï × Ø . In conclusion @ % � ` Ü ï�� ñ �g� @ � ï`| ã û | ï ñ × Ø . ÞProof of Lemma 3.15. Secondly, we provide the details for the translation of thefamily ½ � . Let ³ � ´#ï Á ð ø ñ represent ² in a neighborhood of

i. By assumption,½ � ï�� ñ � ï`¿ ï�� ñ ð À ï�� ð � � ñsñ is attached to ² � , with ½ � ï ÷ ñb� i

. Equivalently,the two equations (3.17) hold. Since ½ � ½ � @ � � ú is an embedding, the vector³ ú �� ç� X ï ÷ ñ - V ú ² is nonzero. As in � 2.12, we choose a small ïÏýYH û � È ÷ ñ

-dimensional submanifold

¨ ú passing throughi

with

Q ³ ú 9 V ú ¨ ú � V ú�² andwe parametrize it by

Ü »L ï Á ï Ü ñ ð ø�ï Ü ñ û m ´=ï Á ï Ü ñ ð ø�ï Ü ñsñsñ , where

Ü - Q ï Á � z û ã issmall, @ Ü @ � Ü ã , Ü ã " �

. Then the translation½ � � Ý ï�� ñ�� ¿ ï�� ñ û Á ï Ü ñ ð À ï�� ð � � ð Ü ñ is constructed by perturbing the two equations (3.17), requiring only that½ � � Ý ï ÷ ñ0� ï Á ï Ü ñ ð ø�ï Ü ñ û m ´#ï Á ï Ü ñ ð ø�ï Ü ñsñsñ ÄThis is easily done:{ È ï � ��X ð � � ð Ü ñ0� Ú�b¿ ï � �(X ñ û Á ï Ü ñ ð Æ ï � ��X ð � � ð Ü ñ ð � � ðÆ ï � ��X ð � � ð Ü ñ0� ø�ï Ü ñ È U ã 5 Úix¿ ïtÑ ñ û Á ï Ü ñ ð Æ ïtÑ ð � � ð Ü ñ ð � � å7 ï � �(X ñ ÄThe non-tangency of ³ ú with

¨ ú ati

then insures that for every small fixed � � ,the mapping ïÍÌ ð Ü ñ »L ½ � � Ý ï � ��X ñ is a diffeomorphism onto a neighborhood of

iin² . Þ� 4. HOLOMORPHIC EXTENSION

ON GLOBALLY MINIMAL GENERIC SUBMANIFOLDS

4.1. Structure of the proof of Theorem 3.8. Let ² be a D ï � é globally minimalgeneric submanifold of ¾ ¡ . For clarity, we begin by a summary of the main stepsof the proof of Theorem 3.8.

(a) Since ² is globally minimal, the distribution H#»L V 5� ² must be some-where not involutive, namely there must exist a point

i -K² and a sec-tion 9 of V ã � ú ² defined in an open neighborhood Æ ú of

iin ² with9�ï i ñ J� �

such that 5 9 ð 9 7 ï i ñ J- V ã � úú ² 9 V ú � ãú ² .


(b) Thanks to an easy generalization of the Lewy extension theorem ( � 2.10),there exists a manifold ² ãú attached to ² at

iwith � ) �A² ãú � ÷ û � ) �N²

to which D ú6 Á ïÏ² ñextends to be CR.

(c) Thanks to the main propagation Proposition 3.14, CR extension to asimilar manifold ² ã� attached to ² holds at every point HÉ-�² �× 6 Á ïÏ² ð i ñ .

(d) Since there are as many manifolds with boundary as points in ² , it maywell happen that at some point

i -(² which belongs to the edge of twodifferent manifolds ² ãú and ² ãú , the tangent spaces V ú ² ãú and V ú ² ãú are distinct. Refering to the diagram of � 4.5 below, we may then imme-diately profit of such a situation, if it occurs.

(e) Indeed, in this case, an appropriate version of the edge-of-the-wedge the-orem guarantees that D ú6 Á ïÏ² ñ

extends to be CR on a D ï � é�û ú CR-wedgeÀ 6 Á � rú ati

whose dimension � is ü ÷ û ÷ � ý .

(f) To reason abstractly, let ��¦ � % be the maximal integer � with÷S� � � �

such that there exists a pointi -K² and a D ï � é�û ú CR-wedge À 6 Á � rú ati

of dimension ýYH û � û � to which D ú6 Á ïÏ² ñextends to be CR. Thanks

to the main propagation Proposition 3.14, CR extension to a D ï � é�û ú CR-wedge À 6 Á � r|§8¨ª©� holds at every point H$-}² � × 6 Á ïÏ² ð i ñ .

(g) If ��¦ � % � � , we are done, Theorem 3.8 is proved. Assuming �5¦ � % �� È ÷, we must construct a contradiction in order to complete the proof.

(h) Since ��¦ � % is maximal, again because of the edge-of-the-wedge theorem,the transversal situation (d) cannot occur; in other words, every pointi -&² that belongs to the edges of two different CR-wedges À 6 Á � r|§8¨ª©ú and À 6 Á � r�§8¨ª©ú has the property that V úYÀ 6 Á � r|§8¨ª©ú � V ú-À 6 Á � r�§8¨ª©ú .

(i) It follows that, asi

runs in ² , the ïÏýYH û � û � ñ -dimensional tangentplanes V ú À 6 Á � r §8¨ª©ú V V ú ² glue together and they define a D ã � é�û ú sub-distribution

¨ ² of the tangent bundle V ² , of dimension ýYH û �5¦ � % ,which contains V 5 ² .

(j) Since ² is globally minimal, such a distributioni »L ¨ ² ï i ñ must be

somewhere not involutive, namely there must exist a pointi -i² such

that / ¨ ² ð ¨ ² 1 ï i ñ J! ¨ ²zï i ñ .(k) The D ï � é�û ú CR-wedge À 6 Á � r §8¨ª©ú may be included in some D ï � é�û ú local

generic submanifold � r�§8¨ª©ú passing throughi

and containing ² in aneighborhood of

i.

(l) Multiplication by m gives V 5ú � r�§8¨ª©ú � ¨ ² ï i ñ û m ¨ ² ï i ñ and the non-degeneracy / ¨ ² ð ¨ ² 1 ï i ñ J! ¨ ² ï i ñ implies that the Levi-form of� r�§8¨ª©ú is not identically zero at

i, namely 5 V 5ú � r|§8¨ª©ú ð V 5ú � r�§8¨ª©ú 7 ï i ñ J!V 5ú � r §8¨ª©ú .


(m) Then a version of the Lewy-extension theorem on conic generic mani-folds having a generic edge guarantees that D ú6 Á ïÏ² ñ

extends to be CR ona CR-wedge «À 6 Á � ã�� r�§8¨ª©ú of dimension ýYH û � û ÷ û ��¦ � % at

i. This new

CR-wedge is constructed by means of discs attached to ² ÃKÀ 6 Á � r §8¨ª©ú ,exploiting the nondegeneracy of the Levi form of � r�§8¨ª©ú . This contra-dicts the assumption that � ¦ � % � � È ÷

was maximal, hence completesthe proof of Theorem 3.8.

The remainder of Section 4 is devoted to provide all the details of the proof.

4.2. Lewy extension in arbitrary codimension. As observed in (a) above, thereexists a point

i -�² and a local section 9 of V ã � ú ² with 9�ï i ñ J� �such that5 9 ð 9 7 ï i ñ J! ¾ Í V 5ú ² .

Lemma 4.3. ([We1982, BPo1982]) There exists a manifold with boundary ² ãúattached to a neighborhood of

iin ² with � ) �º² ãú � ÷ û � ) �º² to whichD ú6 Á ïÏ² ñ

extends to be CR.

We shall content ourselves with only one direction of extension, since this willbe sufficient for the sequel. Nevertheless, we mention that finer results expressedin terms of the Levi-cone of ² at

imay be found in [BPo1982, Bo1991]. Any-

way, all the extension results that are based on pointwise nondegeneracy condi-tions as the openness of Levi-cone or the finite typeness of ² at a point are by farless general than Theorem 3.8, in which propagational aspects are involved.

Proof. The arguments are an almost straightforward generalization of the proof ofthe Lewy extension theorem (hypersurface case), already exposed in � 2.10 above.Here is a summary.

By linear algebra reasonings, we may find local coordinates ï Á ðKh ñ -X¾ Á Ã ¾ zvanishing at

iwith 9�ï i ñ�� ®® ú , with ² given by ³ � ´#ï Á ð ø ñ , where ´#ï � ñ0� �

,�q´#ï � ñE� �, and with first equation given by³ ã � ´ ã � Á ã ùÁ ã ûQP ï @ Á ã @ ï � é ñ ûmP ï @ tÁ @ ñ ûmP ï @ Á @a@ ø @ ñ ûQP ï @ ø @ ï ñ ð

where we have split further the coordinates as ï Á ã ð tÁ ðKh ã ð th ñ , withtÁ - ¾ Á û ã andth -X¾ z û ã . For × " �

small, we introduce the disc defined by½ Ö ï�� ñ�� n ×�ï ÷ È � ñ ð t � ð À ãÖ ï�� ñ ð «À Ö ï�� ñ p ðwhere

À Ö ï�� ñv� Æ Ö ï�� ñ û m È Ö ï�� ñ is uniquely defined by requiring that ½ Ö is at-tached to ² and satisfies ½ Ö ï ÷ ñE� i . As in � 2.10, one verifies that

È ÿ È ãÖÿ�Û ï ÷ ñE� ý�× ï ûmP ï~× ï � é ñ ÄHence the exit vector of ½ Ö at

÷ - ÿ©ÿ is nontangential to ² ati

, provided × " �is small enough and fixed. By translating ½ Ö , we construct the desired manifoldwith boundary ² ãú . Þ


4.4. Maximal dimension for CR extension. As in � 4.1(f), let �¬¦ � % be the max-imal integer � � � such that there exists a point

i -�² and a D ï � é�û ú CR-wedgeÀ 6 Á � rú ati

of dimension ýYH û � û � to which D ú6 Á ïÏ² ñextends to be CR. By

the above Lewy extension, we have �¦ � % ü ÷. Thanks to the main propagation

Proposition 3.14, it immediately follows that CR extension to a D ï � é�û ú CR-wedgeÀ 6 Á � r|§8¨ª©� holds at every point H�-7² � × 6 Á ïÏ² ð i ñ . If ��¦ � % � � , Theorem 3.8is proved, gratuitously.

Assuming that÷�� i¦ � % � � È ÷

, in order to establish Theorem 3.8, we mustconstruct a contradiction. In the sequel, we shall simply denote � ¦ � % by � .

To proceed further, we must reformulate with high precision how were con-structed all the CR-wedges obtained by the propagation Proposition 3.14.

For every pointi -�² , there exists a local CR-wedge À 6 Á � rú attached to a

neighborhood ofi

in ² which is described by means of a family of analytic discs½#ú � � � Ý ï�� ñ , where � and

Üare parameters. Here, the subscript

iis not a parameter, it

indicates only thati

is the base point of ½6ú � � � Ý , namely ½#ú � � � ú ï ÷ ñ�� i . The family½ ú � � � Ý enjoys properties that are listed below. In this list, the conditions are moreuniform than those formulated in Definition 3.4, but one immediately verifies thatboth formulations are equivalent, up to a shrinking of � ã ï i ñ " �

, of

Ü ã ï i ñ " �, ofÌ ã ï i ñ " �

and of÷ È Û ã ï i ñ " �

.! The rotation parameter �$- Q r û ã runs in 4 @ � @ � � ã ï i ñ 5 , for some small� ã ï i ñ " �.! The translation parameter

Ü - Q ï Á � z û ã runs in 4 @ Ü @ � Ü ã ï i ñ 5 , for somesmall

Ü ã ï i ñ " �.! The point H�ï i ñ�� ½@ú � ú � ú ï È ÷ ñ -}² is close to

i.! At H�ï i ñ , there is a CR-wedge À 6 Á � r� ý ú þ .! The family ½#ú � � � Ý satisfies ½@ú � � � Ý ï�ÿ©ÿ ñ ! ²ßÃ À 6 Á � r� ý ú þ .! A small angle Ì ã ï i ñ " �

and a radius Û ã ï i ñ " �close to

÷are chosen.! A family

( ú of linear subspaces of V ú ¾ ¡ satisfying V ú ( ú 9 V ú ² �V ú ¾ ¡ for alli ��-C² in a neighborhood of

iis chosen.! For every � with @ � @ �Ê� ã ï i ñ , every

Üwith @ Ü @ � Ü ã ï i ñ and every Ì with@ Ì @ �NÌ ã ï i ñ , the exit vector of ½ ú � � � Ý ï � ��X ñ at � ��X is not tangent to ² :

ex ïÍ½#ú � � � Ý ñ ï � �(X ñE�� < ö<;>=A@ Ð O � � � F � V sxu � � m ÿ ½=ú � � � Ýÿ Ì ï � �(X ñ|� J� � Ä! For every fixed

Üwith @ Ü @ � Ü ã ï i ñ and every fixed Ì with @ Ì @ �NÌ ã ï i ñ , the

normalized exit vector mappingQ r û ã ¹��» È L n-ex ïÍ½#ú � � � Ý ñ ï � �(X ñ -®B z û ãis of rank ï � È ÷ ñ at every ��- 4 @ � @ �N� ã ï i ñ 5 .

196 JOEL MERKER AND EGMONT PORTEN! For some � ï ï i ñ , Ü ï ï i ñ , Ì ï ï i ñ and Û ï ï i ñ satisfying� �A� ï ï i ñ �i� ã ï i ñ , � �Ü ï ï i ñ � Ü ã ï i ñ , � ��Ì ï ï i ñ ��Ì ã ï i ñ and

� � ÷ È Û ï ï i ñ � ÷ È Û ã ï i ñ � ÷,

the CR-wedge is precisely defined as:À 6 Á � rú ��À¿ ½ ú � � � Ý ï³Û � ��X ñ�� @ � @ �A� ï ï i ñ ð @ Ü @ � Ü ï ï i ñ ð @ Ì @ �NÌ ï ï i ñ ð Û ï ï i ñ � Û�� ÷ ÃvÄ! Finally, the CR-wedge À 6 Á � rú is contained in a D ï � é�û ú local generic sub-manifold � rú of the same dimension ýYH û � û � that contains a neigh-

borhood ofi

in ² . At a pointi � � ½ ú � � � Ý ï � ��X ñ -�² of the edge ofÀ 6 Á � rú , the tangent space of � rú is:V ú � rú � V ú�² 9 Q � m ÿ ½ ú � � � Ý ÿ�Ì ï � �(X ñ|� ¯ãKJ � J r û ã Q � m ÿ ï ½ ú � � � Ý ÿ Ì�ÿ � � ï � ��X ñ|� Ä

4.5. An edge-of-the-wedge theorem. There are as many generic submanifolds� 6 Á � rú of codimension � È � as pointsi �©-�² . At a point

i � ½ ú � � � Ý ï � �(X ñ that

belongs to the edge of such an � 6 Á � rú , we may define a linear subspace of V ú�²by ¨ ² ú ï i ñ�� V 5ú � rú V V ú�² ÄSince � rú is generic and contains ² in a neighborhood of

i, this space

¨ ² ú ï i ñcontains V 5ú ² and is ïÏýYH û � ñ -dimensional. Also, multiplication by m induces anisomorphism

¨ ² ú ï i ñ g V 5ú ² ë V úð� rú g V ú�² .In general, two different

¨ ² ú ï i ñ and¨ ² ú ï i ñ need not coincide, or equiv-

alently, two different tangent spaces V úS� rú and V úS� rú are unequal.

ç O � � � � � F � ý ` þ

J° ^ ý O � � � � � � � F � � Ó<��ÕJ J ± Ó J Õ ± Ó J Õ· ·

K RUT � VO � �K R�T � VO �

Two non-tangent CR-wedges at JMore precisely, there is a dichotomy.

(I) Either for every two pointsi � ð i � � -�² such that there exists a point

ibelonging to the intersection of the edges of the two CR-wedges À 6 Á � rú


and À 6 Á � rú , namely of the form:i � ½ ú � � � Ý ï � �(X ñ0� ½ ú � � � Ý ï � ��X ñ ðfor some values@ � � @ �i� ï ï i � ñ ð @ Ü � @ � Ü ï ï i � ñ ð @ Ì � @ �NÌ ï ï i � ñ ð@ � � � @ �i� ï ï i � � ñ ð @ Ü � � @ � Ü ï ï i � � ñ ð @ Ì � � @ �NÌ ï ï i � � ñ ðthe two spaces V úS� rú and V úS� rú coincide. Equivalently,

¨ ² ú ï i ñ��¨ ² ú ï i ñ .(II) Or there exist two points

i � ð i � �-�² and a pointi � ½ ú � � � Ý ï � �(X ñ»�½ ú � � � Ý ï � ��X ñ in the intersection of the edges of the two CR-wedgesÀ 6 Á � rú and À 6 Á � rú such thatV úð� rú J� V úS� rú Ä

Lemma 4.6. The case V ú � rú J� V ú � rú implies that D ú6 Á ïÏ² ñextends to be

CR on a CR-wedge «À 6 Á � ã�� rú ati

whose dimension equals ýYH û � û ÷ û � ,contradicting the maximality of � � �¦ � % .

Of course, this lemma follows by a known CR version of the edge-of-the-wedgetheorem ([Ai1989]), but for completeness, we summarize a shorter proof that ex-ploits the existence of the discs ½ ú � � � Ý , as in [Po2004].

Proof. By construction, the family ½ ú � � � Ý ï�� ñ covers the CR-wedge À 6 Á � rú . The

pointi

belongs to the edge of À 6 Á � rú .

Since V úS� rú J� V úS� rú , there exists a manifold ² ãú ! À 6 Á � rú attached to ²ati

with � ) �Ö² ãú � ÷ û � ) �Ö² such that÷ û � � � ) � n � V ú�² ãú û V ú-À 6 Á � rú � U V ú�² p Ä²´³5µ¬¶ ·\ �

ç O � � � � F�� \ ý ` þ�¸\º¹ ² ³5µ�¶ ·\ � �!

ç O � � � � F�� \ ý ã þ²´³5µ�¶ ·!ª» \ �½¼

¾² ³µ�¶ ¸ l ·\

Translating ý O � � � � F along · �O


We may deform the family ½ ú � � � Ý by translating it along ² ãú , as in the diagram.So we introduce a supplementary parameter ì " �

and we require that the point½ ú � � � Ý � í ï ÷ ñ should cover a one-sided neighborhood ofi

in ² ãú as ì runs in ï � ð ì ã ñ ,for some small ì ã " �

, and as the previous translation parameter

Ü - Q ï Á � z û ãruns in 4 @ Ü @ � Ü ï ï i � ñ 5 . Thanks to Theorem 3.7(IV), the corresponding Bishop-type equation has D ï � é�û ú solutions.

If we choose � ± " �with @ �B� @ û � ± �µ� ï ï i � ñ , Ü ± " �

with @ Ü � @ û Ü ± � Ü ï ï i � ñ ,Ì ± " �with @ Ì�� @ û Ì ± ��Ì ï ï i � ñ , ì ± " �

with ì ± ��ì ã and Û ± � ÷with Û ï ï i � ñ �Û ± � ÷

, the set:« À 6 Á � ã�� rú �� ó ½ ú � � � Ý � í ï³Û � �(X ñ�� @ � È � � @ �N� ± ð @ Ü È Ü � @ � Ü ± ð@ Ì È Ì � @ �NÌ ± ð Û ± � Û � ÷ ð � �Nì(�Nì ± ôwill constitute a CR-wedge of dimension ýYH û � û ÷ û � at

i. By a technical

adaptation of the approximation Theorem 5.2(III) (cf. Lemma 2.19), D ú6 Á ïÏ² ñextends to be CR on «À 6 Á � ã�� rú . Þ4.7. Definition of the (non-integrable) subbundle

¨ ² ! V ² . Consequently,case (II) cannot occur, because of the definition of � � �5¦ � % . Thus, case (I) holds.In other words, as

i � runs in ² , the D ã � é�û ú distributionsi »L ¨ ² ú ï i ñ defined

fori

in the edge of À 6 Á � rú (a neighborhood ofi � in ² ) glue together in a well-

defined D ã � é�û ú vector subbundle of V ² . Observe that V 5 ² is a subbundle of¨ ² of codimension � . For every pointi -}² , we have:V 5ú ² ! ¨ ² ï i ñ�� V 5ú � rú V V ú�² Ä

As in � 4.1(j), since ² is globally minimal and since¨ ² is of codimension� È � ü ÷

in V ² , there must exist a pointi -K² such that / ¨ ² ð ¨ ² 1 ï i ñ J!¨ ² ï i ñ .

Lemma 4.8. At such a pointi

, the Levi form of � rú does not vanish identically:5 V 5 � rú ð V 5 � rú 7 ï i ñ J! V 5ú � rú ÄProof. We reason by contradiction, assuming that 5 V 5 � rú ð V 5 � rú 7 ï i ñ ! V 5ú � rú .Let

¨ ãand

¨ ïbe two arbitrary D ã � é�û ú sections of

¨ ² defined in a small neigh-borhood Æ ú of

iin ² . Since

¨ ² @ E O is a subbundle of V ² @ E O , we have5 ¨ ã ð ¨ ï 7 ï i ñ - V ú�² ÄWe may extend

¨ ãand

¨ ïto a neighborhood ¿�ú of

iin � rú that contains Æ ú as

sections À ã and À ï of V 5 � rú @ Á O . Since¨ ã

and¨ ï

are tangent to ² V Æ ú , oneverifies that, independently of the extension:5 ¨ ã ð ¨ ï 7 ï i ñ�� 5 À ã ð À ï 7 ï i ñ - V 5ú � rú ð


where the second Lie bracket belongs to V 5ú � rú , because we assumed that theLevi form of � rú vanishes at

i. We deduce5 ¨ ã ð ¨ ï 7 ï i ñ - V 5ú � rú V V ú�² � ¨ ² ï i ñ Ä

This contradicts / ¨ ² ð ¨ ² 1 ï i ñ J! ¨ ² ï i ñ . Þ4.9. Lewy extension on CR-wedges. To contradict the maximality of � � ��¦ � %at a point

iat which / ¨ ² ð ¨ ² 1 ï i ñ J! ¨ ² ï i ñ , we formulate a Lewy extension

theorem on the conic manifold with edge À 6 Á � rú .

Proposition 4.10. Leti -y² and assume that 5 V 5ú � rú ð V 5ú � rú 7 ï i ñ J! V 5ú � rú .

Then there exists a ïÏýYH û � û ÷ û � ñ -dimensional local CR-wedge «À 6 Á � ã�� rú ofedge ² at

ito which D ú6 Á ïÏ² Ã À 6 Á � rú ñ

extends to be CR.

Thus, this proposition concludes the proof of Theorem 3.8.

Proof. There exists a local section 9 of V ã � ú � rú with 9�ï i ñ J� �such that5 9 ð 9 7 ï i ñ J- V ã � úú � rú 9 V ú � ãú � rú . It is appropriate to distinguish two cases.

Firstly, assume that 9Oï i ñ - V ã � ú ² . Then as in � 4.2, we may construct asmall analytic disc ½ Ö attached to ² in a neighborhood of

ihaving exit vectorÈ � ç Ê� é ï ÷ ñ approximately directed by 5 9 ð 9 7 ï i ñ J-«¾¾Í V úð� rú . So this disc has

exit vector nontangential to � rú ati

. By translating it along ² and along the �supplementary directions offered by À 6 Á � rú , we deduce CR extension to a ïÏýYH û� û ÷ û � ñ -dimensional CR wedge « À 6 Á � ã�� rú .

Ä úQ RUT � VOÄ

ÂQ R�T � � h VO

Translating a disc non-tangent to _ VO alongK R�T � VO

Secondly, assume that 9�ï i ñ J- V ã � úú ² for every local section 9 of V ã � ú ² suchthat 5 9 ð 9 7 ï i ñ J- V ã � úú � rú 9 V ú � ãú � rú .

We explain the case � � ý , � � ÷first, since this case is easier to understand.

Under this assumption, � ãú is a hypersurface of ¾ ¡ divided in two parts by ² ,

one part being À 6 Á � ãú . We draw a diagram.


ßiÃ ÷ Ã ÷ Û

··° ^

ý ÊJÛ

K R�T � �O_ �O

DISC ATTACHED TO A HALF HYPERSURFACE AND HAVING NONTANGENT EXIT VECTOR

There exist coordinates ï Á ðKh � ðKh � � ñ -v¾ ¡ û ï Ã ¾ Ã ¾ centered ati

in which� ãú is given by ³ � � � � ï Á ðKh � ð ø� � ñ , with � ï � ñ-� �and � � ï � ñ � �

and in which² is given by a supplementary equation ³ � � ´ � ï Á ð ø � ð ø � � ñ with ´ � ï � ñw� �and�q´ � ï � ñ��

. Changing the orientation of the ³ � -axis if necessary, it follows À 6 Á � ãúis given by the equation ³ � � � � ï Á ðKh � ð ø� � ñ and the inequation ³ � " ´�� ï Á ð ø� ð ø� � ñ ,with ´��ï � ñ��

and �q´�� ï � ñ � �. In the diagram, V 5ú � ãú is the direct sum of theÁ

-coordinate space with the ø � û m ³ � -coordinate axis.The Levi form of � ãú is represented by a scalar Hermitian form( ï Á ðKh � ð ùÁ ð ùh � ñ . By assumption, its restriction to V 5ú ² vanishes (other-

wise, the first case holds), so( ï Á ð � ð ùÁ ð � ñ v �

. The assumption that the Leviform of � ãú does not vanish identically insures that

(is nonzero. To proceed

further, we need( ï � ðKh � ð � ð ùh � ñ Jv �

. If( ï � ðKh � ð � ð ùh � ñ v �

, since(

isnonzero, by a linear coordinate change of the form

th � � h � , tÁ � � Á � û � � h � ,� � ÷ ð Ä Ä Ä ð È ý ,th � � � h � � , we may insure that

( ï � ðKh � ð � ð h � ñ Jv �. Observe

that such a change of coordinates stabilizes both V ú�² and V úS� ãú . After a realdilation, we can assume that the equation of � ãú is of the form:³ � � � h � ùh � ûQP ï @ h � @ ï � é�û ú ñ ûQP ï @ Á @ @ ï Á ðKh � ñ @ ñ ûmP ï @ ø � � @ @ ï Á ðKh � ñ @ ñ ûQP ï @ ø � � @ ï ñ ÄTo the hypersurface � ãú , we attach a disc ½ Ö ï�� ñ with zero

Á-component, with h � -

component equal to m ×=ï ÷ È � ñ and with h � � -component ï Æ � �Ö ï�� ñ û m È � �Ö ï�� ñsñ of classD ï � é�û ú satisfying the corresponding Bishop-type equation. Exactly as in the Lewyextension theorem ( � 2.10), for × " �

small enough and fixed, the exit vector of ½ Öati

is nontangent to � ãú (this is uneasy to draw in the diagram above, but imaginethat the disc drawn in � 2.10 is attached to a half-paraboloid). Furthermore, usingthe inequality ³ � ï � ��X ñ�� ×�ï ÷ È ��Ì ñ ü\× X ¥� for @ Ì @ � � together with the property�q´ � ï � ñZ� �

, it is elementary to verify that ½ Ö ï�ÿPÿ 3�4 ÷{5 ñ is contained in the openhalf-hypersurface 4 ³ � " ´�� 5 , as shown in the diagram.

Since the exit vector of ½ Ö is nontangent to � ãú , in order to get holomorphic ex-tension to a wedge at

i, it suffices to translate the disc ½ Ö in the half-hypersurfaceÀ 6 Á � ãú .


However, if we translate ½ Ö as usual by requiring that the base point ½ Ö � Ý ï ÷ ñ0�i Ý , with

Ü - Q ï ¡ û ï small, covers a neighborhood ofi

in ² , it may well hap-pen that, due to the curvature of ² in a neighborhood of

i, the boundary of the

translated disc enters slightly the other side of � ãú , which is forbidden.To remedy this imperfection, two equally good options present themselves.

The first option would be to rotate slightly the translated disc ½ Ö � Ý in order thatit becomes tangent to ² at the point

i Ý � ½ Ö � Ý ï ÷ ñ . Then adding a small pa-rameter ì " �

, we would translate it slightly in the positive direction of À 6 Á � ãú ,essentially along the positive ³ � -direction.

The second option is to introduce a family of complex affine biholomorphisms¸ Ý that transferi Ý -µ² to the origin and transfer the tangent spaces at

i Ý of� ãú and of ² to 4 ³ � � � ��5and to 4 ³ � � � ³ � � ��5

. So ¸ Ý ï)� ãú ñ is given by³ � � � � � � ï Á ðKh � ð ø � � � Ü ñ with � of class D ï � é�û ú with respect to all variables andwith the map ï Á ðKh � ð ø�� ñ »L � � � ï Á ðKh � ð ø� � � Ü ñ vanishing to second order at theorigin for every

Ü - Q ï ¡ û ï small. Also, ¸ Ý �À 6 Á � ãú is given by a supplementaryinequation ³ � " ´ � ï Á ð ø � ð ø � � � Ü ñ , with ´ � of class D ï � é (the smoothness of ² )with respect to all variables and with ï Á ð øw� ð ø�� ñ »L ´#ï Á ð ø� ð ø�� Ü ñ vanishing tosecond order at the origin.

To the hypersurface ¸ Ý ï)� ãú ñ , we attach the family of discst½ Ö � Ý � í ï�� ñÊ� n � ð m ì û m ×=ï ÷ È � ñ ð tÆ � �Ö � Ý � í ï�� ñ û m tÈ � �Ö � Ý � í ï�� ñ phaving zero

Á-component and h � -component equal to m ì û m ×#ï ÷ È � ñ , whereì - Q

with @ ì @ ��ì ã , ì ã " �, is a small parameter of translation along the³ � -axis. Of course:{ tÆ � �Ö � Ý � í ï � ��X ñ�� È

U ã�5 � � ð m ì û m × ï ÷ È Ñ ñ ð tÆ � �Ö � Ý � í ïtÑ ñ�� Ü å7 ï � ��X ñ ðtÈ � �Ö � Ý � í ï � ��X ñ�� U ã 5 tÆ � �Ö � Ý � í 7 ï � �(X ñ ÄBy means of elementary computations involving Taylor’s formula, we verify twofacts. ! If × " �

is sufficiently small and fixed,t½ Ö � Ý � ú ï�ÿ©ÿ 3�4 ÷{5 ñ is contained in

the open half-hypersurface 4 ³ � " ´ � ï Á ð ø � ð ø � � � Ü ñ 5 , for all

Ü - Q ï ¡ û ïwith @ Ü @ � Ü ã , Ü ã " �

small.! Furthermore, for all ì with� �yì � ì ã , and all

Üwith @ Ü @ � Ü ã , the disc

boundaryt½ Ö � Ý � í ï�ÿPÿ ñ is contained in the open half-hypersurface 4 ³ � "´ � ï Á ð ø � ð ø � � � Ü ñ 5 .

Coming back to the old system of coordinates, it follows that the family of discs½ Ö � Ý � í �� ¸ û ãÝ F t½ Ö � Ý � í has base point ½ Ö � Ý � í ï ÷ ñ covering a neighborhood ofi

inthe half-hypersurface À 6 Á � ãú , as

Üand ì vary. Since the exit vector of ½ Ö is not

tangent to � ãú ati

, this family of discs covers a ý{ -dimensional wedge «À 6 Á � ï ¡ú


of edge ² ati

. This completes the proof of the second case of the propositionwhen � � ÷

and � � ý .

Based on these explanations, we may now summarize the general case. Thereexist coordinates ï Á ðKh � ðKh � � ñ -X¾ Á Ã ¾ r Ã ¾ z û r vanishing at

iin which the D ï � é�û ú

generic submanifold � rú is represented by ³ � � � � ï Á ðKh � ð ø�� ñ , with � ï � ñE� �and� � ï � ñ��

. After killing the second order pluriharmonic quadratic terms in everyright hand side � î ï Á ðKh � ð � ñ , Ä>� � � ÷ ð Ä Ä Ä ð � È � , we may assume that the quadraticterms are Hermitian forms

( î ï Á ðKh � ð ùÁ ð ùh � ñ .After a linear change of coordinates in the h � -space, V ú�² � 4 ³ � � ³ � � � ��5

,the D ï � é generic edge ² is defined by ³ � ´#ï Á ð ø ñ with ´#ï � ñv� �

, �q´=ï � ñ � �and the conic open submanifold À 6 Á � rú of � rú is defined by ³ � � � � ï Á ðKh � ð ø�� ñtogether with the inequations³ �î " ´ �î ï Á ð ø � ð ø � � ñ ð Ä � � ÷ ð Ä Ä Ä ð � ðwhere ´ � ï`´ � ð ´ � � ñ . In fact, we may assume that the cone defining the CR-wedgeon the tangent space is slightly larger than the salient cone ³ �î " �

, Ä�� ÷ ð Ä Ä Ä ð � .The nonvanishing of the Levi form of � rú at

ientails that at least one Her-

mitian form( î ï Á ðKh � ð ùÁ ð ùh � ñ is nonzero. After renumbering,

( ã is nonzero.Also, since V 5ú ² is the

Á-coordinate space, we have

( ã ï Á ð � ð ùÁ ð � ñ v �(oth-

erwise, the first case holds). After a complex linear coordinate change of theform

th � � h � , tÁ � � Á � û y rî � ã � î � h �î , th � � � h � � , we may insure that( ã ï � ðKh � ð � ð h � ñ Jv �. Then the set of vectors ï � ðKh � ñ on which

( ã vanishes isa proper real quadratic cone of ¾ r . Consequently, for almost every real vectorï � ð m ³ � ñ , the quadratic form

( ã is nonzero on the complex line ¾6ï � ð m ³ � ñ . Sincethe cone defining À 6 Á � rú is open and may be slightly shrunk, we can assume that( ã does not vanish on ¾6ï � ð m ³ �ã ñ , with ³ � ã � ï ÷ ð Ä Ä Ä ð ÷ ñ - Q r

. It follows thatthe disc ½ Ö attached to � rú having zero

Á-component and h � -component equal toï m ×=ï ÷ È � ñ ð Ä Ä Ä ð m ×=ï ÷ È � ñsñ is nontangent to � rú at

i.

Furthermore, letting a pointi Ý -:² of coordinates

Ü �� ï Á ð ø ñ vary in asmall neighborhood of

iin ² , we may construct a family of biholomorphisms ¸ Ý

sendingi Ý to the origin and normalizing the equations of ² , of � rú and of À 6 Á � rú

under the form ³ � ´#ï Á ð ø � Ü ñ , ³ � � � � ï Á ðKh � ð ø � � � Ü ñ and ³ �î " ´ �î ï Á ð ø � ð ø � � � Ü ñ ,with ´ being D ï � é and with � being D ï � é�û ú with respect to all variables and bothvanishing to second order at the origin.

Let ì¾- Q r, @ ì @ �¨ì ã , be a small parameter of translation along the ³ � -

coordinate space. To the generic submanifold ¸ Ý ï)� rú ñ , we attach the family ofdiscs t½ Ö � Ý � í ï�� ñÊ� � ð À �Ö � í ï�� ñ ð Æ � �Ö � Ý � í ï�� ñ û m È � �Ö � Ý � í ï�� ñ ðwhere À �Ö � í ï�� ñ�� m ì ã û m ×#ï ÷ È � ñ ð Ä Ä Ä ð m ì r û m ×=ï ÷ È � ñ ð


and where{ tÆ � �Ö � Ý � í ï � ��X ñ�� ÈU ã 5 � � ð m ì û m × ï ÷ È Ñ ñ ð tÆ � �Ö � Ý � í ïtÑ ñ�� Ü å7 ï � ��X ñ ðtÈ � �Ö � Ý � í ï � ��X ñ�� U ã 5 tÆ � �Ö � Ý � í 7 ï � �(X ñ Ä

By means of elementary computations involving Taylor’s formula, we may verifythat for all ìÖ- Q r

with� �öì î � ì ã , Ä>� � ÷ ð Ä Ä Ä ð � , and all

Ü - Q ï Á � z ,@ Ü @ � Ü ã , the disc boundaryt½ Ö � Ý � í ï�ÿPÿ ñ is contained in 4 ³ �î " ´��î ï Á ð ø�� ð ø� � �Ü ñ ð Ä�� ÷ ð Ä Ä Ä ð � 5 .

Coming back to the old system of coordinates, it follows that the family ofdiscs ½ Ö � Ý � í �� ¸ û ãÝ F t½ Ö � Ý � í has base point ½ Ö � Ý � í ï ÷ ñ covering a neighborhood ofi

in the CR-wedge À 6 Á � rú , as

Üand ì vary. Since its exit vector is not tangent to� ãú at

i, this family of discs covers a ïÏýYH û � û ÷ û � ñ -dimensional CR-wedge«À 6 Á � ã�� rú of edge ² at

i.

The proofs of the proposition and of Theorem 3.8 are complete. Þ4.11. Wedgelike domains. On a globally minimal ² , at every point

i -7² , wehave constructed a local wedge ÀVú by gluing deformations of discs. It may wellhappen that at a point

ithat belongs to the edges of two different wedges À � andÀ � , the wedges have empty intersection in ¾ ¡ (imagine two thin opposite cones

having vertex at� - Q ï

). Fortunately, by means of the translation trick presentedin � 4.5 (cf. the diagram), we can fill in the space in between. Achieving thissystematically, by a sort of gluing-shrinking processus, we obtain some connectedopen set À attached to ² containing possibly smaller wedges À`�ú ! À ú at everypoint.

To set-up a useful definition, by a wedgelike domain À attached to ² wemean a connected open set that contains a local wedge of edge ² at every point.Geometrically speaking, the requirement of connectedness prevents jumps of thedirections of local wedges in the normal bundle V ¾ ¡ @ Ä g V ² .

We may finally conclude this section by the formulation of a statement that isthe very starting point of the study of removable singularities for CR functions([MP1998, MP1999, MP2000, MP2002, MP2006a]).

Theorem 4.12. ([Me1997, MP1999]) If ² is a globally minimal D ï � é genericsubmanifold of ¾ ¡ , there exists a wedgelike domain À attached to ² such thatevery continuous CR function �°-ND ú6 Á ïÏ² ñ

possesses a holomorphic extensionÂÎ-C×Qï³À ñ VSD ú ïÏ² Ã À ñwith Â @ Ä � � .

Its 9 ; version deserves special attention. Let À be a wedgelike domain at-tached to ² . A holomorphic function Âº-(×Qï³À ñ

is said to belong to the Hardyspace

( ;ä%4 5 ï³À ñif, for every

i -C² , for every local coordinate system centered atiin which ² is given by ³ � ´#ï�¶ ð · ð ø ñ , for every local wedge of edge ² at

i


contained in À of the formÀ � À ï�¸ ð ì ð | ñ�� ó ï�¶ û m · ð ø û m ³ ñ -wÿ Áß Ã Þ zß Ã m Þ zí �³ È ´#ï�¶ ð · ð ø ñ -C| ô ðas defined in � 4.29(III), for every cone | � ! Q z with | � VCB z û ã !é! | V®B z û ãand for every ¸ ��N¸ , the supremum:��¬]jX § 6 � ` F� o ]sÅ � ®® Â�Ï¶ û m · ð ø û m ´#ï�¶ ð · ð ø ñ û m Ì � ®® ; ��¶ E� · E��øÖ� >is finite. An adaptation of the proof of the preceding theorem yields its 9 ; version.

Theorem 4.13. ([Po1997, Po2000]) If ² is a globally minimal D ï � é generic sub-manifold of ¾ ¡ , there exists a wedgelike domain À attached to ² such that everyfunction �K-�9 ;ä%4 5 � 6 Á ,

÷��µ<Õ� > , possesses a Hardy space holomorphic exten-sion ÂÉ- � ;ä%4 5 ï³À ñ

.

To conclude, we would like to mention that arguments similar to those of The-orem 4.12 yield a mild generalization, worth to be mentioned: the CR extensiontheory is valid for D ï � é CR manifolds that are only locally embeddable.

However, for concreteness reasons, we preferred to set up the theory in a glob-ally embedded context. In the remainder of the memoir, not to enter superficialcorollaries, we will formulate all our results under the paradigmatic assumption ofglobal minimality. Thus, Theorems 4.12 and 4.13 will be our basic main startingpoint.

The two monographs [Trv1992, BCH2005] deal not only with embedded struc-tures but also with locally integrable structures. Nevertheless, most topics exposedhere are not yet embraced in a comprehensive theory (cf. � 3.29(III)). So it is anopen direction of research to transfer the theory of holomorphic extension of CRfunctions (including removable singularities) to locally integrable structures.


VI: Removable singularities

Table of contents

1. Removable singularities for linear partial differential operators . . . . . . . . . . 206.2. Removable singularities for holomorphic functions of one or several complexvariables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210.3. Hulls and removable singularities at the boundary . . . . . . . . . . . . . . . . . . . . . . . 227.4. Smooth and metrically thin removable singularities for CR functions . . . . . 241.5. Removable singularities in CR dimension Ô . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.

[7 diagrams]

Removable singularities for general linear partial differential operators � �y Ø § F � Ø ï�¶ ñ ÿ ØÌ on domains Æ ! Q ¡having order H ü ÷

and Dx* coeffi-cients have been studied by Harvey and Polking (1970) in a general setting. As-sumptions of metrical thinness of singularities, in the sense of Minkowski contentor of Hausdorff measure, insure automatic removability. For instance, relativelyclosed sets | ! Æ whose ï� È H ñ -dimensional Hausdorff measure is null areï � ð 9 *ä%4 5 ñ -removable. For structural reasons, these general results (valid whateverthe structure of the operator) necessitate a control of growth when dealing with9 ãä%4 5 -removability. In addition, when � is an embedded complex-tangential op-erator, this approach does not convey to the adequate results, because removablesingularities for holomorphic or CR functions must take advantage of automaticextension to larger sets.

Since almost two decades, thanks to the impulse of Stout, removable singulari-ties have attracted much attention in several complex variables. A natural questionis whether the Hartogs-Bochner extension Theorem 1.9(V) holds when consider-ing CR functions that are defined only in the complement ÿGÆ 3 ¨ of some compactset

¨ ! ÿÆ of a connected smooth boundary ÿÆ ! ¾ ¡ . In complex dimension � ý , Stout showed that the answer is positive if and only if¨

is convex withrespect to the space of functions that are holomorphic in a neighborhood of Æ . Incomplex dimension Jü . , a complete cohomological characterization of differ-ent nature was obtained by Lupacciolu (1994).

In another direction, by means of the above-cited global continuity principle,Joricke (1995) generalized Stout’s theorem to weakly pseudoconvex domains. Re-cently, Joricke and the second author were able to remove the pseudoconvexity as-sumption by applying purely geometrical constructions without integral formulas,controlling uniqueness of the extension (monodromy) by fine arguments.

Within the general framework of CR extension theory (exposed in Part V), thestudy of removable singularities has been endeavoured by Joricke in the hypersur-face case since 1988, and after by the two authors in arbitrary codimension since1995. The notions of CR-, À - and 9 ; -removability, although different, may be


shown to be essentially equivalent, thanks to technical deformation arguments.All the surveyed results hold in 9 ;ä%4 5 with

÷i�¾<J� > , including< � ÷

andwithout any growth assumption near the singularity. On a generic globally min-imal D ï � é generic submanifold ² of ¾ ¡ , closed sets | ! ² having vanishingï ��) �J² È ý ñ -dimensional Hausdorff measure are CR-, À - and 9 ; -removable.As an application, CR meromorphic functions defined on an everywhere locallyminimal ² do extend meromorphically to a wedgelike domain attached to ² .

In conjunction with the Harvey-Lawson complex Plateau theorem, singulari-ties | that are a priori contained in a ý -codimensional D ï � é submanifold D of astrongly pseudoconvex D ï � é boundary ÿÆ ! ¾ ¡ ( 0ü«. ) are shown by Joricke tobe not removable if and only if D is a maximally complex cycle. The conditionthat D be somewhere generic was shown by the two authors to be sufficient forits removability in arbitrary codimension.

Concerning more massive singularities, a compact subset¨

of a one-codimensional submanifold ² ã ! ÿGÆ ! ¾ ¡ is CR-, À - and 9¯; -removableprovided the CR dimension of ÿGÆ is ü�ý (viz. oü�. ) and provided

¨does

not contain any CR orbit of ² ã(Joricke, 1999). The second author generalized

this theorem to higher codimension, assuming that ² is globally minimal ofCR dimension H üÖý . The main geometric argument (called sweeping out bywedges) being available only in CR dimension H¾üÛý , the more delicate case ofCR dimension H � ÷

is studied extensively in the research article [MP2006a],placed in direct continuation to this survey.� 1. REMOVABLE SINGULARITIES FOR

LINEAR PARTIAL DIFFERENTIAL OPERATORS

1.1. Hausdorff measure. Let ² be a D ã abstract manifold of dimension yü ÷equipped with some Riemannian metric. For a�- Q

with�}� a � , we remind

([Ch1989]) the definition of the notion of a -dimensional Hausdorff measure� � on² , that generalizes the notion of integer dimension of submanifolds.

If | ! ² is an arbitrary subset and if 6 " �is small, we define� � Ü ï`| ñ0� )*W�p ab c *öî � ã Û �î � | is covered by geodesic balls

s î of radius Û�î � 6£ÆÈÇÉ ÄClearly,

� � Ü ï`| ñ � � � Ü ï`| ñ , for 6 � � 6 , so the limit� �0ï`| ñÊ� '*) �Ü + ú h � � Ü ï`| ñ exists in/ � ð > 1 . This limit is called the a -dimensional Hausdorff measure of | . The value

of� �'ï`| ñ depends on the choice of a metric, but the two properties

� �'ï`| ñ � �and

� �'ï`| ñ � > are independent. The most significant property is that thereexists a critical exponent a 6 ü �

, called the Hausdorff dimension of | , such that� � ï`| ñw� > for all a�� a 6 and such that� � ï`| ñw� �

for all a " a 6 . Then thevalue

� � R ï`| ñ may be arbitrary in / � ð > 1 .


Proposition 1.2. ([Fe1969, Ch1989]) The following properties hold true:

(1)� ú ï`| ñ�� a¯lðR � ï`| ñ ;

(2)� ¡ ï`| ñ coincides with the outer Lebesgue measure of | ! ² ;

(3) a D ã submanifold D ! ² has Hausdorff dimension a C � ��) �1D ;

(4) if� ¡ û ã ï`| ñ��

, then ² 3 | is locally connected;

(5) if � � ² L D is a D ã map and if | ! ² satisfies� � ï`| ñ��

for someaCü � ) �1D , then for almost every H0- D , it holds that� �RûAÊ Z ¦ C�ï`| V� û ã ï)H ñsñ0� �

.

(6)� ��ï`| ñÊ� �

if and only if� �0ï ¨ ñ��

for each compact set¨ ! | .

1.3. Metrically thin singularities of linear partial differential operators. LetÆ be a domain in

Q ¡, where �ü ÷

. We shall denote the Lebesgue measure by� ¡. Consider a class of Ë ï�Æ ñ of distributions defined on Æ , for instance 9 'Ì _�^ ï�Æ ñ ,D è � éwï�Æ ñ ( F�-yG ,

�K�ÖIÎ� ÷) or D * ï�Æ ñ . Consider a linear partial differential

operator � � � ï�¶ ð ÿ Ì ñ�� öØ § ^ � ó Ø ó J Á � Ø ï�¶ ñ ÿ ØÌof order H�ü ÷

, defined in Æ and having DÇ* coefficients � Ø ï�¶ ñ .Definition 1.4. A relatively closed subset | of Æ is called ï � ð Ë ñ -removable ifevery �N-ÍË ï�Æ ñ satisfying � � � �

in Æ 3 | does satisfy � � � �in all of Æ , in

the sense of distributions.

For instance, according to the classical Riemann removability theorem, discretesubsets 4 i � 5 �'§ of a domain Æ in ¾ are ï ÿ ð 9�* ñ -removable. In fact, since everydistribution solution of ÿ is holomorphic (hypoellipticity), functions extend to betrue holomorphic functions in a neighborhood of each

i � . The Riemann remov-ability theorem also holds under the weaker assumption that ��-�×Qï�Æ 3�4 i � 5 �0§ ñsatisfies �*ï Á È i � ñÊ� ��ï @ Á È i ��@ û ã ñ as

Áapproaches

i � .In several complex variables, the classical Riemann removability theorem may

be stated as follows.

Theorem 1.5. ([Ch1989]) Let ½ be a complex analytic subset of Æ . Holomorphicfunctions in Æ 3 ½ extend uniquely through ½ either if � ) � ] ½ � È ý or if��) � ] ½ � È ÷ and they belong to 9�*ä%4 5 ï�Æ ñ .

The second case also holds true for functions that belong to 9 ïä�4 5 ï�Æ ñ . Theproofs are elementary and short: in one or several complex variables, everythingcomes down to observing that

ã� is a trueP ï ãó � ó ñ near

Á � �and does not belong

to 9 ïä%4 5 .These preliminary statements are superseded by more general removability the-

orems, exposed in [HP1970], that we shall now restitute. Some of the (elemen-tary) proofs will be surveyed to give the flavour of the arguments.


In 1956, S. Bochner ([Bo1956, HP1970]) established remarkable removabilitytheorems, valid for general linear differential operators � , in which the metricalconditions on the size of the singularity | depend only on the order H of � . Somepreliminary material is needed.

Lemma 1.6. Let¨ ! Q ¡

be a compact set. For every × " �, there exists a

function ´ Ö -�D+*5 ï Q ¡ ñ with ´ Ö v ÷in a neighborhood of

¨and with �,¬]j j ´ Ö !¨ Ö such that @ ÿ ØÌ ´ Ö ï�¶ ñ @ � | Ø × û ó Ø ó for all ¶7- Q ¡

and all Ù�-HG ¡ .

Proof. Denote by Î � ïtÑ ñ the characteristic function of a sets ! Q ¡

. It sufficesto define the (rescaled) convolution integral ´ Ö ï�¶ ñ-�� × û ¡ % � ^ Î É Ê & ¥ ï · ñ � ïsï�¶ È· ñ gY× ñ � · , where � -HD *5 ï Q ¡ ñ has support contained in 4 @ ¶ @ �)÷ g-. 5 and satisfies% � ï · ñ � · � ÷

. ÞIt may happen that | is not ï � ð Ë ñ -removable, whereas | is removable for

some individual function ��-ÏË ï�Æ ñ satisfying certain supplementary conditions.In this case, we shall say that | is an illusory singularity of � .

Theorem 1.7. ([Bo1956, HP1970]) Let �Ê-�9 ãä%4 5 ï�Æ ñ . If, for each compact set¨ ! | , we have '*) � )*W�pÖR+ ú h 5 × û Á @ @ �ÐÎ É Ê @ @ ) � 7 � � ðthen | is an illusory singularity of � .

Whenever the integral is meaningful, for instance if �o-Ó9 ãä�4 5 ï�Æ ñ and ´�-D *5 ï�Æ ñ , we define ï�� ð ´ ñZ�� % Ç � ´ , where the integral is computed with respectto the Lebesgue measure. The formal adjoint of � , denoted by

� � , satisfies therelations ï � ´ ð � ñ�� ´ ð � � � for all ´ ð � -EDx*5 ï�Æ ñ , and these relations define ituniquely as � � ï`´ ñ�� öó Ø ó J Á ï È ÷ ñ ó Ø ó ÿ ØÌ ï � Ø ´ ñ ÄProof of Theorem 1.7. Let

¨ �� ïÍ�,¬ j jO´ ñ VC| and let ´ Ö be the family of func-tions constructed in Lemma 1.6. Since ��¬]j j � � ! | , we have ï � � ð ´ ñ1�ï � � ð ´ Ö ´ ñ»� ï�� ð � � ï`´ Ö ´ ñsñ . Lemma 1.6 entails that ®® ®® � � ï`´ Ö ´ ñ ®® ®® )7Ñ � |µ× û Á ,for some quantity | " �

that is independent of × . We deduce that @ ï � � ð ´ ñ @ �|µ× û Á @ @ �$Î É Ê @ @ ) � for all × " �. Thanks to the main assumption, this implies thatï � � ð ´ ñE� �

. ÞIf< - Q

with÷ �Û<$� > is the exponent of an 9 ; -space, we denote by

< � ��;; û ã -0/ ÷ ð > 1 the conjugate exponent, also defined by the relation÷ � ã; û ã; . By

Holder’s inequality, we have @ @ �ÒÎ É Ê @ @ ) � � � ¡ ï ¨ Ö ñ ã c ; @ @ �ÐÎ É Ê @ @ ) B .Corollary 1.8. Let ��-°9 ;ä�4 5 ï�Æ ñ , where

÷7�8<&� > . If, for each compact set¨ ! | , '*) � )dW�pÖR+ ú h � × û Á ; � ¡ ï ¨ Ö ñ ã c ; @ @ �$Î É Ê @ @ ) B � � � ð


then | is an illusory singularity of � .

The next theorem translates Corollary 1.9 in terms of Hausdorff measures, afiner concept than the Minkowski content.

Theorem 1.9. ([HP1970]) (i) Let÷ � < �Û> and assume that È H < �Âü �

. If� ¡ û Á ; ï ¨ ñ �y> for every compact set¨ ! | , then | is ï � ð 9 ;ä�4 5 ñ -removable.

(ii) Let< � > and assume that È H ü �

. If� ¡ û Á ï`| ñ � �

, then | isï � ð 9E*ä%4 5 ñ -removable.

(iii) Let< � > and assume that È H ü �

. If,� ¡ û Á ï ¨ ñ �Ö> for each

compact set¨ ! | , then � � is a measure supported on | , for every �J-A9 *ä%4 5

satisfying � � � �on Æ 3 | .

An application of (ii) to � � ÿ in one or several complex variables yields theRiemann removability Theorem 2.31 below.

Proof. We survey only the proof of (i). Let ´K-�D *5 ï�Æ ñ and set¨ �� |�VO�,¬]j j ´ .

Lemma 1.10. ([HP1970]) Let¨ ! Q ¡

be a compact set. Let< � with

÷ ��< ��y>and assume È H < � ü �

. For every × " �, there exists ´ Ö -�D *5 ï Q ¡ ñ with´ Ö v ÷

in a neighborhood of¨

and with ��¬]j j ´ Ö ! ¨ Ö such that for all ÙK-HG ¡with @ Ù @ � H , we have®® ®® ÿ ØÌ ´ Ö ®® ®® ) B � |4× Á û ó Ø ó � ¡ û Á ; ï ¨ ñ û × ã c ; ðwhere | " �

is independent of × .With such cut-off functions ´ Ö , since ��¬]j j � � ! | , we have ï � � ð ´ ñ �ï � � ð ´ Ö ´ ñ�� ï�� ð � � ï`´ Ö ´ ñsñ . By Holder’s inequality and the preceding lemma:@ ï � � ð ´ ñ @ � @ @ �$Î É Ê @ @ ) B ®® ®® � � ï`´ Ö ´ ñ ®® ®® ) B � | @ @ �ÐÎ É Ê @ @ ) BS � ¡ û Á ; ï ¨ ñ û × ã c ; Ä

The theorem follows from '*) �Ö + ú h @ @ �$Î É Ê @ @ ) B � � ðsince

� ¡ ï ¨ Ö ñ L �(remind

� ¡ û Á ; ï ¨ ñ �y> ). ÞIt seems impossible to get 9 ã removability without an assumption of growth.

At the opposite, in a CR context, the techniques introduced in [Jo1999b, MP1999]that are developed in Section 5 and in [MP2006a] will exhibit 9 ã -removabilityof certain closed subsets of generic submanifolds with only metrico-geometricassumptions.

210 JOEL MERKER AND EGMONT PORTEN� 2. REMOVABLE SINGULARITIES FOR HOLOMORPHIC FUNCTIONS OF ONE

OR SEVERAL COMPLEX VARIABLES

2.1. Painleve problem, zero length and analytic capacity. The classicalPainleve problem ([Pa1888, Ah1947]) is to find metric or geometric characteri-zations of compact sets

¨ ! ¾ that are ï ÿ ð 9 * ñ -removable, i.e. such that every��-7×Qï~¾ 3 ¨ ñ V�9E*Kï~¾ 3 ¨ ñ extends holomorphically through¨

.Theorem 1.9(ii) says that

� ã ï ¨ ñb� �suffices. It is also known ([Ma1984,

Pa2005]) that if� ã�� Ö ï ¨ ñ " �

for some × " �, then

¨has positive ana-

lytic capacity (definition below) and is never ï ÿ ð 9 * ñ -removable. Furthermore,Garnett ([Gar1970]) constructed a self-similar Cantor compact set

¨ ! ¾ with� � � ã ï ¨ ñ � û > which is ï ÿ ð 9�* ñ -removable. Consequently, Hausdorff mea-sure is not fine enough.

Under a geometric tameness assumption a converse to the sufficiency of( ã ï ¨ ñ�� holds and is usually called the solution to Denjoy’s conjecture.

Theorem 2.2. ([Cal1977, CMM1982]) A compact set¨ ! ¾ that is a priori

contained in a Lipschitz curve is ï ÿ ð 9 * ñ -removable if and only if it has zero÷-dimensional Hausdorff measure.

Classically, this statement is an application of the celebrated result of Calderon,Coifman, McIntosh and Meyer about the 9 ï -boundedness of the Cauchy integralon Lipschitz curves. Let us survey one of the simplified proofs ([MV1995]) whichinvolves Menger curvature, a concept useful in a recent answer to Painleve’s prob-lem obtained in [To2003].

Let Ó �� ó ï�¶ ð · ñ - Q ï � · � ´#ï�¶ ñ ô be a (global) Lipschitz graph; here´�-�D ú � ã is locally absolutely continuous and ´�� exists almost everywhere (a.e.)with @ @ ´ � @ @ ) Ñ � û > .

Theorem 2.3. ([Cal1977, CMM1982, MV1995]) If ��-}9 ï ïªÓ ñ , the Cauchy prin-cipal value integral 1 ú �*ï Á�ñ�� 'd) �ÖR+ ú ÷ý ��m � ó � û � ó ý Ö �*ï�� ñ� È Á �<�exists for almost every

Á -�Ó and defines a function1 ú �*ï Á�ñ on Ó , the Cauchy

transform of � , which belongs to 9 ï ïªÓ ñ and satisfies in addition@ @ 1 ú � @ @ ) ¥ ý 8 þ � | ã @ @ � @ @ ) ¥ ý 8 þ ðfor some positive constant | ã � | ã @ @ ´�� @ @ )7Ñ .Parametrizing Ó by ��ï�� ñÊ� � û m ´#ï�� ñ , dropping the innocuous factor

÷ û m ´ � ï�� ñand setting

ÁQ�� ¶ û m ´=ï�¶ ñ , one has to estimate the 9 ï -norm of the truncatedintegral 1 �Ö ï�� ñ ï�¶ ñ�� ó � û Ì ó ý Ö �*ï�� ñ��ï�� ñ È ��ï�¶ ñ �� ð


with a constant independent of × . Even more, interpolation arguments reduce thetask to a single estimate of the form�� ®® 1 �Ö ï�bdÔ ñ ®® ï � | ã @ } @ ðwhere | ã � | ã @ @ ´�� @ @ ) Ñ and where bdÔ is the characteristic function of an interval} ! Q

of length @ } @ . Following [MV1995], a symmetrization of the (implicitelytriple) integral % Ô 1 �Ö ï�b Ô ñ 1 �Ö ï�b Ô ñ provides÷ � Ô ®® 1 �Ö ï�bÕÔ ñ ®® ï � �Ö�Ö�7Ö Ê áâ öí §¬× ; ÷�ï�¶ í ý ï þ ñ È ��ï�¶ í ý ã þ ñ ÷�ï�¶ í ý ± þ ñ È ��ï�¶ í ý ã þ ñ æç ÑÑ~��¶ ã ��¶ ï ��¶ ± ûQP ï @ } @ ñ ðwhere B Ö �� ó ï�¶ ð · ð � ñ - } ± � @ · È ¶ @ " × ð @ � È ¶ @ " × ð @ � È · @ " × ô and whereØ ± is the permutation group of 4 ÷ ð ý ð . 5 .

Then a “magic” ([Po2005]) formula enters the scene:� j BÝï Á ã'ð Á ï ð Á ± ñ@ Á ã È Á ï @a@ Á ã È Á ± @a@ Á ï È Á ± @ � ï � öí §5× ; ÷��ï�¶ í ý ï þ ñ È �ï�¶ í ý ã þ ñ ÷��ï�¶ í ý ± þ ñ È ��ï�¶ í ý ã þ ñ ðwhere BÝï Á ã�ð Á ï ð Á ± ñ denotes the enclosed area; the left hand side measures the“flatness” of the triangle. This crucial formula enables one to link rectifiabilityproperties to the Cauchy kernel.

Definition 2.4. The Menger curvature of the triple 4 Á ã ð Á ï ð Á ± 5 is the square rootof the above k>ï Á ã'ð Á ï ð Á ± ñ�� j BÝï Á ã'ð Á ï ð Á ± ñ@ Á ã È Á ï @a@ Á ã È Á ± @a@ Á ï È Á ± @�Ùone sets k ��

if the points are aligned. One also verifies that k-ï Á ã�ð Á ï ð Á ± ñ �÷ gÚ ï Á ã'ð Á ï ð Á ± ñ , where Ú is the radius of the circumbscribed circle.

Thanks to the nice formula and to the basic inequalityk ��ï�¶ ñ ð ��ï · ñ ð ��ï�� ñ � ý ®®®®®®y ý Íþ û y ý Ì þÍ û Ì È y ý � þ û y ý Ì þ� û Ì@ � È · @ ®®®®®®the previous symmetric Cauchy triple integral is transformed to an integral involv-

ing geometric Lipschitz properties of Ó . After some computations ([MV1995]),one gets % Ô ®® 1 �Ö ï�b Ô ñ ®® ï � | ã @ @ ´�� @ @ ) Ñ Ñ @ } @ .

Menger curvature also appears in a recent result, considered to be an answer toPainleve’s problem.

Theorem 2.5. ([To2003, Pa2005]) A compact set¨ ! ¾ is not removable for×Qï~¾ 3 ¨ ñ VM9�*:ï~¾ 3 ¨ ñ if and only if there exists a nonzero positive Radon measureÛ with ��¬]j j Û ! ¨ such that

212 JOEL MERKER AND EGMONT PORTEN! there exists | ã " �with Û uÿXï Á ð ¸ ñ � | ã ¸ for every

Á -X¾ and ¸ " �;! %�%#% 5 k>ï�¶ ð · ð Á�ñ 7 ï � Û ï�¶ ñ � Û ï · ñ � Û ï Á�ñ � û > .

The first condition concerns the size of¨

; the second one is of quantitative-geometric nature.

We conclude by mentioning a classical functional characterization due toAhlfors, usually considered to be only a reformulation of Painleve’s problem,but which has already found generalizations in locally integrable structures ( � 2.16below). The analytic capacity of a compact set

¨ ! ¾ is21

an-cap ï ¨ ñ�� ,¬]j ó @ � � ï`> ñ @ � ��- ( * ï~¾ 3 ¨ ñ ð @ @ � @ @ ) Ñ � ÷ ô ðwhere

( *Kï~¾ 3 ¨ ñ denotes the space of bounded holomorphic functions definedin ¾ 3 ¨ (or defined in the complement of

¨in the Riemann sphere ¾vÃ 4 > 5

,because 4 > 5

is removable).

Theorem 2.6. ([Ah1947, Ma1984, HT1997]) A compact set¨ ! ¾ is removable

for ×Qï~¾ 3 ¨ ñ VE9 * ï~¾ 3 ¨ ñ if and only if an-cap ï ¨ ñ�� .

2.7. Rado-type theorems. A classical theorem due to Rado ([Ra1924, Stu1968,RS1989, Ch1994]) asserts that a continuous function � defined in a domain Æ ! ¾that is holomorphic outside its zero-set � û ã ï � ñ is in fact holomorphic everywhere.By a separate holomorphicity argument, this statement extends directly to severalcomplex variables. In [Stu1993], it is shown that � û ã ï � ñ may be replaced by� û ã ï~¦ ñ , where ¦ ! ¾ is compact and has null analytic capacity. In [RS1989],it is shown that a continuous function defined in a strongly pseudoconvex D ï hy-persurface ² ! ¾ ¡ ( µüºý ) that is CR outside its zero-set is CR everywhere;a thin subset of weakly pseudoconvex points is allowed, but the case of generalhypersurfaces is not covered. In [Al1993], it is shown that closed sets � û ã ï~¦ ñare removable in the same situation, wehere ¦ ! ¾ is a closed polar set, viz.¦ ! 4 ø � È > 5

for some subharmonic function ø Jv È > . Chirka strength-ens these results in the following theorem, where no assumption is made on thegeometry of the hypersurface.

Remind ([Ch1989, De1997]) that ¦ ! ¾ Á is called complete pluripolar if¦ � 4 ´ � È > 5for some plurisubharmonic function ´«Jv È > on ¾ Á .

Theorem 2.8. ([Ch1994]) Let ² ! ¾ ¡ ( �üÊý ) be hypersurface that is a localLipschitz ( D ú � ã ) graph at every point, let | be a closed subset of ² and let � �² 3 |ÉLÞ¾ Á 3 ¦ be a continuous mapping satisfying @ @ � @ @ Y/ ý Ä È 6 þ ��> such thatthe set of limit values of � from ² 3 | up to | is contained in a closed complete

21If Ü Ü Ý�Ü Ü ÞUßáà ½ , setting â��<¤-�ÕãÅç�äåÝ_�ªæ � â Ý_��¤Y�&ç>èéäÿ½ â Ý_��æ��¬Ý_�<¤-�ªç , we have â<�ªæ �¤ç=¹ ,Ü Ü â8Ü Ü Þ ß�à ½ and Ü â¬�³��æ��Ü'çêÜ Ý7�³��æ��Ü èÕë ½ â Ü Ý_��æ��Ü �(ì�í Ü Ý7�u�ªæ �(Ü , so that in the definition ofanalytic capacity, we may restrict to take the supremum over functions â �Ïî ¢ � á äåá �with Ü Ü âAÜ Ü Þ ßïà?½ and â��æ��¤çz¹ .


pluripolar set ¦ ! ¾ Á ( HÔü ÷). Then the trivial extension

t� of � to | definedby

t� �� on | is a CR mapping of class 9"* on the whole of ² .

In higher codimension, nothing is known.

Open question 2.9. Let ² ! ¾ ¡ ( Küv. ) be a generic submanifold of codimen-sion �Süyý and of CR dimension H ü ÷

that is at least D ã . Let �7-ED ú ïÏ² ñthat is

CR outside its zero-set � û ã ï � ñ . Is � CR everywhere ?

Remind that condition (P) (Definition 3.5(III)) for a linear partial differentialoperator � of principal type assures local solvability of the equations � � �ñð

.Remind also that nowhere vanishing vector fields are of principal type.

Theorem 2.10. ([HT1993]) Let Æ ! Q ¡( Éü:ý ) be a domain and let 9 be a

nowhere vanishing vector field on Æ having D * complex-valued coefficients andsatisfying condition (P) of Nirenberg-Treves. If ��-JD ú ï�Æ ñ satisfies 9E� � �

inÆ 3 � û ã ï � ñ in the sense of distributions, then � is a weak solution of 9�� all

over Æ .

2.11. Capacity and partial differential operators having constant coefficients.The preceding results admit partial generalizations to vector field systems. LetÆ ! Q ¡

be an open set and let � � � ï�ÿ Ì ñ�� y Ø § ^ � Ø ÿ ØÌ be a linear partialdifferential operator having constant coefficients � Ø -«¾ . By a theorem due toMalgrange, Ehrenpreis and Palamodov ([Ho1963]), such a � always admits afundamental solution, namely there exists a distribution ¦à- �P��ï Q ¡ ñ such that� ï�ÿ Ì ñ ¦ � 6 ú is the Dirac measure at the origin.

Let Ë ! � �~ï�Æ ñ be a Banach space, e.g. Ë � 9 ; ï�Æ ñ with÷7��<�� > , orË � 9 * ï�Æ ñ VSD ú ï�Æ ñ , or Ë � D ú � é¤ï�Æ ñ with

� � IK� ÷.

Definition 2.12. For each relatively closed set | ! Æ , the Ë -capacity of | withrespect to � isË -cap òÝï`| ð ÿÆ ñ�� ,¬ j ¿ ®® ï � � ð Î Ç ñ ®® � �7-óË ð @ @ � @ @ ô � ÷ ð �,¬]j jéï � � ñ õy| Ã+Ä

If a closed set | ! Æ is ï � ð Ë )-removable, by definition � � � �everywhere,

hence Ë -Cap ò ï`| ð Æ ñv� �. The following theorem establishes the converse for

a wide class of differential operators having constant coefficients. For Ùµ-NG ¡ ,denote by

| ý Ø þ ï�¶ ñ�� ÿ ØÌ | ï�¶ ñ the Ù -th partial derivative of a polynomial| ï�¶ ñ -Q / ¶ 1 .

Theorem 2.13. ([HP1972]) Assume that � possesses a fundamental solution ¦Î-� � ï Q ¡ ñ such that � ý Ø þ ï�ÿ Ì ñ ¦ is a regular Borel measure on

Q ¡for every Ù0-HG ¡ .

Let Æ`õ Q ¡be a bounded domain and let | ! Æ be a relatively closed subset.

Then ! | is ï � ð 9 ;ä%4 5 ñ -removable,÷ � < � > , if and only if 9 ; -cap òÝï`| ð Æ ñE� �

;! | is ï � ð 9 * D ú ñ -removable if and only if 9 * D ú -cap ò ï`| ð Æ ñE� �;

214 JOEL MERKER AND EGMONT PORTEN! | is ï � ð D ú � é ñ -removable,� � IN�[÷

, if and only if D ú � é -cap ò ï`| ð Æ ñ"��.

This hypothesis about � is satisfied by elliptic, semi-elliptic, and parabolicoperators and also by the wave operator in

Q ï([HP1972]). The theorem (whose

proof is rather short) also holds true if � � � ï�¶ ð ÿ Ì ñ has real analytic coefficientsand admits a fundamental solution ¦ such that � ý Ø þ ¦ is a regular Borel measurefor every Ù�-HG ¡ . But it is void in 9 ã .Theorem 2.14. ([HP1972]) There is a unique function, called a capacitary ex-tremal, � ®"õ ; -P9 ; ï�Æ ñ with @ @ � ®"õ ; @ @ ) B �ß÷

and � � ®"õ ; � �in Æ 3 ¨ such that � � ®"õ ; ð Î Ç � 9.; -cap ò ï ¨ ð Æ ñ .

We observe that the definition of 9 ; -cap ò@ï ¨ ð Æ ñ is inspired from Ahlfors’notion of analytic capacity and we mention that the capacitary extremal � ®"õ ; islinked to the Riemann uniformization theorem.

Example 2.15. In fact, with Æ � ¾ and � � ÿ2g-ÿ ùÁb�»� ÿ , the 9Z* -capacity of acompact set

¨ ! ¾ with respect to ÿ may be shown to be equal, up to the constant� , to the analytic capacity of¨

, namely9 * -cap � ï ¨ ð ¾ ñ-� � an-cap ï ¨ ñ ÄIndeed, letting �P-É9 * ï~¾ ñ , assuming that ÿ�� is supported by

¨, choosing a

big open disc ¢ õ ¾ containing¨

, integrating by parts (Riemann-Green) andperforming the change of variables h �� ÷ g Á , we may compute ÿ�� ð Î ] � ÿ�� ð ÎUö � ÷ý m �#� ö ÿ��ÿ ùÁ � ùÁ �� Á � ÷ý m � � ö �*ï Á�ñ � Á � � � � ï`> ñ ÄRemind (footnote) that in the definition of an-cap ï ¨ ñ given in � 2.1, one mayassume that �*ï`> ño� �

. If in addition the complement of¨

in the Riemannsphere ¾¹Ã 4 > 5

is simply connected, the unique solution � ®�õ ; of ÿ�� ®"õ ; ð Î ] �9 * -cap� ï ¨ ð ¾ ñ asserted by Theorem 2.14, viz. the unique solution of the ex-

tremal problem�,¬]j ¿ @ � � ï`> ñ @ � ��-�9 * ï~¾ ñ ð ÿ~�«g-ÿ ùÁ � �in ¾ 3 ¨ ð �*ï`> ñÊ� �

and @ @ � @ @ )7Ñ � ÷ Ãis the (unique) Riemann uniformization map � ®"õ ; � ¾¹Ã 4 > 5 3 ¨ L ÿ satisfy-ing � ®"õ ;>ï`> ñÊ� �

and ÿ � � ®"õ ;-ï`> ñ " �.

2.16. Removable singularities of locally solvable vector fields. Let÷ ï Q ¡ ñE�� ¿ ��-ED * ï Q ¡ ñE� '*) �ó Ì ó + * ®® ¶ é ÿ ØÌ �*ï�¶ ñ ®® � � ðx3 I ð Ù�-EG ¡ Ãbe the space of Dx* functions defined in

Q ¡and having tempered growth. As is

known ([Ho1963]), the Fourier transform� �*ï�ø ñE�� ^ � û ï �Ø��ù Ì � ú>û �*ï�¶ ñ ��¶ ð �C- ÷ ï Q ¡ ñ ð

HOLOMORPHIC EXTENSIONS AND REMOVABLE SINGULARITIES 215ü ¶ ð ø�ý �� y ¡ � � ã ¶ � ø � , is an automorphism of÷ ï Q ¡ ñ having as inverse� û ã �*ï�ø ñE�� ^ � ï �Ø��ù Ì � ú>û �*ï�¶ ñ ��¶ �í� �*ï È ø ñ Ä

Equipping÷ ï Q ¡ ñ with the countable family of semi-norms

i é¢� Ø ï�� ñ ��¬]j Ì § � ^ ®® ¶ é ÿ ØÌ �*ï�¶ ñ ®® , the space÷ ��ï Q ¡ ñ of tempered distributions consists of

linear functionals

Uon

÷ ï Q ¡ ñ that are continuous, viz. there exists | " �andI ð Ù�-HG ¡ such that ®® ü

U ð �þý ®® � | i é¢� Ø ï�� ñ for every �7- ÷ ï Q ¡ ñ .For

< - Qwith

÷H�Ö<C� > and for ì�- Q, we remind the definition of the

Sobolev space9 ;í ï Q ¡ ñ�� ¿ U - ÷ � ï Q ¡ ñ�� @ @ U @ @ ) B\ �� ®® ®® � ûí U ®® ®® ) B �y> Ã ð

where � û í U ï�¶ ñ��í� û ã 5 ï ÷ û @ ø @ ï ñ û í c ï � U ï�ø ñ 7 ï�¶ ñ . For ì � FC-�G and<

in therange

÷ � < �º> , the space 9 ;è ï Q ¡ ñ is exactly the subspace of functions ø�-9 ; ï Q ¡ ñ whose partial derivatives of order� F (in the distributional sense) belong

to 9 ; ï Q ¡ ñ . This space is equivalently normed by @ @ ø @ @ ) B ÿ �� y ó Ø ó J è ®® ®® ÿ ØÌ ø ®® ®® ) B .Let Æ ! Q ¡be a domain and let � � � ï�¶ ð ÿ Ì ñ � y ó Ø ó J Á � Ø ï�¶ ñ ÿ ØÌ be a

linear partial differential operator of order H ü ÷defined in Æ and having D *

coefficients � Ø ï�¶ ñ .Definition 2.17. We say that � is locally solvable in 9 ; with one loss of derivativeif every point

i - Æ has an open neighborhood Æ ú ! Æ such that for everycompactly supported

U -P9 ;í ï Æ ú ñ , the equation �� Uhas a solution �É-9 ;í � Á û ã ï Æ ú ñ .

Theorem 2.18. ([BeFe1973, HP1996]) Let 9 be a nowhere vanishing vector fieldhaving D+* coefficients in a domain Æ ! Q ¡

( Kü�ý ) and assume that 9 satisfiescondition (P) of Nirenberg-Treves. Then for every

< - Qwith

÷ � < �Ó> , theoperator 9 is locally solvable in 9 ; with loss of one derivative.

Example 2.19. As discovered in [HT1996], local solvability fails to hold in 9�*for the (locally solvable) vector field

�� Ì È �Ì ¥ � û ã c ó Ì ó ��Í

satisfying (P) on

Q ï.

Removable singularities for vector fields in 9O; have been studied in [HT1996,HT1997]. Because of the example, results in 9 ; with

÷ � < �Ö> differ fromresults in 9�* .

Definition 2.20. A relatively closed set | ! Æ of an open set Æ ! Q ¡is

everywhere ï � ð 9 ; ñ -removable if for every open subset Æ ! Æ and for every��-}9 ; ï Æ ñ satisfying � � � �in Æ 3 | , then � also satisfies � � � �

in all of Æ .

Theorem 2.21. ([HT1996, HT1997]) Let 9 be a nowhere vanishing vector fieldhaving Dx* coefficients in an open subset Æ ! Q ¡

( üÖý ) and assume that 9satisfies condition (P). Let

< - Qwith

÷ � < �µ> . Then a relatively closed set

216 JOEL MERKER AND EGMONT PORTEN| ! Æ is everywhere ïÍ9 ð 9 ; ñ -removable if and only if there is an open coveringof | by open sets Æ î , ÄÕ-uê , such that9 ; -cap ) x|mVQÆ$î ð Æ î � � ðfor every ÄS- ê .

In 9�* , when trying to perform the proof of this theorem, local solvability ofpositive multiples of 9 is technically needed. Observing that � � � �

is equiva-lent to ��

, the following notion appeared to be appropriate to deal withïÍ9 ð 9 * ñ -removability.

Definition 2.22. ([HT1996, HT1997]) The full 9Z* -capacity of a relatively closedset | ! Æ with respect to 9 is

full- 9 * -cap ) ï`| ð Æ ñ�� ,¬]j�) ¿ 9 * -cap �) ï`| ð Æ ñ Ã ðwhere the supremum is taken over all vector fields

t9 � � � 9 with � -AD * ï�Æ ñsatisfying �,¬]j Ç ®® ÿ éÌw� ï�¶ ñ ®® � ÷

for @ I @ � ÷.

By a fine analysis of the degeneracies of 9 and of the structure of the Suss-mann orbits of ó ËZÌ 9 ð cB�N9 ô , Hounie-Tavares were able to substantially gener-alize Ahlfors’ characterization.

Theorem 2.23. ([HT1996, HT1997]) A relatively closed set | ! Æ is everywhereïÍ9 ð 9E* ñ -removable if and only if there is an open covering of | by open sets Æ6î ,ÄS- ê , such that

full- 9 * -cap ) ï`|mVXÆ î ð Æ î ñÊ� � ðfor every ÄS- ê .

On orbits of dimension one, 9 behaves as a multiple of a real vector field (one-dimensional behavior); on orbits of dimension two, 9 has the behavior of ÿ ona Riemann surface ½ ! × , but on the complement × 3 ½ which is a union ofcurves with different endpoints along which

Ë-Ì 9 and cB�N9 are both tangent (de-generacy), 9 behaves again as a multiple of a real vector field (one-dimensionalbehavior). As shown in [HT1996] (main Theorem 7.3 there) a relatively closedset | ! Æ is everywhere removable if and only if | does not disconnect almostevery curve on which 9 has one-dimensional behavior and furthermore, the inter-section of | with almost every (reduced) orbit of dimension two has zero analyticcapacity for its natural holomorphic structure.

Open problem 2.24. Study removability of a D * locally integrable involutivestructure of rank

ø üyý in terms of analytic capacity.


2.25. Cartan-Thullen argument and a local continuity principle. The Behnke-Sommer Kontinuitatssatz, alias Continuity Principle, states informally as follows([Sh1990]). Let ï ½ Å ñ Å § be a sequence of complex manifolds with boundaryÿ_½ Å contained in a domain Æ of ¾ ¡ . If ½ Å converges to a set ½ * ! Æ andif ÿ_½ * is contained in Æ , then every holomorphic function ��-�×Qï�Æ ñ extendsholomorphically to a neighborhood of the set ½ * in ¾ ¡ . The geometries of the½ Å and of ½ * have to satisfy certain assumptions in order that the statement becorrect; in addition, monodromy questions have to be considered carefully. Forapplications to removable singularities in [MP2006a], the rigorous Theorem 2.27below is formulated, with the ½ Å being embedded analytic discs.

We denote byÁ � ï Á ã ð Ä Ä Ä ð Á ¡ ñ the complex coordinates on ¾ ¡ and by @ Á @ ��Õl{Ë ãKJ � J ¡ @ Á � @ the polydisc norm. If ¦ ! ¾ ¡ is an arbitrary subset, for ¸ " �

, wedenote by ù ß ï~¦ ñ�� Úú §�� 4 Á -X¾ ¡ � @ Á È i @ �i¸ 5the union of all open polydiscs of radius ¸ centered at points of ¦ .

Lemma 2.26. ([Me1997]) Let Æ be a nonempty domain of ¾ ¡ and let ½ � ÿÎL Æ ,½Ó-�×Qïsÿ ñ VSD ã ï ÿ ñ , be an analytic disc contained in Æ having the property thatthere exist two constants k and | with

� �\k �y| such thatk @ � ï È � ã @ � @ ½ ï�� ï ñ È ½ ï�� ã ñ @ ��| @ � ï È � ã @ ðfor all distinct points � ã�ð � ï - ÿ . Set¸ �� )*W�p ó @ Á È ½xï�� ñ @ �0Á - ÿGÆ ð ��-�ÿPÿ ô ðnamely ¸ is the polydisc distance between ½ ï�ÿ©ÿ ñ and ÿGÆ , and set ì �� ¸=kðg�ý�| .Then for every holomorphic function �7-C×Qï�Æ ñ , there exists a holomorphic func-tion ÂÉ-C×( ù í ïÍ½ ï ÿ ñsñ such that Â � � on

ù í `½xï�ÿ©ÿ ñ .�� ç ý � ` þ] ^

Ç

] ^û \�� ç ý ` þ��

Continuity principle

Ç

ç ý ` þ ç F ý � ` þ û \ � ç F ý ` þ �

The inequalities involving k and | are satisfied for instance when ½ is D ãembedding of ÿ into Æ . Whereas ½ ï ÿ ñ is contained in Æ , the neighborhoodù í ïÍ½xï ÿ ñsñ is allowed to go beyond. We do not claim that the two functions�:-P×Qï�Æ ñ and Â -o× ù í ïÍ½ ï ÿ ñsñ stick together as a holomorphic function


globally defined in Æ9Ã ù í ïÍ½xï ÿ ñsñ . In fact, Æ�V ù í ïÍ½ ï ÿ ñsñ may have severalconnected components.

��

] ^ ] ^

The intersection �� \ Ó ý Ó ��Õ�Õ may be not connected

Ç Ç ÇÇ

û \ ý ç ý ` þ£þÇ Ç

û \ ý ç ý ` þuþ ÇÇÇ

In the geometric situations we encounter in [Me1997, MP1999, MP2002,MP2006a], after shrinking Æ somehow slightly to some subdomain Æ � , we shallbe able to insure that the intersection Æ ��V ù í ïÍ½ ï ÿ ñsñ is connected and that theunion Æ � Ã ù í ïÍ½xï ÿ ñsñ is still significantly “bigger” than Æ .

Proof of Lemma 2.26. Let �8-:×Qï�Æ ñ . For �Ó- ÿ arbitrary, we consider thelocally converging Taylor series y é § ^ � é ï Á È ½xï�� ñsñ é of � at ½xï�� ñ . For ¸�� with� �i¸��©�i¸ arbitrarily close to ¸ , since

ù ß ïÍ½xï�ÿPÿ ñsñ õÙÆ , the quantity² ß ï�� ñ�� ,¬]j ó @ �*ï Á�ñ @ ��Á - ù ß ïÍ½xï�ÿ©ÿ ñsñ ô��y> ðis finite (it may explode as ¸ � L=¸ ). Thus, Cauchy’s inequality on a polydisc ofradius ¸�� centered at an arbitrary point ½ ï � �(X ñ of ÿ©ÿ yields÷I � ®®®®® ÿ

ó é ó �ÿ Á é `½xï � ��X ñ ®®®®® � ² ß ï�� ñ¸ � é ðuniformly for all � ��X -�ÿ©ÿ . Then the maximum principle applied to the function��»L �� % � :� � % ïÍ½ ï�� ñsñ holomorphic in ÿ provides the crucial inequalities (Cartan-Thullen argument):@ � é @ � ÷I � ®®®®® ÿ

ó é ó �ÿ Á é ½xï�� ñ ®®®®® �÷I �à�,¬]jr s`u § � ` ®®®®® ÿ

ó é ó �ÿ Á é ½xï � ��X ñ ®®®®®� ² ß ï�� ñ¸ � é ÄConsequently, the Taylor series of � converges normally in the polydisc ÿ ¡ß ïÍ½ ï�� ñsñof center ½xï�� ñ and of radius ¸ , this being true for every ½xï�� ñ -°½ ï ÿ ñ . Theseseries define a collection of holomorphic functions Â ç ý � þ � ß -õ×Qïsÿ ¡ß ïÍ½ ï�� ñsñsñparametrized by �µ- ÿ . We claim that the restrictions of all these functions


to the smaller polydiscs ÿ ¡ í ïÍ½xï�� ñsñ stick together in a well defined holomorphicfunction ÂÓ-}× ù í ïÍ½xï ÿ ñsñ .

Indeed, assume that two distinct points � ãåð � ï - ÿ are such that the intersectionof the two small polydiscs ÿ ¡ í ïÍ½xï�� ã ñsñ V/ÿ ¡ í ïÍ½ ï�� ï ñsñ is nonempty, so @ ½xï�� ï ñ È½xï�� ã ñ @ �ÖýÂì . It follows that for every � belonging to the segment /q� ã ð � ï 1 , wehave: @ � È � ã @ � @ � ï È � ã @ � @ ½xï�� ï ñ È ½xï�� ã ñ @ gYkñ�JýÂì gYkwhence @ ½ ï�� ñ È ½xï�� ã ñ @ �y| @ � È � ã @ �JýM|ñì gYk � ¸ ÄThis means that the curved segment ½xïR/q� ã ð � ï 1 ñ is contained in the connectedintersection of the two large polydiscs ÿ ¡ß ïÍ½ ï�� ã ñsñ V ÿ ¡ß ïÍ½xï�� ï ñsñ . In a smallneighborhood of ½ ï�� ã ñ and of ½xï�� ï ñ , the two holomorphic functions Â ç ý � � þ � ßand Â ç ý � ¥ þ � ß coincide with � by construction. Thanks to the principle of ana-lytic continuation, it follows that they even coincide with � in a thin connectedneighborhood of the segment ½xïR/q� ã�ð � ï 1 ñ . Again thanks to the principle of an-alytic continuation, Â ç ý � � þ � ß and Â ç ý � ¥ þ � ß coincide in the connected intersectionÿ ¡ß ïÍ½xï�� ã ñsñ V�ÿ ¡ß ïÍ½ ï�� ï ñsñ . It follows that they stick together to provide a welldefined function Â ç ý � � þ � ç ý � ¥ þ � ß that is holomorphic in ÿ ¡ß ïÍ½xï�� ã ñsñ Ã3ÿ ¡ß ïÍ½ ï�� ï ñsñ .In conclusion, the restriction Â ç ý � � þ � ç ý � ¥ þ � ß ®® ` \ ý ç ý � � þ£þ � ` \ ý ç ý � ¥ þuþ is holomorphic in

the union of the two small polydiscs ÿ í ïÍ½ ï�� ã ñsñ Ã�ÿ í ïÍ½ ï�� ï ñsñ , whenever theintersection ÿ í ïÍ½xï�� ã ñsñ V/ÿ í ïÍ½xï�� ï ñsñ is nonempty. This proves that all the re-strictions Â ç ý � þ � ß ®® ` ^\ ý ç ý � þ£þ stick together in a well defined holomorphic functionÂÎ-C× ù í ïÍ½ ï ÿ ñsñ . Þ

In the next theorem (a local continuity principle often used in [Me1997,MP1999, MP2002, MP2006a]), ½ ã ï ÿ ñ ! Æ , but contrary to Lemma 2.26, ½ Ý ï ÿ ñmay well be not contained in Æ for

Ü � ÷; nevertheless, the boundaries ½ Ý ï�ÿ©ÿ ñ

must always stay in Æ .

Theorem 2.27. ([Me1997]) Let Æ be a nonempty domain in ¾ ¡ and let ½ Ý � ÿÓL¾ ¡ , ½ Ý -µ×Qïsÿ ñ V�D ã ï ÿ ñ , be a one-parameter family of analytic discs, whereÜ -0/ � ð ÷ 1 . Assume that there exist two constants k Ý and | Ý with� �\k Ý ��| Ý such

that k ÝY@ � ï È � ã @ � @ ½xï�� ï ñ È ½ ï�� ã ñ @ �y| Ý @ � ï È � ã @ ðfor all distinct points � ã�ð � ï - ÿ . Assume that ½ ã ï ÿ ñ ! Æ , set¸ Ý �� )*W�p ó @ Á È ½ Ý ï�� ñ @ ��Á -�ÿÆ ð �Õ- ÿ©ÿ ô ðnamely ¸ Ý is the polydisc distance between ½ Ý ï�ÿ©ÿ ñ and ÿÆ , and set ì Ý ��¸ Ý k Ý g�ý�| Ý . Then for every holomorphic functions �7-C×Qï�Æ ñ , and for all

Ü -�/ � ð ÷ 1 ,there exist holomorphic functions Â Ý -É× ù í F ïÍ½ Ý ï ÿ ñsñ such that Â Ý � � inù í F ïÍ½ Ý ï�ÿ©ÿ ñsñ .


Proof. Let � ! / � ð ÷ 1 be the connected set of real

Ü ú such that the statement is truefor all

Üwith

Ü ú � Ü � ÷. By Lemma 2.26, we already know that

÷ -�� . We wantto prove that � � / � ð ÷ 1 . It suffices to prove that � is both open and closed.

The fact that � is closed follows by “abstract nonsense”. We claim that � isalso open. Indeed, let

Ü ú -�� and let

Ü ã � Ü ú be such that ½ Ý � ï ÿ ñ is contained inù í F / ïÍ½ Ý / ï ÿ ñsñ . Since Â Ý / � � in

ù í F / ïÍ½ Ý / ï�ÿ©ÿ ñsñ and since the polydisc distancebetween ½ Ý � ï�ÿ©ÿ ñ and ÿGÆ is equal to ¸ Ý � , it follows as in the first part of the proofof Lemma 2.26, that the Taylor series of Â Ý / at arbitrary points of the form ½ Ý � ï�� ñ ,�K- ÿ , converges in the polydisc ÿ ¡ß F � ïÍ½ Ý � ï�� ñsñ . This gives holomorphic func-

tions Â ç ý � þ � ß F � -C×�uÿ ¡ß F � ïÍ½ Ý � ï�� ñsñ , for every ��- ÿ . Reasoning as in the second

part of the proof of Lemma 2.26, we obtain a function Â Ý � -�×� ù í F � ïÍ½ Ý � ï ÿ ñsñ with Â Ý � � � in

ù í F � ïÍ½ Ý � ï�ÿ©ÿ ñsñ . This shows that � is open, as claimed andcompletes the proof. Þ2.28. Singularities as complex hypersurfaces. Let Æ ! ¾ ¡ ( Êüºý ) be a do-main. A typical elementary singularity in Æ is just the zero set ¿ : �� 4 � � ��5

ofa holomorphic function ��-(×Qï�Æ ñ since for instance, the functions

÷ gð� � , }ü ÷,

and � ã c : are holomorphic in Æ 3 ¿ : and singular along ¿ : . Because ¾ is alge-braically closed, the closure in Æ of such ¿ : ’s necessarily intersects ÿGÆ . Earlyin the twentieth century, the italian mathematicians Levi, Severi and B. Segre([Se1932]) interpreted Hartogs’ extension theorem as saying that compact sets¨ ! Æ are removable, confirming the observation ¿ : VKÿÆ¿J�"! .Definition 2.29. (i) A relatively closed subset | of a domain Æ ! ¾ ¡ is calledremovable if the restriction map ×Qï�Æ ñ Lõ×Qï�Æ 3 | ñ is surjective.

(ii) Such a set | is called locally removable if for everyi -�| , there exists an

open neighborhood Æ ú ofi

in Æ such that the restriction map ×0Dï Æ ú$ÃxÆ ñ 3 | L×Qï�Æ 3 | ñ is surjective.

Under the assumption that | is contained in a real submanifold of Æ , the gen-eral philosophy of removable singularities is that a set too small to be a ¿ : (viz.a complex ï� È ÷ ñ

-dimensional variety) is removable. The following theoremcollects five statements saying that | is removable provided it cannot containany complex hypersurface of Æ . Importantly, our submanifolds D of Æ will al-ways be assumed to be embedded, namely for every

i -UD , there exist an openneighborhood Æ ú of

iin Æ and a diffeomorphism � ú � Æ ú°L Q ï ¡

such that� ú�ï)D V Æ ú ñ0� Q Ê Z ¦ C Ã 4 ��5 .Theorem 2.30. Let Æ be a domain of ¾ ¡ ( AüÛý ) and let | ! Æ be a relativelyclosed subset. The restriction map ×Qï�Æ ñ L ×Qï�Æ 3 | ñ is surjective, namely | isremovable, under each one of the following five circumstances.

(rm1) | is contained in a connected submanifold D ! Æ of codimension üÙ. .(rm2)

� ï ¡ û ï ï`| ñ�� .


(rm3) | is a relatively closed proper subset of a connected D ï submanifoldD ! Æ of codimension ý .

(rm4) | � D is a connected D ï submanifold D ! Æ that is not a complexhypersurface of Æ .

(rm5) | is a closed subset of a connected D ï real hypersurface ² ã ! Æ thatdoes not contain any CR orbit of ² ã

.

In (rm1) and in (rm2), | is in fact locally removable. In (rm5), two kinds ofCR orbits coexist: those of real dimension ïÏý{ È ý ñ , that are necessarily complexhypersurfaces, and those of real dimension ïÏý{ È ÷ ñ

, that are open subsets of² . It is necessary to exclude them also. Indeed, if for instance Æ is divided intwo connected components Æ 2 by a globally minimal ² ã

, taking | � ² ã,

any locally constant function on Æ 3 ² ãequal to two distinct constants k 2 inÆ 2 does not extend holomorphically through ² ã

. The proof of the theorem iselementary22 and we will present it in � 2.34 below, as a relevant preliminary toTheorems 4.9, 4.10, 4.31 and 4.32, and to the main Theorem 1.7 of [MP2006a]).

We mention that under the assumption of local boundedness, more massive sin-gularities may be removed. An application of Theorem 1.9(ii) to several complexvariables deserves to be emphasized.

Theorem 2.31. ([HP1970]) If Æ ! ¾ ¡ is a domain and if | ! Æ is a rela-tively closed subset satisfying

� ï ¡ û ã ï`| ñZ� �, then | is removable for functions

holomorphic in Æ 3 | that are locally bounded in Æ .

Following [Stu1989] and [Lu1990], we now provide variations on (rm2). Anyglobal complex variety of codimension one in ¾ ¡ is certainly of infinite ïÏý{ È ý ñ -dimensional area.

Theorem 2.32. ([Stu1989]) Every closed set | ! ¾ ¡ satisfying� ï ¡ û ï ï`| ñ �y>

is removable for ×Qï~¾ ¡ ñ .The result also holds true in the unit ball

ò ¡of ¾ ¡ , provided one computes

the ïÏý{ È ý ñ -dimensional Hausdorff measure with respect to the distance functionderived from the Bergman metric ([Stu1989]). Also, if ½ is an arbitrary com-plex -dimensional closed submanifold of ¾ ¡ , every closed subset | ! ½ with� ï � û ï ï`| ñ ��> is removable for ×Qï ½ 3 | ñ ([Stu1989]).

A finer variation on the theme requires that� ï ¡ û ï ï`|�VCÚ ò ¡wñ does not grow

too rapidly as a function of the radius Ú L > . For instance ([Stu1989]), aclosed subset | ! ¾ ï that satisfies

� ï ï`| V Ú ò ¡ ñ ��k£Ú ï for all large Ú isremovable, provided k � � ¥¤$# ï . It is expected that k¼� � is optimal, since the line9 �� 4 ï Á ã�ð � ñ 5 satisfies

� ï ïÍ9uV[Ú s ï ñÊ� � Ú ï .22Some refinements of the statements may be formulated, for instance assuming in

(rm3) and (rm4) that % is �~¸ and has some metrically thin singularities ([CSt1994]).


Yet another variation, raised in [Stu1989], is as follows. Consider a closedset | in the complex projective space � ¡ ï~¾ ñ ( Pü�ý ) such that the HausdorffïÏý{ È ý ñ -dimensional measure (with respect to the Fubini-Study metric) of | isstrictly less than that of any complex algebraic hypersurface of � ¡ ï~¾ ñ . Is it truethat | is a removable singularity for meromorphic functions, in the sense thatevery meromorphic function on � ¡ ï~¾ ñ 3 | extends meromorphically through | ?This question was answered by Lupacciolu.

Let �'&)(<ï Á ðKh ñ denote the geodesic distance between two pointsÁ ðKh - � ¡ ï~¾ ñ

relative to the Fubini-Study metric and let� � &$( denote the a -dimensional Haus-

dorff measure in � ¡ ï~¾ ñ computed with �)&)( . Given a nonempty closed subset| ! � ¡ ï~¾ ñ , define:¸Gï`| ñE�� #l{Ë � § ò ^ ý ] þ �*&$(�ï Á ð | ñ�#l{Ë � � Â § ò ^ ý ] þ �*&)( ï Á ðKh ñ � �#l{Ë � § ò ^ ý ] þ �'&)( ï Á ð | ñ� ) l>�+&)(<ï � ¡ ï~¾ ñ ñ �)÷ ÄIf the Fubini-Study metric is normalized so that k � ' � ¡ ï~¾ ñ7� ÷

, the ïÏý{ È ý ñ -dimensional volume of an irreducible complex algebraic hypersurface ½ ! � ¡ ï~¾ ñis equal to � Ì ¦ È and

� ï ¡ û ï&)( ï`| ñ�� ï j g � ñ ¡ û ã ï� È ÷ ñ�� Ì ¦ È . It follows that theminimum value of

� ï ¡ û ï ï ½ ñ is equal to ï j g � ñ ¡ û ã ï� È ÷ ñ��and is attained forÈ equal to any hyperplane of � ¡ ï~¾ ñ . Let � denote the sheaf of meromorphic

functions on � ¡ ï~¾ ñ .Theorem 2.33. ([Lu1990]) Let | ! � ¡ ï~¾ ñ be a closed subset such that� ï ¡ û ï&)( ï`| ñ � 5 ¸Gï`| ñ 7 ¤ ¡ û ¤ ï j g � ñ ¡ û ã ï� È ÷ ñ��Then the restriction map � ï � ¡ ï~¾ ñ ñ È L � ï � ¡ ï~¾ ñ 3 | ñ is onto.

2.34. Proof of Theorem 2.30. We claim that we may focus our attention only on(rm2) and on (rm5). Indeed, since a submanifold D ! Æ of codimension ü`.satisfies

� ï ¡ û ï ï)D ñ�� , (rm1) is a corollary of (rm2).

In both (rm3) and (rm4), we may include D in some D ï hypersurface ² ãof¾ ¡ , looking like a thin strip elongated along D . We claim that | then does not

contain any CR orbit of any such ² ã, so that (rm5) applies. Indeed, CR orbits of² ã

being of dimension ïÏý{ È ý ñ or ïÏý{ È ÷ ñ and | being already contained in theïÏý{ È ý ñ -dimensional D ! ² ã, it could only happen that | � D � ½ identifies

as a whole to a connected (CR orbit) complex hypersurface ½ ! ² ã. But this is

excluded by the assumption that | J� D in (rm3) and by the existence of genericpoints in (rm4).

Firstly, we prove (rm2). We show that | is locally removable. Leti - |

and lets ú ! Æ be a small open ball centered at

i. By a relevant application of

Proposition 1.2(5), one may verify that for almost every complex line a passingthrough

i, the intersection a�V s ú»V�| is reduced to 4 i 5 . Choose such a linea ã . Centering coordinates at

iand rotating them if necessary, we may assume


that a ã � 4 ï Á ã�ð � ð Ä Ä Ä ð � ñ 5 , whence for × " �small and fixed, the disc ½ Ö ï�� ñ��ï~×,� ð � ð Ä Ä Ä ð � ñ satisfies ½ Ö ï�ÿ-, ñ V�| �"!

.Fix such a small × ú "/. and set ¸ ú �� ) �Re10b½ Ö / ï�ÿ2, ñ ð |43 "/. . For l �ï l ï ð Ä Ä Ä ð l ¡<ñ -è¾ ¡ û ã satisfying @ l @65 ãï ¸ ú , set ½ Ö / � n ï�� ñ �� ï~× ú � ð l ï ð Ä Ä Ä ð l ¡<ñ .

Letting

Ü4798 . ð ÷;: , we interpolate between ½ Ö / � ú and ½ Ö / � n by defining½ Ö / � n � Ý ï�� ñE�� 0 × ú � ð Ü l ï ð Ä Ä Ä ð Ü l ¡ 3ÒÄSince < l < 5 ãï ¸ ú , these discs all satisfy =?>A@CBD0FE Ö / � n � G ï�ÿ2, ñ ðIH 3Sü ãïKJ ú . Since theembedded disc E Ö / � n � ã ï , ñ is ý -dimensional and since L ïNM û ï ï H ñ � . , Proposi-tion 1.2(5) assures that for almost every l with < l < 5 ãï J ú , its intersection withH is empty. Thus, we may apply the continuity principle Theorem 2.27, settingk G � ãï × ú , H G � ý�× ú and J G �� ãïKJ ú uniformly for every O 7P8 . ð ÷;: , whenceì G � ãQ J ú independently of the smallness of l : for every � 7SR ï�Æ�T H ñ , thereexists Â ú 7UR 0 ù � /V ïWE Ö /IX n X ú ï , ñsñ 3 with Â ú � � in

ù � /V ïWE Ö /;X n X ú ï�ÿ2, ñsñ . But since( ïNM û ï ï H ñÊ� . , for every connected open set

ù ! Æ , the intersection

ù VXï�Æ�T H ñis also connected (Proposition 1.2(4)), so Â ú and � stick together as a well definedfunction holomorphic in ÆXÃ ù � /V ïWE Ö /IX n X ú ï , ñsñ . If l was chosen sufficiently small,

it is clear thati � . 7 ¾ M is absorbed in

ù � /V ïWE Ö /;X n X ú ï , ñsñ , hence removable.

Secondly, we prove (rm5). Let H !ZY ãcontaining no CR orbit and define[ � �� ¿ H � ! H closed ð]\ � 7^R ï�Æ�T H ñ ðI_ � � 7`R ï�Æ�T H � ñ with � �Naa Ç�È 6 � �EÃ Ä

Lemma 2.35. If H �b ðIH �ï 7 [ � , then H �b V H �ï 7 [ � .Proof. Let �_�î 7cR ï�Æ�T H �î ñ , Ä � ÷ ð ý , with �_�î aa Ç�È 6 � � . We claim that �_�b and �G�ïmatch up on H T<ï H �b Ã H �ï ñ , hence define together a holomorphic function � �b ï 7R 0 Æ�T<ï H �b V H �ï ñ 3 with �_�b ï aa Ç�È 6 � � . Indeed, choose an arbitrary point

i 7H T<ï H �b Ã H �ï ñ . There exists a small open ball

s ú centered ati

withs ú.V¹ï H �b ÃH �ï ñ0�"! . Since � �b � � �ï � � at least in the dense subset

s ú'T Y b ofY b V s ú , by

continuity of �_�b and of �G�ï ins ú , necessarily �_�b ï i ñ0� �G�ï ï i ñ . d

Next, we define tH �� e6 § H � ÄIntuitively,

tH is the “nonremovable core” of H . By the lemma, for every� 7PR ï�Æ�T H ñ , there existst� 7fR ï�ÆgT tH ñ with

t� aa Ç�È 6 � � . To prove (rm5),

we must establish thattH �h!

. Reasoning by contradiction, we assume thattH J�i!and we apply to H �� tH the lemma below, which is in fact a corol-

lary of Trepreau’s Theorem 2.4(V). Of course,tH cannot contain any CR orbit ofY b

. Remind ( j 1.27(III), j 4.9(III)) that for us, local CR orbits are not germs, butlocal CR submanifolds of a certain small size.


Lemma 2.36. LetY b ! ¾ M ( k0üJý ) be a

[mlhypersurface and let H !ZY b

be aclosed subset containing no CR orbit of

Y b. Then there exists at least one pointi 7 H such that for every neighborhood n bú ofi

inY b

, we have:n bú V R�o 4 56 Á ï Y b ð i ñ J! HOðand in addition, all such points

iare locally removable.

Thent� extends holomorphically to a neighborhood of all such points

i 7 tH ,contradicting the definition of

tH , hence completing the proof of (rm5).

Proof. If

R o 4 56 Á ï Y b ðCp ñ ! H for every p 7 H , then small complex-tangential

curves issued from p necessarily remain in

R o 4 56 Á ï Y b ðCp ñ , hence in H , and pursuing

from point to point, global complex-tangential curves issued from p remain in H ,whence

R 6 Á ï Y b ðCp ñ ! H , contrary to the assumption.So, let

i 7 H with

R o 4 56 Á ï Y b ð i ñ J! H . To pursue, we need that

Y bis minimal

ati

. SinceY b

is a hypersurface, it might only happen that

R o 4 56 Á ï Y b ð i ñ is a local

complex hypersurface, a bad situation that has to be changed in advance.Fortunately, without altering the conclusion of the lemma (and of (rm5)), we

have the freedom of perturbing the auxiliary hypersurfaceY b

, leaving H fixed ofcourse. Thus, assuming that

R 6 Á ï Y b ð i ñ is a complex hypersurface, we claimthat there exists a small (in

[qlnorm) deformation

Y br ofY b

supported in a neigh-borhood of

iwith

Y br > H such thatY br is minimal at

i.

Indeed, let ï p � ñ �0§ts be a sequence of points tending toi

in

R o 4 56 Á ï Y b ð i ñ withp � J7 H . To destroy the local complex hypersurface

R o 4 56 Á ï Y b ð i ñ , it suffices to

achieve, by means of cut-off functions, small bump-deformations ofY

centeredat all the p � ; it is easy to write the technical details in terms of a local graphedrepresentation u �wv b ï Á ðyx ñ for

Y b. Outside a small neighborhood of the union

of the p � , Y br coincides withY b

. Then the resultingY br is necessarily minimal

ati

, since if it where not, the uniqueness principle23 for complex manifolds wouldforce

R o 4 56 Á ï Y b ð i ñ�� R o 4 56 Á ï Y br ð i ñ , but

Y br does not contains the p � .So we can assume that

Y bis minimal at every point

i 7 H at whichR o 4 56 Á ï Y b ð i ñ J! H . Let

s ú ! ¾ M be a small open ball centered ati

withs ú V Y b ! R o 4 56 Á ï Y b ð i ñ . We will show that

R ï�Æ�T H ñ extends holomorphi-cally to

s ú . By assumption,s ú�V Y b J! H , hence

s ú'T H is connected, a fact thatwill insure monodromy.

Fixing uåú 7{z ú>¾ M T'| .$} with uåúKJ7{z ú Y b , we consider the global translationsY bG �� Y b�~ OquÒú ð O 7�� ðofY b

. Let � 7�R ï�Æ�T H ñ be arbitrary. For small O"�� . , Y bG V s ú does notintersect

Y b, hence the restriction �1< Ä��F W � O is a

[ lCR function on

Y bG V s ú (but�1< Ä �� W � O has possible singularities at points of H V s ú ).23Minimalization at a point takes strong advantage of the rigidity of complex hyper-

surfaces in this argument.


With �Gú �� Y b V s ú , Theorem 2.4(V) says that[q�6 Á ï��Gú ñ extends holomorphi-

cally to some one-sided neighborhood � 2ú ofY b

ati

. Reorienting if necessary,we may assume that the extension side is ��ú and that

i ~ Oqu ú 7 s��ú for O{� .small. The statement and the proof of Theorem 2.4(V) are of course invariant bytranslation. Hence

[q�6 Á ï��_ú ~ O]uåú ñ extends holomorphically to ��ú ~ OquÒú , forevery O4� . . It is geometrically clear that for O4� . small enough, � �ú ~ Oquåú con-tains

i. Thus � aa Ä��F W � O extends holomorphically to a neighborhood of

ifor suchO . Monodromy of the extension follows from the fact that

s��ú T H is connected forevery open ball

s �ú centered ati

. This completes the proof of the lemma. d2.37. Removability and extension of complex hypersurfaces. Let Æf�)¾ M bea domain. Theorem 2.30 (rm4) shows that a connected � -codimensional subman-ifold D�� Æ is removable provided it is not a complex hypersurface, or equiv-alently, is generic somewhere. Conversely, assume that Æ is pseudoconvex andlet( � Æ be a (not necessarily connected) closed complex hypersurface. ThenÆgT ( is (obviously) locally pseudoconvex at every point, hence the characteriza-

tion of domains of holomorphy yields a function � 7^R ï�ÆgT ( ñ whose domain ofexistence is exactly ÆgT ( . Thus,

(is nonremovable. But in a nonpseudoconvex

domain, closed complex hypersurfaces may be removable.

Example 2.38. For ×^� . small, consider the following nonpseudoconvex sub-domain of

ò l , defined as the union of a spherical shell together with a thin rod ofradius × directed by the � l -axis:Æ Ö �� ó ÷�� 5 < Á b < l ~ < Á l < l 5 ÷ ô Ú 0 ò l V ó*� l b ~ � lb ~ � ll 5 × l ô 3 ÄThen the intersection of the

Á b -axis with the spherical shell, namely( �� óï Á b�ð . ñ�� ÷�� 5 < Á b < 5 ÷ ô ðis a relatively closed complex hypersurface of Æ Ö homeomorphic to an open an-nulus. We claim that

(is removable.

Indeed, applying the continuity principle along discs parallel to theÁ b -axis,R ï�Æ Ö ñ extends holomorphically to

ò l T'| Á l � .$} . Since the open small disc|�ï Á b'ð . ñ � < Á b < 5 × } , considered as a subset of the closed complex hypersurface�(ofò l defined by �( �� |�ï Á b ð . ñ�� < Á b < 5 ÷ }

is contained in the thin rod, hence in Æ Ö , Theorem 2.30 (rm3) finishes to showthat � ï�Æ Ö T ( ñ�� ï�Æ Ö ñ0� ò l ÄIn such an example, we point out that the closed complex hypersurface

( � Æ Öextends as the closed complex hypersurface

�( � � ï�Æ Ö T ( ñ but that the intersec-tion �( VKÆ Ö � óï Á b�ð . ñ�� < Á b < 5 ×<ô Ú ó�ï Á b�ð . ñ�� ÷�� 5 < Á b < 5 ÷ ô


is strictly bigger than(

.

Problem 2.39. Understand which relatively closed complex hypersurfaces of ageneral domain Æ"�\¾ M are removable.

We thus consider a (possibly singular and reducible) closed complex hypersur-face

(of Æ . Basic properties of complex analytic sets ([Ch1991]) insure that( �w� î §�� ( î decomposes into at most countably many closed complex hyper-

surfaces( î��ÙÆ that are irreducible.

Definition 2.40. We say that( î allows an

(-compatible extension to

� ï�Æ ñ ifthere exists an irreducible closed complex hypersurface

�( î of� ï�Æ ñ extending

( îin the sense that

( î�� ( îZV3Æ whose intersection with Æ remains contained in(: �( îEVKÆ"� Úî�� §�� ( î � ÄThe principle of analytic continuation for irreducible complex analytic sets

([Ch1991]) assures that( î has at most one

(-compatible extension. On the

other hand,�( î may be an

(-compatible extension of several

( î � . In the aboveexample, the removable annulus

(had no

(-compatible extension to

� ï�Æ ñ .Theorem 2.41. ([Dl1977, JaPf2000]) Let Æw� ¾ M ( kU�� ) be a domain and let( � � î §�� ( î be a closed complex hypersurface of Æ , decomposed into irre-ducible components

( î . Setê*�F ¦ ' �� óiÄ 7 ê � ( î allows an(

-compatible extension�( î to

� ï�Æ ñ ô ÄThen � ï�Æ�T ( ñ0� � ï�Æ ñy¡ Úî£¢ ��¤¦¥ §�§ �( î ÄIn particular,

(is removable (resp. nonremovable) if and only if ê?�F ¦ ' �"! (resp.ê �¨ ¦ ' ��©! ).

This statement was obtained after a chain of generalizations originating fromthe classical results of Hartogs [Ha1909] and of Oka [Ok1934]. In [GR1956] itwas proved for the case that

(is of the form ª V �( ,

�( � � ï«ª ñ , and in [Nis1962]under the additional assumption that

� ï«ª�T ( ñ is a subset of� ï«ª ñ (a priori, it is

only a set over� ï«ª ñ ). Actually Theorem 2.41 was stated in [Dl1977] even for

Riemann domains ª . But it was remarked in [JaPf2000] (p. 306) that the proofin [Dl1977] is complete only if the functions of

R ï«ª ñ separate the points of ª ,i.e. if ª can be regarded as a subdomain of

� ï«ª ñ . Actually the proof in [Dl1977]starts from the special case where ª is a Hartogs figure, which can be treated by asubtle geometric examination. Then a localization argument shows that extensionof hypersurfaces which are singularity loci of holomorphic functions cannot stopwhen passing from ª to

� ï«ª ñ . But in the nonseparated case, the global effect


of identifying points of ª interferes nastily, and it is unclear how to justify thelocalization argument. The final step for general Riemann domains was achievedby the second author by completely different methods.

Theorem 2.42. ([Po2002]) Let ¬ �m¯®±° Mbe an arbitrary Riemann domain,

and let( � be a closed complex hypersurface. Denote by ² ��³® � ï 4ñ the

canonical immersion of

into� ï :ñ . Then there is a closed complex hypersur-

face�(

of� ï 4ñ with ² � b ï �( ñ � ( such that� ï T ( ñ0� � ï :ñy´ �( Ä

Let us briefly sketch the main idea of the proof ([Po2002]). The essence of theargument is to reduce extension of hypersurfaces to that of meromorphic func-tions. For every pseudoconvex Riemann domain ¬ �µ ® ° M

, there exists� 7¶R ï :ñ Vc· l ï 4ñ having

as domain of existence whose growth is con-trolled by some power of the polydisc distance to the abstract boundary ¸ÿ . Atboundary points where ¸ÿ can be locally identified with a complex hypersurface,� has just a pole of positive order. One can deduce that those hypersurfaces

(of

along which some holomorphic function on T ( becomes singular can be

represented as the polar locus of some meromorphic functionð

defined in

. Butðextends meromorphically to

� ï 4ñ , and the polar locus of the extension yieldsthe desired extension of

(.j 3. HULLS AND REMOVABLE SINGULARITIES AT THE BOUNDARY

3.1. Motivations for removable singularities at the boundary. As already ob-served in Section 1, beyond the harmonious realm of pseudoconvexity, the generalproblem of understanding compulsory holomorphic (or CR) extension is intrinsi-cally rich and open. Some elementary Baire category arguments show that mostdomains are not pseudoconvex, most CR manifolds have nontrivial disc-envelope,and most compact sets have nonempty essential polynomial hull. Hence, the Grailfor the theory of holomorphic extension would comprise:¹ a geometric and constructive view of the envelope of holomorphy of most

domains, following the Behnke-Sommer Kontinuitatssatz and Bishop’sphilosophy;¹ a clear correspondence between function-theoretic techniques, for in-stance those involving ÿ arguments, and geometric techniques, for in-stance those involving families of complex analytic varieties.

Several applications of the study of envelopes of holomorphy appear, for in-stance in the study of boundary regularity of solutions of the ÿ -complex, in thecomplex Plateau problem, in the study of CR mappings, in the computation ofpolynomial hulls and in removable singularities, the topics of this Part VI andof [MP2006a].


In the 1980’s, rapid progress in the understanding of the boundary behavior ofholomorphic functions led many authors to study the structure of singularities upto the boundary. In j 2.28, we discussed removability of relatively closed subsetsH of domains ªS� ° M , i.e. the problem whether

R ï«ª ñº® R ï«ª�T H ñ is surjective.Typically H was supposed to be lower-dimensional and its geometry near ÿ-ª wasirrelevant. Now we assume ª to be bounded in

° M( k»�³� ) and we consider

compact subsets ¼ of ª , possibly meeting ÿ-ª .

Problem 3.2. Find criteria of geometric, or of function-theoretic nature, assuringthat the restriction map

R ï«ª ñ½® R ï«ª�T¾¼ ñ is surjective.

If ¼ � ª V¹ÿ2ª �¿!and ª�T¾¼ is connected, surjectivity follows from the

Hartogs-Bochner extension Theorem 1.9(V). Since this theorem even gives exten-sion of CR functions on ÿ2ª , it seems reasonable to ask for holomorphic extensionof CR functions on ÿ2ª�T¾¼ , and then it is natural to assume that ¼ is contained inÿ2ª . Hence the formulation of a second trend of questions24.

Problem 3.3. Let ¼ is a compact subset of ÿ-ª such that ÿ2ª�T¾¼ is a hypersurfaceof class at least

[ b. Understand under which circumstances CR functions of class[m�

or ·ÁÀo 4NÂ on ÿ-ªgT¾¼ extend holomorphically to ª .

A variant of these two problems consists in assuming that functions are holo-morphic in some thin (one-sided) neighborhood of ÿ2ª�T¾¼ . In all the theorems thatwill be surveyed below, it appears that the thinness of the (one-sided) neighbor-hood of ÿ2ª�T¾¼ has no influence on extension, as in the original Hartogs theorem.In this respect, it is of interest to immediately indicate the connection of these twoproblems with the problem of determining certain envelopes of holomorphy.

In the second problem, the hypersurface ÿ-ª�T¾¼ is often globally minimal, afact that has to be verified or might be one of the assumptions of a theorem. Forinstance, several contributions deal with the paradigmatic case where ÿ-ª is at least[ml

and strongly pseudoconvex (hence obviously globally minimal). Then thanksto the elementary Levi-Lewy extension theorem (Theorem 1.18(V), lemma 2.2(V)and j 2.10(V)), there exists a one-sided neighborhood

ù ï�ÿ-ªgT¾¼ ñ of ÿ2ª�T¾¼ con-tained in ª to which both

[q�6mÃ ï�ÿ2ª�T¾¼ ñ and · Ào 4NÂ X 6mÃ ï�ÿ-ªgT¾¼ ñ extend holomorphi-cally. The size of

ù ï�ÿ-ª�T¾¼ ñ depends only on the local geometry of ÿ2ª , becauseù ï�ÿ2ª�T¾¼ ñ is obtained by gluing small discs (Part V). In fact, an inspection of theproof of the Levi-Lewy extension theorem together with an application of the con-tinuity principle shows also that the envelope of holomorphy of any thin one-sidedneighborhood

ù � ï�ÿ-ª�T¾¼ ñ (not necessarily contained in ª !) contains a one-sidedneighborhood

ù ï�ÿ2ª�T¾¼ ñ of ÿ-ª�T¾¼ contained in the pseudoconvex domain ª thathas a fixed, incompressible size25.

24CR distributions may also be considered, but in the sequel, we shall restrict consid-erations to continuous and integrable CR functions.

25To be rigorous: for every holomorphic function ÝÅÄ\æ ë¦Æ �¦Ç¦È ã äåá�É�ì , there existsa holomorphic function ÊËÄ:æ ë Æ Ç�È ã ä�á�É ì that coincides with Ý in a possibly smaller


As they are formulated, the above two problems turn out to be slightly too re-strictive. In fact, the final goal is to understand the envelope

� 0 ù ï�ÿ2ª�T¾¼ ñ 3 , or atleast to describe some significant part of

� 0 ù ï�ÿ2ª�T¾¼ ñ 3 lying above ª . Of course,the question to which extent is the geometry of

� 0 ù ï�ÿ2ª�T¾¼ ñ 3 accessible (con-structively speaking) depends sensitively on the shape of ª . Surely, the strictlypseudoconvex case is the easiest and the best understood up to now. In what fol-lows we will encounter situations where

� 0 ù ï�ÿ-ª�T¾¼ ñ 3 contains ª�TDÌ¼ , for some

subset Ì¼Í� ª defined in function-theoretic terms and depending on ¼Î�vÿ ª . Wewill also encounter situations where

� 0 ù ï«ª�T¾¼ ñ 3 is necessarily multisheeted over° M. In this concern, we will see a very striking difference between the complex

dimensions k � � and kÏ�ZÐ .In the last two decades, a considerable interest has been devoted to a subprob-

lem of these two problems, especially with the objective of characterizing thesingularities at the boundary that are removable.

Definition 3.4. In the second Problem 3.3, the compact subset ¼ � ÿ-ª iscalled CR-removable if for every CR function � 7 [ �6mÃ ï�ÿ-ª�T¾¼ ñ (resp. � 7·ÁÀo 4NÂ X 6mÃ ï�ÿ-ª�T¾¼ ñ ), there exists Ñ 7ÒR ï«ª ñ V [ � ï ª�T¾¼ ñ (resp. Ñ 7ÒR ï«ª ñ V( Ào 4NÂ ï ª�T¾¼ ñ ) with Ñ�< Ó�ÔÖÕy× � � (resp. locally at every point Ø 7 ÿ2ª�T¾¼ , the·ÁÀo 4NÂ X 6mÃ boundary value of Ñ equals � ).

Before exposing and surveying some major results we would like to mentionthat a complement of information and different approaches may be found in thetwo surveys [Stu1993, CSt1994], in the two monographs [Ky1995, Lt1997] andin the articles [Stu1981, LT1984, Lu1986, Lu1987, Jo1988, Lt1988, Stu1989,Ky1990, Ky1991, Stu1991, FS1991, Jo1992, KN1993, Du1993, LS1993, Lu1994,AC1994, Jo1995, KR1995, Jo1999a, Jo1999b, JS2000, JP2002, JS2004].

3.5. Characterization of removable sets contained in strongly pseudoconvexboundaries. Taking inspiration from the pivotal Oka theorem, one of the goalsof the study of removable singularities ([Stu1993]) is to characterize removabilityin function-theoretically significant terms, especially in terms of convexity withrespect to certain spaces of functions. In the very beginnings of Several Com-plex Variables, polynomial convexity appeared in connexion with holomorphicapproximation. According to the Oka-Weil theorem ([AW1998]), functions thatare holomorphic in some neighborhood of a polynomially convex compact set¼Í� ° M may be approximated uniformly by polynomials. Later on, holomorphicconvexity appeared to be central in Stein theory ([Ho1973]), one of the seminalfrequently used idea being to encircle convex compact sets by convenient analyticpolyhedra.

one-sided neighborhood Æ � � Ç�È�Ù1Ú£Û ÉÝÜ Æ � Ç�È�Ù1Ú£Û É . Details of the proof (involving adeformation argument) will not be provided here (see [Me1997, Jo1999a]).


The notion of convexity adapted to our pruposes is the following. By

R{Þ ª�ß ,we denote the ring of functions that are holomorphic in some neighborhood of theclosure ª of a domain ªà� ° M . As in the concept of germs, the neighborhoodmay depend on the function.

Definition 3.6. Let ªËá ° M be a bounded domain and let ¼â� ª be a compactset. The

R{Þ ª�ß -convex hull of ¼ isÌ¼äã½å ÔÖæ½çéè ¿ Á 7 ª ç < ê Þ Á ß£<�ëíìïî¾ðÂ ¢ × < ê Þ h ß£< for all ê 7`R{Þ ª�ß Ã ÄIf ¼ è Ì¼ ã½å ÔÖæ , then ¼ is called

R{Þ ª�ß -convex.

If ª is strongly pseudoconvex, a generalization of the Oka-Weil theorem showsthat every function which is holomorphic in a neighborhood of some

R{Þ ª�ß -convex compact set ¼Í� ª may be approximated uniformly on ¼ by functions ofR{Þ ªgß (nevertheless, for nonpseudoconvex domains, this approximation propertyfails26).

We may now begin with the formulation of a seminal theorem due to Stout thatinspired several authors. We state the CR version, due to Lupacciolu27.

Theorem 3.7. ([Stu1981, Lu1986, Stu1993]) In complex dimension k è � , acompact subset ¼ of a

[ lstrongly pseudoconvex boundary ñ-ªòá °ºl

is CR-removable if and only if it is

R{Þ ª�ß -convex.

The “only if” part is the easiest, relies on a lemma due to Słodkowski ([RS1989,Stu1993]) and will be presented after Lemma 3.11. Let us sketch the beautiful keyidea of the “if” part ([Stu1981, Lu1986, Stu1993, Po1997]).

From j 1.7(V), remind the expression of the Bochner-Martinelli kernel:ó1ô Þ«õ�ö Á ß è ÷Þ ��¬ùøúß l < õ È Á < û`üÞ«õ l È Á l ßþý�ÿõ b È Þ«õ b È Á b ß$ý�ÿõ l�� ý õ b � ý õ l Ä

Let � � ° l be a thin strongly pseudoconvex neighborhood of ª . By means ofa fixed function ê 79R{Þ �³ß , it is possible to construct some explicit primitive ofóºô

as follows. This idea goes back to Martinelli and has been exploited by Stout,

26Indeed, consider for instance the Hartogs figure Ù ã��Ü ¤ ¸ ÜYº1½�Ü ¤��ÜÒº� ��<½]àÜ ¤ ¸ ÜYº� ��½"º�Ü ¤ � ÜYº� � iná � . Then the annulus Û ��¤ ¸ �m½��½�à�Ü ¤ � Ü�à� �É��Ü È�Ù is� Ç Ù É -convex. We claim that the function â ã��¾½��¤ � , holomorphic in a neighborhood ofÛ , cannot be approximated on Û by functions ÝäÄ � Ç Ù É . Indeed, by Hartogs extension� Ç Ù É�� ë��¬Ü ¤ ¸ Ü�à� �-Ü ¤ � Ü�à� �� ì , which implies that every Ý�Ä � Ç Ù É has to satisfy the

maximum principle on the disc ��¤ ¸ �¾½��Ü ¤ � Ü�à� �É �"! Û . Rounding off the corners, weget an example with ÈþÙ Ä$#&% .

27Said differently, the envelope of holomorphy of an arbitrarily thin (interior) one-sidedneighborhood of È�Ù1Ú£Û is one-sheeted and identifies with Ù .


Lupacciolu, Leiterer, Laurent-Thiebaut, Kytmanov and others. By a classical re-sult ([HeLe1984]), ê admits a Hefer decompositionê Þ«õ ß È ê Þ Á ß è ê b Þ«õ�ö Á ß 8 õ b È Á b : ~ ê l Þ«õ�ö Á ß 8 õ l È Á l : öwith ê b ö ê l 7 R{Þ � '(�³ß . Then a direct calculation shows that for

Á 7 � fixed,the

Þ . ö �*ß -form) * X + Þ«õ ß è ê l Þ«õ?ö Á ß Þ«õ b È Á b ß È ê b Þ«õ?ö Á ß Þ«õ l È Á l ßÞ ��¬ùøúß l < õ È Á < l-, ê Þ«õ ß È ê Þ Á ß/. ý õ b � ý õ lsatisfies ñ10 ) * X + Þ«õ ß è ý�0 ) * X + Þ«õ ß è ó1ô Þ«õ�ö32 ß öon | õ 7 � ç ê Þ«õ ßä�è ê Þ42 ß } , i.e. provides a primitive of

óºôoutside some thin

set. In° M

for kÏ�ZÐ , there is also a similar explicit primitive.Let ¼ � ñ-ª be as in Theorem 3.7 and fix

2í7 ª ´ ¼ . By

R{Þ ª�ß -convexityof ¼ , there exists ê 7"R{Þ ª�ß with ê Þ42 ß è ÷ and ìïî¾ð65 ¢ × < ê Þ87 ß£< 5 ÷ . After aslight elementary modification of ê ([Jo1995, Po1997]), one can insure that theset 9 7�7 � ç < ê Þ87 ß£< è ÷;: is a geometrically smooth

[=<Levi-flat hypersurface

of � transverse to ñ-ª . Then the region ª * çéè ª�>?9-< êm<� ÷;: has piecewisesmooth connected boundaryñ2ª * è 0�ñ-ª@>`|$< êù<þ� ÷ } 3BA�0�ª�>`|$< êù< è ÷ } 3and its closure ª * in

° ldoes not intersect ¼ .

Let C be an arbitrary continuous CR function on ñ-ª�T¾¼ . Supposing for awhile that C already enjoys a holomorphic extension Ñ 7wR{Þ ª�ßD> [ � 0 ª�T¾¼ 3 ,the Bochner-Martinelli representation formula then provides for every

2ï7 ª * thevalue Ñ Þ42 ß è�E Ó�ÔGF C Þ«õ ß ó1ô Þ«õ�ö32 ßIHDecomposing ñ-ª * as above and using the primitive

) * X + , we may writeÑ Þ42 ß è�E Ó�Ô&J�K�L * L M bON C Þ«õ ß ó1ô Þ«õ�ö32 ß ~ E Ô&J�K�L * L P bON C Þ«õ ß$ý;0 ) * X + Þ«õ ßIHSupposing C 7 [ b and applying Stokes’ theorem28 to the (Levi-flat) hypersurfaceª@>`|$< êù< è ÷ } with boundary equal to ñ2ª�>`|$< êù< è ÷ } , we getÑ Þ42 ß è E Ó�Ô1J�K�L * L M bON C Þ«õ ß óºô Þ«õ?ö32 ß ~ E Ó�Ô&J�K�L * L P bON C Þ«õ ß ) * X + Þ«õ ßIH

But the holomorphic extension Ñ of an arbitrary C 7 [ �Q Ã 0 ñ-ª�T¾¼ 3 is stillunknown and in fact has to be constructed ! Since the two integrations in the

28In the general case Ý ÄR#6S , one shrinks slightly ÙBT inside Ù , rounds off its cornersand passes to the limit.


above formula are performed on parts of ñ2ª�T¾¼ where C is defined, we are led toset: Ñ * Þ42 ß è�E Ó�Ô&J�K�L * L M bON C Þ«õ ß ó1ô Þ«õ�ö32 ß ~ E Ó�Ô&JUK�L * L P bON C Þ«õ ß ) * X + Þ«õ ß öas a candidate extension of C at every

2Å7 ª * . Since ¼ is

R{Þ ª�ß -convex, ª isthe union of the regions ª * for ê running in

R{Þ ª�ß , but these extensions Ñ * Þ42 ß dodepend on ê , because

) * X + does. The remainder of the proof ([Stu1993, Po1997])then consists in:

(a) verifying that Ñ * is holomorphic (the kernels are not holomorphic withrespect to

2);

(a) showing that two differente candidates Ñ * � and Ñ *3V coincide in fact onª * � >�ª *3V ;(b) verifying that at least one candidate Ñ * has boundary value equal to C on

some controlled piece of ñ-ª�T¾¼ .

The reader is referred to [Stu1981, Lu1986, Stu1993] for complete arguments.

In the above construction, the strict pseudoconvexity of ª insured the existenceof a Stein (i.e. pseudoconvex) neighborhood basis 0��XW�3 W�¢ � of ª which guaran-teed in turn the existence of a Hefer decomposition. It was pointed out by Ortegathat Hefer decomposition (called Gleason decomposition in [Or1987]) holds on[ZY

pseudoconvex boundaries ñ-ªÒá ° M, but may fail in the nonpseudoconvex

realm. So, let ª á ° M be a bounded domain having[ Y

boundary. Denote byE Y Þ ª�ß çéè R{Þ ªgß[> [ Y Þ ª�ß the ring of holomorphic functions in ª that are[ Y

up to the boundary.

Theorem 3.8. ([Or1987]) If ñ2ª á ° M ( kË� ÷ ) is[ Y

and pseudoconvex, thenevery ê 7 E Y Þ ª�ß has a decompositionê Þ42 ß]\ ê Þ87 ß è M^_ P b ê

_ Þ42�ö�7 ß 8 2 _ \ 7 _ : öwith the ê _ 7 E Y Þ ª`'`ª�ß .

This decomposition formula also holds under the assumption that ª is a domainof holomorphy (having possibly nonsmooth boundary), but provided that ª hasa basis of neighborhoods consisting of Stein domains. However, not every

[ Yweakly pseudoconvex boundary admits a Stein neighborhood basis, as is shownby the so-called worm domains ([DF1977, FS1987]).

Example 3.9. Furthermore, the above decomposition theorem fails to hold ongeneral domains. Following [Or1987], consider the union ª bba ª l in

° lof the

two sets ª b çéè |;\dc 5 � b 5 . ö < 2 l < 5fehg � } andª l çéè | . ë � b 5 c ö e � b/i g � 5 < 2 l < 5 ÷ } H


The continuity principle along families of analytic discs parallel to the

2 l -axisshows that the envelope of holomorphy of ª b a ª l contains ª b a ªdj , whereª j çéè | . ë � b 5 c ö < 2 l < 5 ÷ } .The holomorphic mapping k Þ42 b ö32 l ß çéè Þ ehl + � ö32 l ß is one-to-one from ª bZa ª lonto its image k Þ ª bma ª l ß . However, the extension of k to ª b�a ª j is notinjective, because k takes the same value at the two points

Þ�n ¬ ö e � l�o ß 7 ª ba ª j .If Theorem 3.8 were true on the domain k Þ ª b a ª l ß , pulling the decompositionformula back to ª bda ª l , it would follow that every ê 7PR{Þ ª bma ª l ß has adecompositionê Þ42 ßp\ ê Þ87 ß è � ê b Þ e l + ö�7 ß , e l + � \ e l + V . ~ � ê l Þ e l + ö�7 ß 8 2 l \ 7 l :è ê b Þ42�ö�7 ß , e l + � \ e l + V . ~ ê l Þ42�ö�7 ß 8 2 l \ 7 l : HThen the same decomposition would hold for every ê 7^R{Þ ª ba ª j ß , by automaticholomorphic extension of ê ö ê b ö ê l . Choosing

2 è Þ \�¬ ö e � l;o ß , 7 è Þ ¬ ö e � l�o ßand ê such that ê Þ42 ß��è ê Þ87 ß ( ê çéè 2 b will do !), we reach a contradiction.

Corollary 3.10. ([Or1987, LP2003]) Every function holomorphic in a domainª�� ° M enjoys the Hefer division property precisely when the envelope of holo-morphy of ª is schlicht.

The above results mean that a direct application of the integral formula ap-proach sketched above becomes impossible for domains having nonschlichtenvelope. Nevertheless, in [Lt1988], using more general divison methods([HeLe1984]), a Bochner-Martinelli kernel on an arbitrary Stein manifold wasconstructed that enabled to obtain Theorem 3.28 below, valid for nonpseudocon-vex domains.

We conclude our presentation of Theorem 3.7 by exposing the “only if” ofTheorem 3.7.

Lemma 3.11. ([RS1989, Stu1993]) Let ñ2ª©á ° l be a[ml

strongly pseudoconvexboundary and let ¼Î�Uñ-ª be a compact set. Then ª ´ Ì¼�ã1å Ô2æ is pseudoconvex.

Taking for granted the lemma, by contraposition, suppose that ¼ �Îñ2ª isnot

R{Þ ª�ß -convex, viz. ¼ q Ì¼ ã1å Ô-æ and show that ¼ is not removable. It fol-

lows from strict pseudoconvexity of ñ2ª that ªr> Ì¼äã½å Ô?æ is nonempty ([Stu1993]).Leaving ¼ fixed, by deforming ñ2ª away from ª , we may enlarge slightly ª asa domain ª �ts ª with ñ2ª �ts ¼ and ª �ts ñ2ª�T¾¼ , having

[mlboundary ñ-ª �

close to ñ-ª , as illustrated. Since by the lemma, ª ´ Ì¼äã½å Ô-æ is pseudoconvex, it

follows easily ([Stu1993]) that ª � ´ Ì¼�ã1å Ô-æ is also pseudoconvex. Consequently

([Ho1973]), there exists a holomorphic function Ñ � 7©R 0 ª � ´ Ì¼äã½å ÔÖæ 3 that does

not extend holomorphically at any point of the boundary of ª � ´ Ì¼ ã½å Ô-æ . The re-striction of Ñ � to ñ-ªgT¾¼ is a CR function on ñ2ª�T¾¼ for which ¼ is not removable,since ª�> Ì¼�ã1å ÔÖæ �è ! .


3.12. Removability, polynomial hulls and Cantor sets. A generalization ofTheorem 3.7, essentially with the same proof (excepting notational complications)holds in arbitrary complex dimension kÏ�Z� .Theorem 3.13. ([Lu1986, Stu1993]) Let ª á ° M , k �»� , be a bounded pseu-doconvex domain such that ª has a Stein neighborhood basis. If ¼ �³ñ2ª iscompact and

R{Þ ª�ß -convex, and if ñ2ª è ¼ a Y , whereY

is a connected[ b

hypersurface of° M T¾¼ , then ¼ is CR-removable.

Example 3.14. ([CSt1994, Jo1999a]) LetY

be a connected compact orientableÞ �¾ku\ Ð*ß -dimensional maximally complex (Definition 4.7 below) CR manifoldof class

[ bcontained in the unit sphere ñGv M ( kf� � ) with empty boundary in

the sense of currents. Such anY

is called a maximally complex cycle. By atheorem due to Harvey-Lawson (reviewed as Theorem 4.16 below), if

Ysat-

isfies the moments’ condition, thenY

is the boundary of a unique complexÞ kw\ ÷ ß -dimensional complex subvariety xË�yv M . Since the cohomology groupz l Þ v M ö3{ ß vanishes, by a standard Cousin problem, x may be defined as the zero-set of some global holomorphic function C 7ZR{Þ v M ßB> [ � Þ v M ß . The maximumprinciple yields that the compact set ¼ çéè x a Y è x is

R{Þ v M ß -convex. Conse-quently, the envelope of holomorphy of an arbitrarily thin one-sided neighborhoodof ñ6v M T Y is equal to the pseudoconvex domain v M T 0 Y a x 3 .

If in addition ª is Runge ([Ho1973]) or if ª is polynomially convex, then everyC 7cR{Þ ª�ß may be approximated uniformly by polynomials on some sufficientlysmall neighborhood of ª (whose size depends on C ). It then follows that polyno-mial convexity and

R{Þ ª�ß -convexity are equivalent. As a paradigmatic example,this holds when ª è v M is the unit ball.

Corollary 3.15. ([Stu1993]) Let ª and ¼Í�Zª be as in Theorem 3.13 and assumethat ª is Runge in

° M, for instance ª è v M . If ¼ is polynomially convex, then¼ is CR-removable. If k è � , the CR-removability of ¼ is equivalent to its

polynomial convexity.

Although the last necessary and sufficient condition seems to be satisfactory,we must point out that concrete geometric characterizations of polynomial con-vexity usually are hard to provide. In j 5.14 below, we shall describe a class ofremovable compact sets whose polynomial convexity may be established directly.

A compact subset ¼ of | M ( kÏ� ÷ ) is a Cantor set if it is perfect, viz. coincideswith its first derived set ¼ � . It is called tame if there is a homeomorphism of | Monto itself that carries ¼ onto the standard middle-third Cantor set contained inthe coordinate line | g � .

Tame Cantor sets ¼ in a[]l

strongly pseudoconvex boundary ñ-ªfá ° l wereshown to be CR-removable in [FS1991], provided there exists a Stein neighbor-hood } of ¼ in

° lsuch that ¼ is

R{Þ }�ß -convex. By further analysis, this lastassumption was shown later to be redundant and in general, tame Cantor sets are


CR-removable. It was then suggested in [Stu1993] that all Cantor subsets of ñGv M( k �U� ) are removable, or equivalently polynomially convex. Nevertheless, Rudinand then Vitushkin, Henkin and others had constructed Cantor sets ¼ � °Ál hav-ing large polynomial hull Ì¼ , e.g. so that Ì¼ contains a complex curve, or evencontains interior points. Recently, in a beautiful paper, Joricke showed how to putsuch sets in the Ð -sphere ñGv l , thus solving the question in the negative.

Theorem 3.16. ([Jo2005]) For every positive number ~ 5 ÷ , there exists a Cantorset ¼Í�ZñGv l whose polynomial hull Ì¼ contains the closed ball ~ v l .3.17. · À -removability and further results. In the definition of CR-removability,nothing is assumed about the behavior from ª ´ ¼ up to ¼ : the rate of growthmay be arbitrarily high. If, differently, functions are assumed to be tame on ñ2ª(including ¼ ), better removability assertions hold.

Definition 3.18. A compact subset ¼ of a[ b

boundary ñ2ª³á ° M( k�� ) is

called · À -removable ( ÷ ë`�µë?� ) if every function C 7 · À Þ ñ2ª�ß which is CR onñ2ª�T¾¼ is in fact CR on the whole boundary ñ2ª .

Then by the Hartogs-Bochner theorem, C admits a holomorphic extension to ªthat may be checked to belong to

z À Þ ª�ß .Theorem 3.19. ([AC1994]) Let ªPá ° M ( k"�w� ) be a bounded domain having[ml

boundary ñ-ª and letY

be a[]l

totally real embedded submanifold of ñ2ª . If¼Í� Y is a polynomially convex compact subset, then ¼ is · À -removable.

In complex dimension k"�wÐ , the two extension Theorems 3.13 and 3.19 arenot optimal. In general, additional extension phenomena occur, which are princi-pally overlooked by assumptions on the hull of the singularity. A more geometricpoint of view ( j 3.23 below) shows that these theorems may be established bymeans of holomorphic extension along one-parameter families of complex ana-lytic hypersurfaces, whereas the (finer) Kontinuitatssatz holds along families ofanalytic discs, whose thinness offers more freedom to fill in maximal domains ofextension.

Example 3.20. Let ª çéè v j be the unit ball in° j , and let¼ è 9 Þ42 b ö32 l ö . ß 7 ñ6v�j ç < 2 b <þ� ÷ � � :

be a 3-dimensional ring in the intersection of ñ6v j with the

Þ42 b ö32 l ß -plane. Themaximum principle along discs parallel to the

2 l -axis yields:Ì¼ ã1å �;� æ è 9 Þ42 b ö32 l ö . ß 7 v j ç < 2 b <þ� ÷ � � : �è ¼ öso ¼ is not

R{Þ vDj�ß -convex. Nevertheless, this ¼ is removable. Indeed, applyingthe continuity principle, we may first fill in v]j ´ Ì¼ by means of discs parallel tothe

2 l -axis and then fill in the complete ball v j , by means of discs parallel to the2 j -axis.


In higher dimensions k �»Ð , the relevant characterizations of CR-removablecompact sets contained in strongly pseudoconvex frontiers are of cohomologicalnature ( j 3.33 below). In another vein, the assumption that ª possesses a Steinneighborhood basis in Theorem 3.13 above inspired some authors to generalizeStout’s theorem as follows.

Definition 3.21. Let ª be a relatively compact domain of a Stein manifold �and let ¼Í� ª be a compact set. The

R{Þ �³ß -convex hull of ¼ isÌ¼ ã½å�� æ çéè�� 2�7 � ç < ê Þ42 ß£<?ëÅìÝî¾ð5 ¢ × < ê Þ87 ß£< for all ê 7`R{Þ �³ß��HIf ¼ è Ì¼ ã½å�� æ , then ¼ is called

R{Þ �³ß -convex.

In° M

, the

R{Þ �³ß -convex hull coincides with the polynomial hull. Notice thatthe next theorem is valid without pseudoconvexity assumption on ª .

Theorem 3.22. ([Stu1981, Lt1988, Ky1991, Stu1993, Jo1995]) Let � be a Steinmanifold of dimension k"�Ë� , let ªPá�� be a relatively compact domain suchthat �³T ª is connected and let ¼ � ª be a compact set with ¼ è Ì¼ ã½å�� æ�>ñ2ª . Then every CR function C defined on ñ-ª�T¾¼ extends holomorphically toªgT1Ì¼ ã1å�� æ , i.e.:¹ if ñ-ªgT¾¼ is a

[=� X � hypersurface, with � � ÷ and . ë©²Uë ÷ , and if C 7[ � X �Q Ã Þ ñ-ª�T¾¼Ïß , then the holomorphic extension Ñ 79R{Þ ª�T¾¼Ïß belongs to

the class[ � X � 0 ª�T1Ì¼ ã1å�� æ 3 ;¹ if ñ-ª�T¾¼ is a

[ bhypersurface and if C 7 · Ào�� Â Þ ñ2ª�T¾¼Ïß with ÷ ë��ë�� , then at every point � 7 ñ-ª�T¾¼ , the holomorphic extensionÑ 7^R 0 ªgT Ì¼ ã1å�� æ¨3 belongs to the Hardy space

z Ào�� Â Þ ��m>�ª�ß , for somesmall neighborhood � � of � in � .

3.23. E Þ ª�ß -hull and removal of singularities on pseudoconvex boundaries.Following [Jo1995, Po1997, JP2002], we now expose a geometric aspect of someof the preceding removability theorems. Let ªfá ° M with k �w� be a boundeddomain having frontier of class at least

[ b. By E Þ ª�ß è R{Þ ª�ß�> [ � Þ ª�ß , we denote

the ring of holomorphic functions in ª that are continuous up to the boundary.Let ¼Í� ª be a compact set.

Definition 3.24. The E Þ ª�ß -hull of ¼ isÌ¼R� å ÔÖæ çéè � 2ï7 ª ç < ê Þ42 ß£<Öëíìïî¾ð5q¢ Ô < ê Þ87 ß£< for all ê 7��{Þ ªgß � HIf ¼ è Ì¼�� å ÔÖæ , then ¼ is called E Þ ª�ß -convex. If ¼ è ñ2ª�>wÌ¼R� å ÔÖæ , then ¼ iscalled CR-convex.

The next theorem is stronger than Theorem 3.13 in two aspects:

HOLOMORPHIC EXTENSIONS AND REMOVABLE SINGULARITIES 237¹ the inclusion Ì¼�� å Ô-æ � Ì¼äã½å ÔÖæ holds in general and may be strict;¹ it is not assumed that the pseudoconvex domain ª has a Stein neighbor-hood basis.

Theorem 3.25. ([Jo1995]) Let ª be a bounded weakly pseudoconvex domain in°½lhaving frontier of class

[ land let ¼ be a compact subset of ñ2ª with ¼â�è ñ2ª

such that ¼ is CR-convex, namely ¼ è ñ2ªR> Ì¼ � å ÔÖæ . Then the following are true.

1) Let � Þ ñ2ª�T¾¼Ïß be an interior one-sided neighborhood of ñ-ªgT¾¼ with theproperty that each connected component of � Þ ñ-ª�T¾¼Ïß contains in itsboundary exactly one component of ñ2ª�T¾¼ and no other point of ñ-ªgT¾¼ .Then for every holomorphic function C 7UR@� � Þ ñ-ª�T¾¼Ïß � , there exists a

holomorphic function Ñ 7`R{Þ ª�TÌ¼R� å ÔÖæ ß with Ñ è C in � Þ ñ2ª�T¾¼Ïß .2) ([AS1990]) There is a one-to-one correspondence between connected

components of ñ-ªgT¾¼ and connected components of ª�T Ì¼ � å ÔÖæ , namely

the boundary of each component of ª�T Ì¼ � å ÔÖæ contains exactly one con-nected component of ñ2ª�T¾¼ and does not intersect any other component.

3) If the boundary ñ2ª is of class[ Y

, then ª�T Ì¼ � å Ô-æ is pseudoconvex, henceit is the envelope of holomorphy of � Þ ñ-ª�T¾¼Ïß .

If ¼ is not CR-convex, the one-to-one correspondence between the connectedcomponents of ñ2ª�T¾¼ and those of ñ-ªgTDÌ¼�� å Ô-æ may fail.

Example 3.26. Indeed, let ª çéè v l >�9 � b 5 bl : be a truncation of the unit balland let ¼ çéè ñGv l > 9*� b è bl : be the intersection of the three-sphere ñ6v l withthe real hyperplane 9 � b è bl : (see only the left hand side of the diagram).

Relevance of the assumption of CR-convexity

�×B��¡ K g � P �V N

¢ V ×Ó � VÔ

Ó � VÔ �

¢ V × � P �× ��

The Levi-flat Ð -ball v l > 9*� b è bl : being foliated by complex discs, the maxi-

mum principle entails that Ì¼ � å ÔÖæ è v l > 9 � b è bl : è Ì¼ � å Ô-æO>ºñ-ª��è ¼ , hence ¼is not CR-convex. Also, ñ2ª�T¾¼ has two connected components ñ6v l >£9 � b 5 bl :


and v l > 9*� b è bl : , whereas ñ2ª�T Ì¼�� å Ô-æ è ñ6v l > 9*� b 5 bl : is connected. Anyfunction on ñ-ªgT¾¼ equal to two distinct constants on the two connected compo-nents of ñ-ªgT¾¼ is CR and not holomorphically extendable to ª è ªgT Ì¼R� å ÔÖæ .Finally, by smoothing out ñ2ª near the two-sphere ñGv l > 9*� b è bl : , we obtain anexample with

[�Yboundary.

3.27. Hulls and holomorphic extension from nonpseudoconvex boundaries.Since the work [Lu1986] of Lupacciolu, the extension of Theorem 3.7 to nonpseu-doconvex boundaries was a daring open problem ([Stu1993]).

Theorem 3.28. ([Po1997, JP2002, LP2003]) Let ª be a not necessarily pseu-doconvex bounded domain in

° M( k �P� ) having connected

[]lfrontier and let¼±�wñ2ª be a compact set with ñ-ªgT¾¼ connected such that ¼ è ñ2ªf>¶Ì¼ � å ÔÖæ .

Then for every continuous CR function C 7 [ �Q Ã Þ ñ-ªgT¾¼ ß , there exists a holo-morphic function Ñ 7ÍR�� ª�T Ì¼R� å ÔÖæ �¤> [ � ��8 ª�T Ì¼R� å ÔÖæ : a 8 ñ-ª�T¾¼ : � such thatÑä< Ó�ÔÖÕy× è C .

A purely geometrical proof applying a global continuity principle togetherwith a fine control of monodromy may be found in [Po1997, JP2002]; cf.also [MP2007]. By a topological device, a second proof ([LP2003]) derives thetheorem from the following statement, established by means of ñ techniques.

Theorem 3.29. ([Lt1988]) Let � be a Stein manifold of complex dimension kÏ�� , let ¼ �¥� be a compact set that is

R{Þ �³ß -convex and let ª¿�¦� be arelatively compact not necessarily pseudoconvex domain such that ñ-ª�T¾¼ is aconnected

[ bhypersurface of �³T¾¼ . Then for every continuous CR function C

on ñ2ª�T¾¼ , there exists a holomorphic function Ñ 7 R{Þ ª�T¾¼Ïß]> [ � Þ ª�T¾¼Ïß withÑä< Ó�ÔÖÕy× è C .

Contrary to the case where ñ-ª is pseudoconvex (as in Theorem 3.25), even if ¼is CR-convex, the one-to-one correspondence between the connected componentsof ñ2ª�T¾¼ and those of ª�TDÌ¼ � å ÔÖæ may fail to hold. For this reason, ñ2ª�T¾¼ isassumed to be connected in Theorem 3.29.

Example 3.30. ([LP2003]) We modify Example 3.26 so as to get a nonpseu-doconvex boundary as follows (see the right hand side of the diagram above).Let ª � be the unit ball v l from which we substract the closed ball § Þ p ö ÷ ß ofradius ÷ centered at the point p of coordinates

Þ ÷ ö3¨ ß . A computation with defin-ing (in)equations shows that ª � is contained in 9 � b 5 bl : . Notice that ª � isnot pseudoconvex and in fact, its envelope of holomorphy is single-sheeted andequal to the domain ª è v l > 9�� b 5 bl : drawn in the left hand side. Let¼ � çéè v l > 9*� b è bl : � ñ2ª � (this set is the same � -sphere as the set ¼ of thepreceding example). Then ¼ � is CR-convex, since the candidate for its E Þ ª � ß -hull is the three-sphere v l > 9�� b è bl : that lies outside ª � . However, ñ-ª � T¾¼ � hastwo connected components, namely ñ6v l >�9 � b 5 bl : and ñ&§ Þ p ö ÷ ßZ>�9 � b 5 bl : ,


whereas ª � T Ì¼ �� å ÔÖæ è ª � T¾¼ � è ª � is connected. Hence any CR function equal totwo distinct constants on these two components fails to extend holomorphicallyto ª � . Finally, by smoothing out ñ2ª near the two-sphere ñGv l >�9 � b è bl : , weobtain an example with

[ Yboundary.

If we drop CR-convexity of ¼ , viz. if ¼ �è Ì¼ � å Ô-æ > ñ2ª , then monodromy

problems come on scene: the natural embedding of ª�T Ì¼R� å ÔÖæ into the envelope ofholomorphy of a one-sided neighborhood of ñ-ª�T¾¼ may fail to be one-to-one.

Example 3.31. ([LP2003]) Consider the real four-dimensional open cube H çéèÞ \ ÷ ö ÷ ß©'�ø Þ \ ÷ ö ÷ ßd' Þ \ ÷ ö ÷ ß©'{ø Þ \ ÷ ö ÷ ß in°Kl$ª | û .

« V¬

�® �

�V¯° �V° ¯

±

²³ V´ V

´ �

´ �µ V µ �

´ �

Multisheetedness if²

is not CR-convex

¶

Choose · � ¨small and remove from this cube H firstly the narrow tunnel¸ b çéè |$< 2 l < ë�· ö < � b \ ÷ � �?<Kë¹· } having an entrance and an exit and secondly

the (incomplete) narrow tunnel¸ l çéè |$< 2 l <ºëº· ö < � b ~ ÷ � �?<Dë�· ö \ ÷ 5 � b ë÷ � � } having only an entrance, and call ª the obtained domain. Let ¼ çéè ñ H >|�� b è ¨ } . The complete tunnel insures that ñ-ª�T¾¼ is connected. Moreover, the

maximum principle along families of analytic discs parallel to the complex

2 l -axisenables to verify thatÌ¼�� å ÔÖæ è � ª�> |�� b è ¨ } � AÍ¼»A � ñ ¸ b >`|�� b è ¨ } � A � ñ ¸ l > |�� b è ¨ } � HIt follows that ñ2ª�TÌ¼ � å ÔÖæ has three connected components, firstly the part

z b ofñ2ª that lies in the half-space |�� b 5 ¨ } ; secondly the dead-lock part

z l of thesecond tunnel that lies in |�� b � ¨ } ; and thirdly, the remainder

z j of the boundary,that lies in |�� b � ¨ } .

The branch of ¼¾½�¿ 2 b satisfying ¼¾½�¿ ÷ è ¨is uniquely defined in

°Dl T'| Þ � b ö32 l ß ç� b ë ¨ } , hence ¼¾½�¿ 2 b is holomorphic in a neighborhood of ñ2ª�T z l , wherez l çéè ñ ¸ l >U|�� b � ¨ } . In addition, ¼À½�¿ 2 b extends from points near¸ l in

240 JOEL MERKER AND EGMONT PORTEN|�� b 5 ¨ } to a neighborhood of

z l . In sum, it defines a single-valued functionthat is holomorphic in a neighborhood of ñ2ª .

Observe that

� \ bl ~ ll ö3¨ � 7{z l �Uñ-ª . The value of ¼¾½�¿ 2 b thus defined at thispoint is ¼¾½�¿ � bÁ l e � l¾Â o i û � è ¼¾½�¿ bÁ l \+ø Â oû . On the other hand, ¼¾½�¿ 2 b restricted to a

neighborhood of ñ H >Ý|�� b � ¨ } �Zñ2ª extends holomorphically to H >Ý|�� b � ¨ }(by means of unit discs parallel to the

2 l -axis) as ¼¾½�¿ 2 b itself! But the value ofthis extension at

� \ bl ~ ll ö3¨ � is different: ¼¾½�¿ � bÁ l e l j o i û � è ¼À½�¿ bÁ l ~ ø j oû .

To conclude this paragraph, before surveying the cohomological characteriza-tions of removable singularities in dimension kÏ�ZÐ , we reformulate the obtainedcharacterization in complex dimension k è � . It is known that a compact set¼Í� ° M is polynomially convex if and only if the ñ -cohomology group

z � X bÓ Þ ¼Ïßis trivial and holomorphic functions in a neighborhood of ¼ can be approximatedby polynomials uniformly on ¼ . Thus, we can state a complete formulation ofTheorem 3.7, with the supplementary assumption that

R{Þ ª�ß may be approximateduniformly by polynomials. This insures that polynomial convexity coincides withR{Þ ªgß -convexity. As a major example, the theorem holds for ª equal to the unitball v l (Corollary 3.15).

Theorem 3.32. ([Stu1989, Stu1993, Lu1994, CSt1994]) The following four con-ditions for a compact subset ¼ of a

[ lstrongly pseudoconvex compact boundaryñ2ª©á ° l with ª Runge or ª polynomially convex are equivalent:¹ ¼ is

R{Þ ª�ß -convex.¹ ¼ is polynomially convex.¹ z � X bÓ Þ ¼Ïß è ¨and holomorphic functions in a neighborhood of ¼ can be

approximated by polynomials uniformly on ¼¹ ¼ is removable.

Thus, in this situation, removability amounts to polynomial convexity. Nev-ertheless, the problem of characterizing geometrically the polynomial convexityof compact sets hides several fine questions. We shall come back to this topic inSection 5.

3.33. Luppaciolu’s characterizations. An outstanding theorem due to Lupacci-olu provides complete cohomological characterizations of removable sets that arecontained in strongly pseudoconvex boundaries, for general k �Z� .

Let � be a Stein manifold of dimension k �Z� and let ª"á�� be a relativelycompact strongly pseudoconvex domain having

[ lboundary.

Letz � X ÃÓ çéèÅÄ � X ÃÓ � ñ6Æ � X Ã � b denote the usual

Þ � ö p ß -th Dolbeault cohomology

group29. We endow the space Ä M X M � lÓ Þ ¼ ß of ñ -closed

Þ k ö kw\í�*ß -forms defined

29Appropriate background, further survey of Lupacciolu’s results and additional mate-rial may be found in [CSt1994].


in a neighborhood of a compact set ¼ �?� with the standard locally convex in-ductive limit topology derived from the inductive system of the Frechet-Schwartzspaces Ä M X M � lÓ Þ � ß , as � ranges through a fundamental system of open neighbor-hoods of ¼ in � .

Theorem 3.34. ([Lu1994, CSt1994]) Assume that ª is

R{Þ �³ß -convex. A properclosed subset ¼ of ñ-ª is removable if and only if

z M X M � bÓ Þ ¼Ïß è änd the re-

striction map Ä M X M � lÓ Þ �³ß ® Ä M X M � lÓ Þ ¼Ïß has dense image.

For k è � , the two conditions of the theorem reduce to the

R{Þ �³ß -convexityof ¼ ([Lu1994, CSt1994]). For k �ËÐ , the following improvement is valid. ByÇ�È

, we denote the separated space associated to a given topological vector spaceÈ, namely the quotient

È � öf

Èby the closure of

¨.

Theorem 3.35. ([Lu1994, CSt1994]) Assume that kÏ�ZÐ . Without the assumptionthat ª is

R{Þ �³ß -convex, the compact set ¼ � ñ-ª is removable if and only ifz M X M � bÓ Þ ¼Ïß è änd

Ç;z M X M � lÓ Þ ¼Ïß è ¨.

Lupacciolu also obtains an extrinsic characterization as follows. Let É be theparacompactifying family of all closed subsets of �³T¾¼ that have compact clo-sure in � . Let

z � X ÃÊ the Dolbeault cohomology groups with support in É .

Theorem 3.36. ([Lu1994, CSt1994]) For k/� Ð , a compact subset ¼ of theboundary ñ2ª of a

[ml-bounded strongly pseudoconvex domain ª áË� is remov-

able if and only ifz � X bÊ Þ �³T¾¼ ß è ¨

.

Notice that, for k9�UÐ , this theorem has the striking consequence that the con-dition that ¼ be removable in a strongly pseudoconvex boundary does not dependon the domain in question, but rather on the situation of ¼ itself in the ambi-ent manifold. Also, Lupacciolu provides analogous characterizations for weakremovability ([Lu1994, CSt1994]).j 4. SMOOTH AND METRICALLY THIN REMOVABLE SINGULARITIES

FOR CR FUNCTIONS

4.1. Three notions of removability. We formulate the concerned notions of re-movability directly in arbitrary codimension. Let

Y � ° M be a[ml X � generic sub-

manifold of positive codimension ý9� ÷ and of positive CR dimension Ìh� ÷ .SuchY

will always be supposed connected. In the sequel, not to mention super-ficial corollaries, we will systematically assume that

Yis globally minimal.

Definition 4.2. ([Me1997, MP1998, Jo1999a, Jo1999b, MP1999, MP2002]) Aclosed subset H of

Yis said to be:¹ CR-removable if there exists a wedgelike domain Í attached to

Yto

which every continuous CR function C 7 [ �Q Ã Þ Y T H ß extends holomor-phically;

242 JOEL MERKER AND EGMONT PORTEN¹ Í -removable if for every wedgelike domain Í b attached toY T H , there

is a wedgelike domain Í l attached toY

and a wedgelike domain ÍÎj��Í b >@Í l attached toY T H such that for every holomorphic functionC 7�R{Þ Í b ß , there exists a holomorphic function Ñ 7wR{Þ Í l ß which

coincides with C in Í j ;¹ · À -removable, where ÷ ëÅ� ëÏ� , if every locally integrable functionC 7 · Ào�� Â Þ Y ß which is CR in the distributional sense onY T H is in fact

CR on all ofY

.

A few comments are welcome. CR-removability requires at leastY T H to

be globally minimal, in order that the main Theorem 4.12(V) applies, yieldinga wedgelike domain Í b attached to

Y T H . Then Í -removability of H impliesits CR-removability. In both CR- and Í -removabililty, after the removal of H ,nothing is demanded about the growth of the holomorphic extension to a globalwedgelike domain Í l attached to

Y. Such extensions might well have essential

singularities at some points of H , although they are holomorphic in Í l . On thecontrary, for · À -removability of H , CR functions on

Y T H should really extendto be CR through H .

Notwithstanding this difference, the sequel will reveal that · À -removability isalso a consequence of Í -removability, thanks to some Hardy-space control ofthe holomorphic extension Ñ 7�R{Þ Í l ß . In fact, functions are assumed to be· Ào�� Â (a variant is to assume continuity on

Yinstead of integrability) even near

points of H . This strong assumption enables to get a control of the growth of thewedge extension. Before providing more explanations, we assert in advance thatÍ -removability is the most general notion of removability, focusing the questionon envelopes of holomorphy.

In codimension ý è ÷ , wedgelike domains identify to one-sided neighbor-hoods. Then Í -removability of H means that the envelope of holomorphy ofevery (arbitrarily thin) one-sided neighborhood of

Y T H contains a complete one-sided neighborhood of the hypersurface

Yin° M

. IfY è ñ2ª is the boundary of

a bounded domain ªP� ° M (having connected boundary), then Í -removabilityof a compact set ¼ �àñ-ª entails its removability in the sense of Problem 3.2,thanks to Hartogs Theorem 1.8(V).

As in [Jo1999b, MP1999], we would like to emphasize that all the general the-orems presented in Sections 3 and 4 are void for · bo�� Â functions, or require a strongassumption of growth. On the contrary, the results that will be presented belowhold in all spaces · Ào�� Â with ÷ ëÐ�ïë�� , without any assumption of growth. Theconcept of Í -removability, interpreted as a result about envelopes of holomorphy,yields a (crucial) external drawing near the illusory singularity, an opportunity thatis intrinsically attached to locally embeddable Cauchy-Riemann structures, but isof course absent for general linear partial differential operators.


4.3. Removable singularities on hypersurfaces. In [LS1993], it is shown thatif ª � ° M is a pseudoconvex bounded domain having

[ lboundary, then every

compact subset ¼ � ñ2ª with L l M � j Þ ¼Ïß è ¨is removable in the sense of Def-

inition 3.4. In fact, Lemma 4.18(III) shows that ñ-ª is globally minimal and thenext lemma shows that in codimension ý è ÷ , metrically thin singularities do notperturb global minimality.

Lemma 4.4. ([MP2002]) IfY � ° M is a globally minimal

[]lhypersurface, then

for every closed set H � Y with L l M � j Þ H ß è ¨, the complement

Y T H is alsoglobally minimal.

Example 4.5. However, this is untrue if L l M � j Þ H ß�� ¨. Let kZ�S� and

v Þ42�ö x ßbe[ l

defined for < 2 < ö < x < 5 ÷ and satisfyingv Þ42?ö3¨ ßÒÑ ¨

for Ó�Ô 2 b ë ¨. LetY � ° M be the graph u è v Þ42�ö x ß and define H çéè | Þ ø*� b ö32 l ö HhHhH ö32 M � b ö3¨ ß } .

Clearly =?>Aì H è �¾k�\íÐ , L l M � j Þ H ßï� ¨and | Þ42�ö3¨ ß ç Ó�Ô 2 b 5 ¨ } is a single

CR orbit

R � ofY T H . Also, the function

vmay be chosen so that

Yis of finite

type at every point ofY T R � , whence

Y T H consists of exactly two CR orbits,namely

R � andY T ÞFR � a�H ß . It follows that

Yis globally minimal.

Theorem 4.6. ([LS1993, CSt1994, MP1998, MP2002]) IfY � ° M is a globally

minimal[ml X � (

¨ 5 ² 5 ÷ ) hypersurface, then every closed set H � YwithL l M � j Þ H ß è L�Õ�ÖØ×ÚÙ � l Þ H ß è ¨

is locally CR-, Í - and · À -removable.

Sometimes, we shall say that H is of codimension � � � inY

. This is a versionof (rm1) and of (rm2) of Theorem 2.30 for CR functions on general hypersur-faces. Except for · À -removability, refinements about smoothness assumptionsmay be found in [CSt1994].

The smallest (Hausdorff) dimension of H � Y � ° M for which its removabil-ity may fail is equal to �¾kr\ Ð . Indeed, if H è Y >Òx is equal to the intersectionofY

with some local complex hypersurface x è |C è ¨ } , the functions ÷ � C_,Û � ÷ and e b/i�Ü restrict to be CR on

Y T H , but not holomorphically extendableto a one-sided neighborhood at points of H , since x visits both sides of

Y. In

such a situation, the real hypersurfaceY >Òx of the complex hypersurface x has

dimension

Þ �¾kr\ Ð*ß and CR dimension

Þ kÒ\ �*ß .Definition 4.7. A CR submanifold Ýi� ° M is called maximally complex if it isof odd dimension satisfying =?>Aì�Ý è ÷ ~ �pÞ©Ó�=?>Aì�Ý .

Every real hypersurface of a complex manifold is maximally complex. Thenext step in to study singularities H contained in

Þ �¾kr\ Ð*ß -dimensional submani-folds Ý � Y .

Example 4.8. We show the necessity of assuming thatY T H is also globally min-

imal ([MP1999]). Take the complex hypersurface

R � of the preceding examplehaving boundary ñ R � è H è Ý . Applying Proposition 4.38(III) to ß çéè R � ,we may construct a measure on

Y T H supported by

R � that is CR onY T H but


does not extend holomorphically to a wedge at any point of

R � è R � a^H , forthe same reason as in Corollary 4.39(III).

Because of this example, we shall systematically assume thatY T H is also

globally minimal, if this is not a consequence of other hypotheses. Here is a CRversion of (rm3) and of (rm4) of Theorem 2.30. It says that true singularitiesshould be maximally complex. Before stating it, we point out that all subman-ifolds of given manifolds will constantly be assumed to be embedded subman-ifolds. Also, all subsets H of a submanifold Ý of manifold

Ythat are called

closed are assumed to be closed both inY

and in Ý .

Theorem 4.9. ([Jo1992, Me1997, Jo1999a, Jo1999b]) LetY � ° M

be a[ml X �

(

¨`à ² à ÷ ) globally minimal hypersurface and let Ý � Ybe a connected[ml X � embedded submanifold of dimension

Þ �¾kw\ Ð*ß , viz. of codimension � inY

.A closed set H �áÝ such that

Y T H is also globally minimal is CR-, Í - and· À -removable under each one of the following two circumstances:

(i) k �U� and H �è Ý ;

(ii) k �UÐ and H è Ý is not maximally complex, viz. there exists at least onepoint �ãârÝ at which Ý is generic.

One may verify ([Jo1999a, MP1999]) that generic points of Ý are locally re-movable and then after erasing them by deforming slightly

Yinside the exten-

sional wedge existing above, (ii) is seen to be a consequence of (i). For vari-ous smoothness refinements, the reader is referred to [Jo1992, CSt1994, MP1998,Jo1999a, Jo1999b, MP1999]. One may also combine Theorem 4.6 and 4.9, as-suming that the submanifold Ý is smooth, except perhaps at all points of somemetrically thin closed subset. The proof will not be restituted.

The study of more massive singularities contained in

Þ �¾k�\©�*ß -dimensionalsubmanifolds has been initiated by Joricke ([Jo1988]), having in mind some gen-eralization of Denjoy’s approach to Painleve’s problem.

Theorem 4.10. ([Jo1999a, Jo1999b]) LetY � ° M

be a[ml X � (

¨�à ² à ÷ )globally minimal hypersurface and letY b � Y be a connected

[ l X � embeddedsubmanifold30 of dimension

Þ �¾kÒ\c�*ß , viz. of codimension ÷ inY

, that is genericin° M

. If kc�"Ð , a closed set H � Y b is CR-, Í - and · À -removable provided itdoes not contain any CR orbit of

Y b.

It may be established

�see e.g. Lemma 3.3 in [MP2006a] � that

Y b T H � is alsoglobally minimal for every closed H � � Y b containing no CR orbit of

Y b.

30We believe that # �åä æ -smoothness of çfè is required in the proof built there, since themap éÎêëÏìí Ç éÉ appearing in equation (3.12) of [Jo1999a] î that corresponds essentiallyto the singular integral ï ÇÀð É defined in (3.20)(V) ñ already requires ç è to be # è ä æ withòÚó£ôróuõ

to exist; then to compute the differential of é�êë ì í Ç éDÉ , one must require ç�èto be at least # �åä æ .


We would like to mention that the removability of two-codimensional singu-larities (Theorem 4.9) is not a consequence of the removability of the bigger one-codimensional singularities (Theorem 4.10). Indeed, it may happen that

z � Ýcontains

z6Â� Y at several points ��â@Ý in Theorem 4.9, preventing the existenceof a generic

Y b � Y containing Ý . In addition, even if

z ��Ý¯�s zgÂ� Y for every�fâuÝ , Theorem 4.9 is not anymore a corollary of Theorem 4.10. Indeed, withÌ è � and ý è ÷ , choosing a local hypersurfaceY � ° j containing a complex

curve x , choosing Ý � Y of dimension Ð containing x and being maximallyreal outside x , and choosing an arbitrary generic

Y b � Y containing Ý (someexplicit local defining equations may easily be written), then x is a CR orbit ofY b

, so Ý s x is not considered to be removable by Theorem 4.10, whereasTheorem 4.9(ii) asserts that Ý is removable.

Although singularities are more massive in Theorem 4.10, the assumption kÏ�Ð in it entails that the CR dimension

Þ kf\ ÷ ß ofY

is � � , whenceY b

haspositive CR dimension � ÷ . This insures the existence of small analytic discswith boundary in

Y b. Section 5 below and [MP2006a] as a whole are devoted to

the more delicate case whereY b

has null CR dimension.

Example 4.11. ([Jo1999a]) In° j , let

Y è ñ6v j and letY b çéè 9 Þ42 b ö32 l ö32 j ß ç¨�à � b à ÷ � � ö � b è ¨ : . Clearly,

Y bis foliated by the Ð -spheresß jgö � çéè 9 2 b è �ø÷ b ödù 2 l ù l ~ ù 2 j ù l è ÷ \ ù ��÷ b ù l : ö� ÷ b â Þ4¨þö ÷ � �*ß , that are globally minimal compact Ð -dimensional strongly pseu-

doconvex maximally complex CR submanifolds of CR dimension ÷ bounding the� -dimensional complex ballsv l X gö � çéè 9 2 b è � ÷ b ödù 2 l ù l ~ ù 2 j ù l à ÷ \ ù � ÷ b ù l : HTheorem 4.10 asserts that a compact set ¼ � Y b is removable if and only if itdoes not contain a whole sphere ß g ö � , for some � ÷ b â Þ4¨þö ÷ � �*ß . If ¼ contains sucha sphere ß jgö � , the complex � -ball v l X g ö � coincides with the E Þ ª�ß -hull of ß g ö � andis nonremovable. More generally, an application of both Theorems 4.10 and 3.25yields the following.

Corollary 4.12. Let ¼ be a compact subset ofY b

. For every (interior)one-sided neighborhood �� ñ6v j T¾¼ � that is contained in v j and every func-tion C holomorphic in � � � ñGv j T¾¼ � , there exists a function Ñ holomorphic inv�j ´ � g ö �3úhû �ü ö �hý × v l X g ö � with Ñ è C in � � � ñGv�j$T¾¼w� .

By means of the complex Plateau problem, the next paragraph discusses thenecessity for Ý not to be maximally complex in Theorem 4.9 and for

Y bnot to

contain any CR orbit in Theorem 4.10, in a more general context thanY è ñ6v M .


4.13. Complex Plateau problem and nonremovable singularities contained instrongly pseudoconvex boundaries. Let � be a complex manifold of dimen-sion kS�� . If x»�þ� is a closed pure

Û-dimensional complex subvariety, we

denote by ÿ�x�� the current of integration on x , whose existence was established byLelong in 1957 ([Ch1989, De1997]).

Definition 4.14. ([HL1975, Ha1977]) A current � on � is called a holomorphicÛ-chain if it is of the form � è ^�� Ö �� k&W;ÿ�x]W� ö

where the x W denote the irreducible components of a pureÛ

-dimensional complexsubvariety x of � and where the multiplicity k=W of each xpW is an integer.

The complex Plateau problem consists in filling boundaries Ý by complex sub-varieties x , or more generally by holomorphic chains � . Maximal complexity ofthe boundary Ý is naturally required and since Ý might encounter singular pointsof x , it should be allowed in advance to be “scarred” somehow. Also, the bound-ary Ý inherits an orientation from x and as the boundary of x , it should haveempty boundary.

Definition 4.15. A scarred[��

( ÷ ë��`ë?� ) maximally complex cycle of dimen-sion

Þ �¡Ì ~ ÷ ß , Ì³� ¨, is a compact subset ÝÎ�f� together with a thin compact

scar set sc �fÝ such that¹ L l�� b Þ sc ß è ¨;¹ Ý T sc is an oriented

Þ �¡Ì ~ ÷ ß -dimensional embedded maximally com-plex

[ �submanifold of �/T sc having finite

Þ �¡Ì ~ ÷ ß -dimensionalHausdorff measure;¹ the current of integration over Ý T sc , denoted by ÿ Ý � , has no boundary:ý1ÿ Ý � è ¨

.

This definition was essentially devised by Harvey-Lawson and appears to beadequately large, but sufficiently stringent to maintain the possibility of filling amaximally complex cycle by a complex analytic set.

Theorem 4.16. ([HL1975, Ha1977]) Suppose Ý is a scarred[ �

( ÷ ë»�íëþ� )maximally complex cycle of dimension

Þ �¡Ì ~ ÷ ß , Ì³� ¨, in a Stein manifold � .¹ If Ì è ¨

, assume that Ý satisfies the moment condition, viz. � �� è ¨for every holomorphic ÷ -form � è�� M _ P b � _ Þ42 ß$ý 2 _ having entire coef-ficients �

_ â R{Þ ° M ß .¹ If Ì � ÷ , assume nothing, since the corresponding appropriate momentcondition follows automatically from the assumption of maximal com-plexity ([HL1975]).


Then there exists a unique holomorphic

Þ Ì ~ ÷ ß -chain � in �/T¡Ý havingcompact support and finite mass in � such thatý�� è ÿ Ý �in the sense of currents in � . Furthermore, there is a compact subset ¼ of Ý withL l�� b Þ ¼Ïß è ¨

such that every point of Ý T � ¼ a sc �� possesses a neighborhoodin which

Þ��ß a Ý is a regular

[ �complex manifold with boundary.

A paradigmatic example, much considered since Milnor studied it, consists inintersecting a complex algebraic subvariety of

° Mpassing through the origin with

a spere centered at

¨; topologists usually require that

¨is an isolated singularity

and that the sphere is small or that the defining polynomial is homogeneous.

We apply this filling theorem in a specific situation. Let ñ2ª á ° M ( kË�àÐ )be a strongly pseudoconvex

[ lboundary and let

Y b �Ëñ2ª be an embedded[ l

one-codimensional submanifold that is generic in° M

. We assume thatY b

hasno boundary and is closed, viz. is a compact submanifold. Since

Y bhas CR

dimension

Þ k \Å�*ß , its CR orbits have dimension equal to either

Þ �¾k£\@c)ß , or toÞ �¾k \äÐ*ß or to

Þ �¾k$\{�*ß . Because of Corollary 4.19(III), no CR orbit ofY b

can bean immersed complex � -codimensional submanifold, of real dimension

Þ �¾k \ c)ß ,since its closure in

Y bwould be a compact set laminated by complex manifolds.

Nevertheless, there may exist

Þ �¾kÒ\9Ð*ß -dimensional CR orbits.

Proposition 4.17. ([Jo1999a]) Every CR orbit

R bQ Ã of a connected[ l

hypersur-face

Y b �Îñ2ª of a[ l

strongly pseudoconvex boundary ñ2ª á ° Mis of the

following types:

(i)

R bQ Ã is an open subset ofY

;

(ii)

R bQ Ã is a closed maximally complex[ b

cycle embedded inY b

;

(iii)

R bQ Ã is a maximally complex[ b

submanifold injectively immersed inY b

whose closure H consists of an uncountable union of similar CR orbits.

In the last situation, H will be called a maximally complex exceptional minimalcompact CR-invariant set. The intersection of H with a local curve transversal toa piece CR orbit in

Y bmay consist of either an open segment or of a Cantor

(perfect) subset.Here is the desired converse to both Theorems 4.9 and 4.10 in a situation where

the Plateau complex filling works.

Corollary 4.18. ([Jo1999a]) Suppose that ñ-ª�â [ l X � contains a compact embed-ded

Þ �¾k�\�Ð*ß -dimensional maximally complex submanifold Ý (without boundary).Then Ý is not removable.

Proof. Indeed, the scar set of Ý is empty and the filling of Ý by a holomorphicchain consists of an irreducible complex subvariety x that is necessarily containedin ª , since ñ2ª is strongly pseudoconvex. Then the domain ª�T�x is seen to be


pseudoconvex and ÌÝ � å ÔÖæ è Ý a x . Theorem 3.25 entails that CR functions onñ2ª�T¡Ý extend holomorphicaly to ª�T�x . dA very natural problem, raised in [Jo1999a] and inspired by a perturbation

of Example 4.11, is to determine for which compact CR-invariant subsets ¼ of astrongly pseudoconvex boundary ñ-ª"� ° M the envelope of holomorphy of ñ-ª�T¾¼is multi-sheeted.

Theorem 4.19. ([JS2004]) LetY b �UñGv M be an orientable

Þ �¾k \��*ß -dimensionalgeneric

[ml X � submanifold of ñGv M ( k �»Ð ) and let ¼ � Y b be a compact CR-invariant subset of

Y bsuch that¹ the boundary of ¼ in

Y bis the disjoint union of finitely many con-

nected compact maximally complex CR manifolds Ý b ö HhHhH ö Ý�� of dimen-sion

Þ �¾kÒ\9Ð*ß that are[ l X � � � CR orbits of

Y b;¹ the interior of ¼ with respect to

Y bis globally minimal.

Then the envelope of holomorphy� � � Þ ñGv M T¾¼Ïß � is multi-sheeted in every

neighborhood � � � v M of every point �ãâ��! ?¼ .

We conclude these considerations by formulating a deeply open problem raisedby Joricke. The complex Plateau problem for laminated boundaries is a virginmathematical landscape.

Open question 4.20. ([Jo1999a]) Let ñ2ª"á ° M , k �UÐ , be a strongly pseudocon-vex boundary of class at least

[ l. Suppose that ñ2ª contains a maximally complex

exceptional minimal compact CR-invariant set H . Does H bound a relativelycompact subset x �Uª laminated by complex manifolds ?

As observed in [DH1997, MP1998, Sa1999, DS2001], removable singular-ities have an unexpected interesting application to wedge extension of CR-meromorphic functions.

4.21. CR-meromorphic functions and metrically thin singularities. For k©�� , a local meromorphic map C from a domain ª/� ° M to the Riemann sphere" b Þ ° ß has an exceptional locus # Ü �Íª , at every point � of which the valueC Þ �mß is undefined. For instance the origin

Þ4¨þö3¨ ß â °Dl with C è + �+ V (notice thatevery complex number in

° a |¡� } is a limit of + �+ V ). This exceptional set # Ü is acomplex analytic subset of ª having codimension �Z� ([De1997]). It is called theindeterminacy set of C .

A meromorphic function may be more conveniently defined as a k -dimensionalirreducibe complex analytic subset $ Ü of ª`' " b Þ ° ß having surjective projectiononto ª , viz. ¬ Ô Þ $ Ü ß è ª . Here, ª might be any complex manifold. Indeter-minacy points correspond precisely to points �yâ©ª satisfying ¬ � bÔ Þ �ùßp>%$ Ü è|�� } ' " b Þ ° ß . So, the generalization of meromorphy to the CR category incorpo-rates indeterminacy points.


Definition 4.22. ([HL1975, DH1997, MP1998, Sa1999]) LetY � ° M

be ascarred

[ bgeneric submanifold of codimension ý¶� ÷ and of CR dimensionÌ è kã\cý^� ÷ . Then a CR meromorphic function on

Ywith values in

" b Þ ° ßconsists of a triple

Þ C ö } Ü ö $ Ü ß such that:

1) } Ü � Y is a dense open subset ofY

and C ç } Ü ® " b Þ ° ß is a[ b

map;

2) the closure $ Ü in° M ' " b Þ ° ß of the graph | Þ � ö C Þ �ùßyß ç �wâ�} Ü } defines

an oriented scarred[ b

CR submanifold of° M ' " b Þ ° ß of the same CR

dimension asY

having empty boundary in the sense of currents.

The indeterminacy locus of C is denoted by

# Ü çéè 9 �ãâ Y ç |�� } ' " b Þ ° ß��&$ Ü : HIn the CR category, # Ü is not as thin as in the holomorphic category (where it hasreal codimension �Ðc ), but it is nevertheless thin enough for future purposes, aswe shall see. A standard argument from geometric measure theory yields almosteverywhere smoothness of almost every level set.

Lemma 4.23. ([Fe1969, HL1975, Ha1977]) LetY � ° M be a scarred

[ bgeneric

submanifold. Let

Þ C ö } Ü ö $ Ü ß be a CR meromorphic function onY

. Then foralmost every

7 â " b Þ ° ß , the level setÝ Ü Þ87 ß çéè 9 �ãâ Y ç Þ � ö�7 ß â'$ Ü :is a scarred � -codimensional

[ bsubmanifold of

Y.

Let � â(# Ü . Since

Þ � ö�7 ßÒâ)$ Ü for every

7 â " b Þ ° ß , it follows that # Ü �Ý Ü Þ87 ß for every

7. Fixing such a

7 â " b Þ ° ß , we simply denote Ý Ü çéè Ý Ü Þ87 ß .In particular, the scar set sc +* of Ý Ü is always of codimension � � � in

Y, namelyL[Õ�ÖØ×$Ù � l Þ-,�. * ß è ¨

.So # Ü �fÝ Ü and by definition # Ü ' " b Þ ° ß �&$ Ü . We claim that, in addition, # Ü

has empty interior in Ý Ü T sc * . Otherwise, there exist a point �wâãÝ Ü T sc * anda neighborhood � � of � in

Ywith � � > sc +* è�/ such that # Ü contains � � > Ý Ü ,

whence Þ � � >�Ý Ü ß©' " b Þ ° ß��&$ Ü HSince

Þ � � >ãÝ Ü ß�' " b Þ ° ß has dimension equal to 021Aì Y è 0�1Aì3$ Ü , it followsthat $ Ü > � � � ' " b Þ ° ß � Ñ Þ � � >�Ý Ü ßD' " b Þ ° ßIHBut � � >ÒÝ Ü having codimension two in � � , this contradicts the assumption that$ Ü is a (nonempty!) graph above the dense open subset ��> } Ü of �� .

Lemma 4.24. ([MP1998, Sa1999]) The indeterminacy set # Ü of C is a closed setof empty interior contained in some � -codimensional scarred

[ bsubmanifold Ý Ü

ofY

. Moreover, the scar set sc * of Ý Ü is always of codimension � � � inY

, viz.L l�� r � l Þ sc +*�ß è ¨.


The statement below and its proof are clear if } Ü è Y; in it, the conditionýøÿ $ Ü � è ¨

helps in an essential way to keep it true when the closure of $ Ü pos-sesses a nonempty scar set.

Proposition 4.25. ([MP1998, Sa1999]) There exists a unique CR measure � Ü onY T4# Ü with � Ü ù 5* coinciding with the

[ bCR function C ç } Ü ® " b Þ ° ß .

It is defined locally as follows. Let �Ëâ Y T4# Ü and let � � be an open neigh-borhood of � in

Y. Since �S�â3# Ü , there exists

7 � â " b Þ ° ß with

Þ � ö�7 � ßä�â6$ Ü .Composing with an automorphism of

" b Þ ° ß and shrinking � � , we may assumethat

7 � è � and that

� � � '{|¡� } � >7$ Ü è8/ . Letting ý:9]½ ¼-;=< be some

Þ �¡Ì ~ ý$ß -dimensional volume form on � � , letting ¬�> * ç $ Ü ®@?

denote the natural pro-jection, the CR measure � Ü aa ;=< is defined by

A � Ü ö vCB çéè E >D* 7FE ¬ ÷>:* Þ v ý!9]½ ¼-; < ß öfor every

v âHGJIÂ Þ � � ß .Thus, on

?LK # Ü , the CR-meromorphic function

Þ C ö } Ü ö $ Ü ß behaves like anorder zero CR distribution. With G �Q Ã , · ÀQ Ã+M o�� Â , it therefore enjoys the extendabil-ity properties of Part V on

?NK # Ü , provided that?

is G l M � . The next theoremshould be applied to O çéè # Ü . Its final proof ([MP2002]) under the most gen-eral assumptions combines both the CR extension theory and the application ofthe Riemann-Hilbert problem to global discs attached to maximally real subman-ifolds ([Gl1994, Gl1996]). We cannot restitute the proof here.

Theorem 4.26. ([MP1998, DS2001, MP2002]) Suppose? � °�P is G l M � (

¨@à² à ÷ ) of codimension ý � ÷ and of CR dimension Ì � ÷ . Then everyclosed subset O of

?such that

?and

?NK O are globally minimal and suchthat Q l�� r � l Þ O4ß è ¨

is CR-, Í - and · À -removable.

However, if C is a CR-meromorphic function defined on such a?

, with G Ireplaced by G l M � in Definition 4.22, the complement

?NK # Ü need not be globallyminimal if

?is, and it is easy to construct manifolds

?and closed sets O�� ?

with Q l�� I Þ Oµß à � which perturb global minimality, cf. Example 4.8. It istherefore natural to make the additional assumption that

?is locally minimal at

every point. This assumption is the weakest one that insures that?LK O is globally

minimal, for arbitrary closed sets O�� ? .

Corollary 4.27. Assume that? âRG l M � is locally minimal at every point and letC be a CR-meromorphic function. Then # Ü is CR-, Í - and · À -removable.

Proof. Lemma 4.24 holds with G I replaced by G l M � . It says that # Ü is a closedsubset with empty interior of some scarred G l M � submanifold Ý Ü of

?. The re-

movability of the portion of # Ü that is contained in the regular part of Ý Ü followsfrom Theorem 4.9(i). The removability of the remaining scar set sc +* followsfrom Theorem 4.26 above. S


Thus the CR measure � Ü on?LK # Ü (Proposition 4.25) extends holomorphically

to some wedgelike domain Í I attached to?LK # Ü . The Í -removability of # Ü en-

tails that the envelope of holomorphy of Í I contains a wedgelike domain Í lattached to

?. Performing supplementary gluing of discs, the CR extension the-

ory (Part V) insures that such a Í l depends only on?

, not on C . As envelopesof meromorphy and envelopes of holomorphy of domains in

°TPcoincide by a

theorem going back to Levi ([KS1967, Iv1992]), we may conclude.

Theorem 4.28. ([MP2002]) Suppose? � ° P

is G l M � and locally minimal atevery point. Then there exists a wedgelike domain Í attached to

?to which

every CR-meromorphic function on?

extends meromorphically.

4.29. Peak and smooth removable singularities in arbitrary codimension. Aclosed set O¶� ? is called a G � M U peak set,

¨�àWVfà ÷ , if there exists a noncon-stant function X âHG � M UQ Ã Þ ? ß such that O è)Y X è ÷ Z and ìïî¾ð � ¢ Ù ù

XÞ �ùß ù�[ ÷ .

Theorem 4.30. ([KR1995, MP1999]) Let?

be G]\ M � (

¨Rà ² à ÷ ) globally mini-mal. Then every G � M U peak set O satisfies QBÕ�ÖØ×$Ù Þ Oµß è ¨

and is · À -removable.

To conclude, we mention two precise generalizations of Theorems 4.9 and 4.10to higher codimension. If x è^Y

2 ç ê Þ42 ß è ¨ Zis a local complex hypersurface

passing through a point � of a generic submanifold? � °�P that is transverse to?

at � , viz. _G�;xa`H_G� ? è _�� °CP , the intersection xR> ? is a two-codimensionalsubmanifold of

?that is nowhere generic in a neighborhood of � and certainly

not (locally) removable, since the CR function I* å + æ�bb Ù Õ ådc J Ù æ is not extendable toany local wedge at � .

Theorem 4.31. ([Me1997, MP1999]) Let? � °eP be a Gf\ M � (

¨�à ² à ÷ ) glob-ally minimal generic submanifold of positive codimension ý � ÷ and of positiveCR dimension Ì èhg \cý`� ÷ . Let Ýâ� ? be a connected two-codimensionalGf\ M � submanifold and assume that

?NK Ý is also globally minimal. A closed setO/�ÐÝ is CR-, Í - and · À -removable under each one of the following two cir-cumstances:

(i) Ì³� ÷ and O»�è Ý ;

(ii) Ì³�Z� and there exists at least one point �ãâ�Ý at which Ý is generic.

In (ii), the assumption that Ì¯�w� is essential. Generally, if Ì è ÷ , whenceý èig \ ÷ and 021Aì ? èjg ` ÷ , a local transverse intersection O è x?> ?has dimension g \ ÷ , hence cannot be generic, and is not (locally) removable byconstruction. In the next statement, the similar assumption that Ì³�Z� is stronglyused in the proof: the one-codimensional submanifold

? I � ?has then CR

dimension Ìá\ ÷ � ÷ , hence there exist small Bishop discs attached to? I .

Theorem 4.32. ([Po1997, Me1997, Po2000]) Let? � °�P be a Gf\ M � (

¨�à ² à ÷ )globally minimal generic submanifold of positive codimension ýä� ÷ . Assume that


the CR dimension Ì èWg \äý of?

satisfies Ì³�Z� . Let? I � ? be a connectedGf\ M � one-codimensional submanifold that is generic in°�P

. A closed set OË� ? Iis CR-, Í - and · À -removable provided it does not contain any CR orbit of

? I .Three geometrically different proofs of this theorem will be restituted in Sec-

tion 10 of [MP2006a]. The next Section 5 and [MP2006a] are devoted to thestudy of the more delicate case where Ì è ÷ and where O is contained in someone-codimensional submanifold

? I � ? .

k5. REMOVABLE SINGULARITIES IN CR DIMENSION ÷

5.1. Removability of totally real discs in strongly pseudoconvex boundaries.In 1988, applying a global version of the Kontinuitatssatz, Joricke [Jo1988] es-tablished a remarkable theorem, opening the way to a purely geometric study ofremovable singularities.

Theorem 5.2. ([Jo1988]) Let ñ2ª©á ° \ be a strongly pseudoconvex Gl\ boundaryand let m �Uñ2ª be a G \ one-codimensional submanifold that is diffeomorphic tothe unit open � -disc of no\ and maximally real at every point. Then every compactsubset ¼ of m is CR-, · Y

and Í -removable.

By maximal reality of m , the line distribution mip(� q®sr � çéè _ � m»>t_ Â� ? isnowhere vanishing and may be integrated. This yields the characteristic foliationu Âv on m . The compact set ¼ is contained in a slightly smaller disc m � á&m hav-ing Gf\ boundary ñwm � . Poincare-Bendixson’s theorem on such a disc m � togetherwith the inexistence of singularities of

u Âv entail that every characteristic curvethat enters into m � must exit from m � . Orienting then the real � -disc m and itscharacteristic foliation, we have the following topological observation (at the verycore of the theorem) saying that there always exists a characteristic leaf that is notcrossed by the removable compact set.

x x x x x xx x x x x xx x x x x xx x x x x x

y y y y y yy y y y y yy y y y y yy y y y y y

zz

Nontransversality of Û to {�|}

~ Û

HOLOMORPHIC EXTENSIONS AND REMOVABLE SINGULARITIES 253u Âv Y ¼ Z ç For every compact subset ¼ � � ¼ , there exists a Jordan curve � çÿ \ ÷ ö ÷ � ® m , whose range is contained in a single leaf of the charac-teristic foliation

u Âv , with �Þ \ ÷ ß �â`¼ � , � Þ4¨ ß�â ¼ � and �

Þ ÷ ß �â`¼ � , suchthat ¼ � lies completely in one closed side of �]ÿ \ ÷ ö ÷ � with respect to thetopology of m in a neighborhood of �]ÿ \ ÷ ö ÷ � .

In the more general context of [MP2006a], we will argue thatu Âv Y ¼ Z is the

very reason why ¼ is removable. We will then remove locally a well chosenspecial point � �� âw¼ � >��Bÿ \ ÷ ö ÷ � . In fact, we shall establish removability ofcompact subsets ¼ of general surfaces ß that are not necessarily diffeomorphicto the unit � -disc, provided that an analogous topological condition holds. Also,getting rid of strong pseudoconvexity, we shall work with a globally minimal G \ M �hypersurface of

° \ . Finally, we shall relax slightly the assumption of total reality,admitting some complex tangencies.

Example 5.3. Let ª è v \ and let" Þ42 ß©â ° ÿ 2 � be a homogeneous polynomial of

degree �Z� having

¨has its only singularity. The intersection ¼ çéè ñ6v \ > Y " è¨ Z

is a finite union of closed real algebraic curvesª ß I that are everywhere

transverse to _Â ñGv \ . We may enlarge each curve of ¼ as a thin G < annulus.

There is much freedom, but every such annulus is necessarily totally real. Denoteby ß the union of all annuli, a surface in ñ6v \ . Clearly, no component of ¼ isremovable. But the theorem does not apply: on each annulus, the characteristicfoliation

u Âû is radial and ¼ crosses each characteristic leaf.

Example 5.4. The theorem may fail with the disc m replaced by a surface ßhaving nontrivial fundamental group, even with ß compact without boundary.For instance, in ñ6v \ è�Y

ù 2Iù\ `

ù 2\ù\ è ÷ Z , the two-dimensional torus _ \ çéè9 � IÁ \ e l�� ö IÁ \ e l�� V � ç�� I ö � \ â3n : is compact and ¼ çéè _ \ is not removable,

since ñGv \ K _ \ has exactly two connected components.

Example 5.5. ([Jo1988]) In the same torus _ \ , consider instead the proper com-pact subset ¼ çéè 9 �

IÁ \ ehl�� ö IÁ \ ehl�� V � ç ù� I

ùl[ j o\ ö � \ â6n : , diffeomorphic toa closed annulus. It is a set fibered by circles (contained in

° + V ) over the curveÌ � çéè 9 IÁ \ e l�� ç ù� I

ù�[ j o\ : that is contained in° + � . One may verify that the

conditionu Â� V Y ¼ Z insuring removability does not hold. In fact, applying The-

orem 2.2 (in the much simpler version due to Denjoy where the curve is realanalytic), the curve Ì � is not

Þ ñ ö · Y ß -removable in° + � . So we may pick a holo-

morphic function ÌC Þ42 I ßãâ�� °oK Ì � � that is bounded in° a Y � Z but does not

extend holomorphically through Ì � . The restriction ÌC bb Ó � V Õy× belongs to · Y Þ ñGv \ ß ,is CR on ñ6v \ K ¼ but does not extend holomorphically to v \ .

Before pursuing, we compare Theorem 5.2 and Theorem 4.10.In codimension �¶� (e.g. for curves in n j ), no satisfactory generalization of

the Poincare-Bendixson theory is known and perhaps is out of reach. This gap is


caused by the complexity of the topology of phase diagrams, by the freedom thatcurves have to wind wildly around limit cycles, and by the intricate structure ofsingular points.

Nevertheless, in higher complex dimension g � Ð , CR orbits are thicker thancurves and often of codimension

[ ÷ . For triples

Þ ? ö ? IöOµß as in Theorem 4.10

with? è ñ2ª being strongly pseudoconvex, one could expect that a statement

analogous to Theorem 5.2 holds true, in which the assumption that? I has simple

topology would imply automatic removability of every compact subset ¼Î� ? I .To be precise, let ñ-ªSá °�P ( g �ZÐ ) be a G+\ M � strongly pseudoconvex boundary

and let? I � ñ2ª be a G \ M � one-codimensional submanifold that is generic in°CP

. Strong pseudoconvexity of ñ2ª entails that CR orbits of? I are necessarily

of codimension

[ ÷ in? I . Remind that Theorem 4.10 says that a compact

subset ¼ of? I is removable provided it does not contain any CR orbit of

? I .Conversely, in the case where

? I has no exceptional CR orbit, if ¼ containsa (then necessarily compact and maximally complex) CR orbit Ý of

? I , then¼ is not removable, since Ý is fillable by some

Þg \ ÷ ß -dimensional complex

subvariety xS� ª with ñøx è Ý . Thus, while comparing the two Theorems 4.10and 5.2, the true question is whether the assumption that

? I �/ñ-ª è ?be

diffeomorphic to the real

Þ � g \+�*ß -dimensional real ball §t\ P ��\6�&ne\ P ��\ preventsthe existence of compact

Þ � g \4Ð*ß -dimensional CR orbits of? I . This would yield

a neat statement, valid in arbitrary complex dimension.For instance, let Ý çéè ñ6v P > z

be the intersection of the sphere ñ6v P ª ß \ P � Iwith a complex linear hyperplane

z � °oP . With such a simple Ý homeomor-phic to a

Þ � g \ Ð*ß -dimensional sphere, one may verify that every G Y subman-ifold

? I �Òñ2ª containing Ý which is diffeomorphic to §�\ P ��\ must containat least one nongeneric point. Nevertheless, admitting that Ý has slightly morecomplicated topology, the expected generalization of Theorem 5.2 appears to fail,according to a discovery of Joricke-Shcherbina. This confirms the strong differ-ences between CR dimension Ì è ÷ and CR dimension Ì³�U� .Theorem 5.6. ([JS2000]) For ·�â8n with

¨fà · à ÷ close to ÷ , consider theintersection Ý�� çéè 9 2 I 2 \ 2 j è · : >'� Ð½ñ6v�jof the complex cubic Y

2I2\2 j è · Z with the sphere � Ð1ñ6v j è�Y

ù 2Iù\ `

ù 2\ù\ `ù 2 j ù \ è Ð Z . Then Ý � is a maximally complex cycle diffeomorphic to ß I '(ß I '(ß I

bounding the (nonempty) complex surface x�� çéè�Y2I2\2 j è · Z >rv�j . Further-

more, there exists a suitably constructed G Y generic one-codimensional subman-ifold

? I �Zñ6v j diffeomorphic to the real

Þ � g \9�*ß -dimensional unit ball §a\ P ��\containing Ý � . Finally, since Ý � bounds x � , every compact subset ¼ � ? Icontaining Ý � is nonremovable.


5.7. Elliptic isolated complex tangencies and Bishop discs. Coming back tocomplex dimension g�è � , we survey known properties of isolated CR singu-larities of surfaces. So, let ß be a two-dimensional surface ß in

° \ of class atleast Gf\ . At a point �fâfß , the complex tangent plane _ � ß is either totally (andin fact maximally) real, viz. _ � ßu>3��_ � ß è�Y

¨ Zor it is a complex line, viz._ � ß è ��_ � ß è _

Â� ß . An appropriate application of the jet transversality the-orem shows that after an arbitrarily small perturbation, the number of complextangencies of ß is locally finite.

If ß has an isolated complex tangency at one of its points � , Bishop ([Bi1965])showed that there exist local coordinates

Þ42�ö�7 ß centered at � in which ß may berepresented by

7 è 2 ÿ2 `7� Þ42 \:` ÿ2 \�ß�`$½ Þ3ù 2øù\¾ß , where the real parameter �râ�ÿ ¨þö ��

is a biholomorphic invariant of ß . The point � is said to be elliptic if �@â�ÿ ¨þö I\ ß ,parabolic if � è I\ and hyperbolic if �`â Þ I\ ö �� . The case � è � should beunderstood as the surface

7 è 2\�` ÿ2 \�`u½ Þ3ù 21ù

\¾ß . The shape of the projection ofsuch a surface onto the real hyperplane Y �Nì 7 è ¨ Z ª n j is essentially ellipsoid-like for

¨¤à�

à ÷ � � and essentially saddle-like for �`� ÷ � � .In the seminal article [Bi1965], Bishop introduced this terminology and showedthat at an elliptic point, ß has a nontrivial polynomial hull Ìß , foliated by a con-tinuous one-parameter family of analytic discs attached to

?. The geometric

structure of this family has been explored further by Kenig and Webster.

Theorem 5.8. ([KW1982, BG1983, KW1984, Hu1998]) Let ß � ° \ be a G �( �c�)� ) surface having an elliptic complex tangency at one of its points � . Thenthere exists a G å � �w� æ i j one-parameter family of disjoint regularly embdedded an-alytic discs attached to ß and converging to � . If ß is G Â , then Ìß is G � M�I . Fur-thermore, every small analytic disc attached to

?near � is a reparametrization

of one of the discs of the family.For � è � , the union of these discs form a G Y hypersurface Ìß with boundaryñDÌß è ß in a neighborhood of � . Furthermore, Ìß is the local hull of holomorphy

of ß at � .

In the case where ß is real analytic, local normal forms may be found thatprovide a classification up to biholomorphic changes of coordinates.

Theorem 5.9. Let ß ç 7 è 2 ÿ2 `W� Þ42 \�` ÿ2 \�ßJ`8� Þ3ù 21ù j ß be a local real analyticsurface in

° \ passing through the origin and having an elliptic complex tangencythere. ¹ ([MW1983]) For every � satisfying

¨Ëà�

à ÷ � � , either ß is locallybiholomorphic to the quadric

7 è 2 ÿ2 `)� Þ42 \ ` ÿ2 \�ß or there exists aninteger ¡`â£¢ , ¡"� ÷ , such that ß is locally biholomorphic to

7 è2 ÿ2 `�ÿ¤�¥`&¦¨§�©�� Þ42 \ ` ÿ2 \ ß , where § è Ó�Ô 7and ¦ è n ÷ .

256 JOEL MERKER AND EGMONT PORTEN¹ ([Mo1985]) For � è ¨, either ß is locally biholomorphic to

7 è 2 ÿ2 `2© ` ÿ2 © `R� Þ3ù 21ù

© � I ß for some integer ¡µ�ZÐ or ß is locally biholomorphicto

7 è 2 ÿ2 .¹ ([HuKr1995]) For � è ¨and ¡

à � , the surface ß is locally biholomor-phic to the surface

7 è 2 ÿ2 ` 2© ` ÿ2 © ` � W � _ M ©�ª W _ 2 W ÿ2

_, with ª W _ è ª _ W .

In all cases, after the straightening, ß is contained in the real hyperplaneY �Nì 7 è ¨ Z.

In the third case � è ¨, ¡

à � , it is still unknown how many biholomorphicinvariants ß can have.

5.10. Hyperbolic isolated complex tangencies. The existence of small Bishopdiscs attached to ß and growing at an elliptic complex tangency impedes localpolynomial convexity. At the opposite, if ß is hyperbolic, Bishop’s constructionfails, discs are inexistent, and in fact ß is locally polynomially convex.

Theorem 5.11. ([FS1991]) Let ß³� ° \ be a Gf\ surface represented by

7 è2 ÿ2 `3� Þ42 \ ` ÿ2 \ ß�`@~ Þ42�ö ÿ2 ß , with a G \ remainder ~ è ½ Þ3ù 21ù\ ß . If � � ÷ � � , viz. if ß

is hyperbolic at the origin, then for every « I � ¨sufficiently small, ß�> �

« I v \ � ispolynomially convex.

The Oka-Weil approximation theorem then assures that continuous functionsin ßÒ> �

« I v \ � are uniformly approximable by polynomials.A local Bishop surface ß is called quadratic if it is locally biholomorphic to

the quadric

7 è 2 ÿ2 `�� Þ42 \ ` ÿ2 \ ß . An isolated complex point � of ß is calledholomorphically flat if there exist local coordinates centered at � in which ß islocally contained in Y �Nì 7 è ¨ Z

. Unlike elliptic points of G < surfaces that arealways flat, hyperbolic complex points of G < surfaces may fail to be flat.

Example 5.12. ([MW1983]) The algebraic hyperbolic surface ( � � ÷ � � )7 è 2 ÿ2 `&� Þ42 \ ` ÿ2 \ ßf`&� 2 j ÿ2cannot be biholomorphically transformed into a real hyperplane.

Theorem 5.11 establishes local polynomial pseudoconvexity of surfaces at hy-perbolic complex tangencies. By patching together local plurisubharmonic defin-ing functions, one may easily construct a Stein neighborhood basis of every sur-face having only finitely many hyperbolic complex tangencies. Unfortunately, inthis way one does not control well the topology of such neighborhoods. A finerresult answering a question of Forstneri ¬c is as follows.

Theorem 5.13. ([Sl2004]) Let ß be a compact real G Y surface embedded in acomplex surface having only finitely many complex points that are all hyper-bolic and holomorphically flat. Then ß possesses a basis of open neighborhoods� � � �¨®�¯ � ¯ ��° , · I�± ¨

, such that:

HOLOMORPHIC EXTENSIONS AND REMOVABLE SINGULARITIES 257¹ ß è�² � M ® � � ;¹ � � è � � � ¯ � � � � ;¹ � � è�² �ú� M � � � � ;¹ each � � has a G Y strongly pseudoconvex boundary ñ&� � ;¹ for every · with

¨¤à · à · I , the surface ß is a strong deformation retractof � � .

It is expected that the same statement remains true without the flatness assump-tion.

5.14. Real surfaces in strongly pseudoconvex boundaries. Coming back to re-movable singularities, let ñ2ªòá ° \ be a Gf\ strongly pseudoconvex boundaryand let ß´³»ñ-ª be a compact surface, with or without boundary. It will be norestriction to assume that ß is connected. Suppose that ß has a finite (possiblynull) number of complex tangencies. These points then constitute the only sin-gular points of the characteristic foliation of ß . At an elliptic (resp. hyperbolic)complex tangency, the phase diagram simply looks like a focus (resp. saddle).

Theorem 5.15. ([FS1991]) Let � be a two-dimensional Stein manifold, let ñ2ª©á� be a strongly pseudoconvex Gl\ boundary and let m be a Gl\ one-codimensionalsubmanifold that is diffeomorphic to the unit open µ -disc of n�\ and is maximallycomplex, except at a finite number of hyperbolic complex tangencies. Then everycompact subset ¼ of m is CR- and Í -removable.

Indirectly, the characterizing Theorem 3.7 of Stout yields the following.

Corollary 5.16. Every compact subset ¼s³&m¶³Zñ2ª is �Þ ªgß -convex. In particu-

lar, such a ¼ is polynomially convex if � è ° \ and if ª is Runge or polynomiallyconvex, e.g. if ª è v \ .

The (short) proof mainly relies upon the (very recent in 1991 and since then fa-mous) works [BK1991] and [Kr1991] by Bedford-Klingenberg and by Kruzhilinabout the hulls of two-dimensional spheres contained in such strictly pseudo-convex boundaries ª�³�� , which may be filled by Levi-flat three-dimensionalspheres after an arbitrarily small perturbation.

Theorem 5.17. ([BK1991, Kr1991]) Let ª á ° \ be a Gf· strongly pseudoconvexdomain and let ß3³Zñ2ª be a two-dimensional sphere of class G · embdedded intoñ2ª that is totally real outside a finite subset consisting of

Ûhyperbolic and

Û `3µelliptic points. Then there exist:

1) a smooth domain §i³)n j Þ � I ö � \ ö � j ß with boundary ñ1§ diffeomorphicto ß such that � j ç § ® n is a Morse function on ñ&§ having

Û `�µextreme points and

Ûsaddle points, whose level sets Y � j èW¸ � hH Z > § are

unions of finite numbers of topological discs; and:


1) a continuous injective map É ç § ® ª sending ñ1§ to ß , the extreme andsaddle points of � j on ñ&§ to the elliptic and hyperbolic points of ß andthe connected components of Y � j è£¸ � hH Z >ã§ to geometrically smoothholomorphic discs.

The set É Þ §ïß is the envelope of holomorphy of ß as well as its �Þ ª�ß -hull, i.e.

its polynomial hull in case ª is polynomially convex.

In [Du1993], motivated by the problem of understanding polynomial convexityin geometric terms, the question of �

Þ ª�ß -convexity (instead of removability) ofcompact subsets of arbitrary surfaces ß6³Zñ2ª (not necessarily diffeomorphic to aµ -disc) is dealt with directly. If ¼ is a compact subset of a totally real surface ß3³ñ2ª , denote by Ì¼ � �-� çéè Ì¼äã½å Ô-æ K ¼ the essential �

Þ ª�ß -hull of ¼ . An application

of Hopf’s lemma shows that if ¼ èº¹Þ ñ�»Ýß is the boundary of a G I analytic

disc ¹ â&� Þ »ÝßB>'G I Þ »Ýß attached to the surface ß , necessarily ¼ è Ì¼ � �� is animmersed GJI curve that is everywhere transversal to the characteristic foliation ofß . If ß has a hyperbolic complex tangency at one of its points � and if ¹

Þ ÷ ß è � ,then ¹

Þ ñw»Ýß must cross at least one separatrix in every neighborhood of � . WhenÌ¼ � �-� contains no analytic disc, similar transversality properties hold.

Theorem 5.18. ([Du1993]) Let ¼ áyßF³©ñ2ª�á ° \ be as above, with ñ2ª¹âRG \strongly pseudoconvex and ßÎâ G \ having finitely many hyperbolic complex tan-gencies. In the totally real part of ß , the essential �

Þ ª�ß -hull Ì¼ � �-� of ¼ crossesevery characteristic curve that it meets. If Ì¼ � �� meets a hyperbolic complex tan-gency, then it meets at least two hyperbolic sectors in every neighborhood of � .

As a consequence ([Du1993]), every compact subset ¼ of a two-dimensionaldisc m¶³Zñ2ª that has only finitely many hyperbolic complex tangencies is �

Þ ª�ß -convex.

5.19. Totally real discs in nonpseudoconvex boundaries. All the above resultsheavily relied on strong pseudoconvexity, in contrast to the removability theoremspresented in Section 6, where the adequate statements, based on general CR ex-tension theory, are formulated in terms of CR orbits rather than in terms of Levicurvature. The first theorem for the non-pseudoconvex situation was establishedby the second author.

Theorem 5.20. ([Po2003]) Let?

be a G Y globally minimal hypersurface of° \

and let m¶³ ? be a G Y one-codimensional submanifold that is diffeomorphic tothe unit open µ -disc of no\ and maximally real at every point. Then every compactsubset ¼ of m is CR-, · À - and Í -removable.

We would like to point out that, seeking theorems without any assumptionof pseudoconvexity leads to substantial open problems, because one loses al-most all of the strong interweavings between function-theoretic tools and geo-metric arguments which are valid in the pseudoconvex realm, for instance: Hopf


Lemma, plurisubharmonic exhaustions, envelopes of function spaces, local max-imum modulus principle, Stein neighborhood basis, etc.

We sketch the proof of the theorem. We first claim that?LK ¼ is (also) globally

minimal. Indeed, if there were a lower-dimensional orbit � of?NK ¼ , we would

obtain a lower-dimensional orbit of?

by adding all characteristic arcs intersect-ing � ([Po2003], Lemma 1; [MP2006a], Lemma 3.5). Then by Theorem 4.12(V),continuous CR functions on

?NK ¼ extend holomorphically to a one-sided neigh-borhood ��¼ Þ ?LK ¼Ïß .

For later application of the continuity principle, similarly as in [MP2002,Po2003, MP2006a], we deform

?LK ¼ in � ¼ Þ ?LK ¼Ïß , so that the functions areholomorphic in some ambient neighborhood ½ of

?LK ¼ in° \ .

The first key idea is to construct an embedded 2-sphere containing a neighbor-hood of ¼ in m and to apply the filling Theorem 5.17. This will give us a Leviflat Ð -ball foliated by analytic discs, which by translations, will enable us to fill ina one-sided neighborhood of ¼ .

In the case where? è ñ2ª is a strictly pseudoconvex boundary, the con-

struction of the µ -sphere is quite direct: we pick an open µ -disc m¿¾ having G Yboundary ñ�m ¾ ª ß I with ¼ ³¶m ¾ á¶m ; translating it slightly and smoothlywithin ñ2ª , we obtain an almost parallel copy m ¾ ¾ ³Pñ2ª ; then we construct theµ -sphere ß ¾ by gluing (inside ñ-ª ) a thin closed strip

ª ÿ \d· ö ·��Z'�ß I to ñ�m ¾ ª ß Iand to ñwm ¾ ¾ ª ß I ; finally, we perturb the strip part of ß ¾ in a generic way toassure that ß ¾ has only (a finite number of) isolated complex tangencies of ellipticor of hyperbolic type31. Then Theorem 5.17 yields a Levi-flat 3-ball § ¾ ³Zª withñ1§ ¾ è ß ¾ .

If?

is not strongly pseudoconvex, the filling of ß ¾ by a Levi-flat ball § ¾ mayfail, because of a known counter-example [FM1995]. As a trick, we modify theconstruction. Using the fact that the squared distance function 021

� ÞÀE öm ¾ ßÀ\ is

strictly plurisubharmonic in a neighborhood of m ¾ (by total reality), for · ± ¨small, the sublevel sets

ª ¾� çéè 9:Á â ° \ ç 0�1 � �Áöm ¾ � à · :

are strongly pseudoconvex neighborhoods of m ¾ intersecting?

transversallyalong the 2-spheres ñ-ª ¾ Â > ? .

31Observe that sincez

is totally real, the last step can be done without changing Ã]Äalong

z Ä .


z Ä

ç

È�Ù Ä Å

Construction of Ã+Ä

z Ä

çÃfÄ

ÈþÙ Ä

Furthermore, a given fixed ª ¾ Â can be slightly isotoped (translated) to a domainª ¾ still strongly pseudoconvex and having boundary transverse to?

so that m ¾ isprecisely contained in the isotoped 2-sphere ñ-ª ¾ > ? . After a very slight genericperturbation, we may insure that ß ¾ has only elliptic or hyperbolic complex tan-gencies (a part of ñ2ª ¾ has also to be perturbed). In sum:

Lemma 5.21. ([Po2003]) There exists a bounded domain ª ¾ ³ ° \ such that:¹ ñ2ª ¾ is G Y , strongly pseudoconvex and diffeomorphic to a Ð -sphere;¹ ñ2ª ¾ intersects?

transversally in a two-sphere ß ¾ çéè ñ2ª ¾ > ? ;¹ ß ¾ hasÛ

hyperbolic andÛ `6µ elliptic points;¹ ñ2ª ¾ contains the open µ -disc m ¾ s ¼ .

Then Theorem 5.17 applies in the strongly pseudoconvex boundary ñ2ª ¾ , yield-ing a Levi-flat Ð -sphere § ¾ ³�ª ¾ with ñ&§ ¾ è ß ¾ . However, the nonpseudocon-vexity of

?obstructs further insights in the position of § ¾ with respect to

?. In

fact, § ¾ may change sides or even be partly contained in?

.In the (simpler) case where

? è ñ-ª is a strongly pseudoconvex boundary,we introduce a foliation of a neighborhood of ß ¾ in

?by G Y µ -spheres ß ¾Æ withß ¾® è ß ¾ . By filling them, we get a family of Levi-flat 3-balls § ¾Æ with ñ1§ ¾Æ è ß ¾Æ .

Denote § ¾Æ è a © » ¾Æ M © the foliation of § ¾Æ by holomorphic discs. For ÇÉÈè ¨, each» ¾Æ M © has boundary ñ�» ¾Æ M © ³ ß ¾Æ ³ ?LK ¼ . Thus, by means of the continuity

principle, we may extend holomorphic functions in the neighborhood ½ of?LK ¼

to a neighborhood of § ¾Æ in°CP

, for all small Ç�Èè ¨. A final simple check shows

that Theorem 2.30 (rm5) applies to remove § ¾® , and we get holomorphic extensionto the union a Æ § ¾Æ , a set containing the strongly pseudoconvex open local side ofª at every point of ¼ .

Without pseudoconvexity assumption on?

, we can still consider a foliationß ¾Æ , but now the global geometry of § ¾Æ is no longer clear. If for instance?

isLevi-flat near ¼ and the ß ¾Æ are contained in the Levi-flat part, then the § ¾Æ justform an increasing family whose union is just a subdomain of

?. Therefore it

seems necessary to deform ß ¾ once again in order to gain transversality of § ¾ and?. Since the global behavior of § ¾ is hard to control, a further localization is

advisable.


As in [Me1997], we consider the set ¼ ��Ê of points Á â ¼ such that�� Þ ?LK ¼Ïß � does not extend holomorphically to a one-sided neighborhood

of Á . So �� Þ ?LK ¼Ïß � extends holomorphically to a one-sided neighorhood��¼ � ¼ K ¼ ��Ê � . By deforming

?at points of ¼ K ¼ �Ê , we come down to the same

situation with ¼ replaced with ¼ �Ê , except that no point of ¼ ��Ê should be remov-able. Assuming ¼ �Ê Èè8/ , to conclude by contradiction, it then suffices to removeonly one point of ¼ �Ê .

To begin with, assume that ¼ �Ê is contained in finitely many of the disc bound-aries ñ�» ¾® M © which foliate ß ¾ è ß ¾® . Then we claim that no ñ�» ¾® M © can be containedin ¼ �Ê . Otherwise, ñ�» ¾® M © ³�¼ ��Ê ³Lm ¾ ³Lm and the µ -disc enclosed by ñw» ® M ©in ß ¾® inside the totally real µ -disc m ¾ contain no complex tangencies, but the fill-ing provided by Theorem 7.17 excludes such a topological possibility. So ¼ �Ê isproperly contained in a finite union of arcs, and hence removable by Theorem 4.9.

Therefore we may assume that ¼ �Ê has nonvoid intersection with infinitelymany of ñw» ¾® M © . Since there is only finitely many complex tangencies, there existsa ñw» ¾® M ©ÌË with ñ�» ¾® M ©ÍË > ¼ �Ê Èè�/ not encountering them. The same argument asabove shows that ñw» ¾® M ©ÍË È³Z¼ �Ê . Let � ® â ñ�» ¾® M ©ÍË >ä¼ �Ê .

If » ¾ ® M ©ÍË and?

meet transversally at � ¾ ® , holomorphic extension to a one-sidedneighborhood at � ¾ ® proceeds as in the strongly pseudoconvex case, by applyingthe continuity principle with discs » ¾Æ M © ³f§ ¾Æ for ÇÎÈè8Ï .

Assume now that » ¾ ® M ©ÍË is tangential to?

in � ¾ ® or equivalently, that ñ�» ¾® M ©ÌËis tangential to the characteristic leaf in � ¾ ® . The idea is to change the angle ofthe discs close to » ¾ ® M ©ÍË , and to apply the above argument to the deformed discpassing through � ¾ ® . Since ñ�» ¾® M ©ÍË È³S¼ �Ê , we may deform slightly ß ¾ near somepoint Á4¾® â`ñw» ¾® M ©ÍË K ¼ �Ê in the direction normal to § ¾ . More precisely, one deformsß ¾ slightly, so that Theorem 5.17 still applies, and then picks up the disc of thedeformed Levi-flat 3-ball that passes through � ¾ ® . In view of known results aboutnormal deformations of small discs (Proposition 2.21(V); [Trp1990, BRT1994,Tu1994a]), the turning of the angle for large discs ([Fo1986, Gl1994]) may alsobe established in such a way (see [Po2003, Po2004]).

There is one final point to be handled carefully. We have to be sure that afterturning the discs, the deformed disc boundary passing through the point � ¾ ® â ¼ �Êis not entirely contained in ¼ �Ê .


ÛÎÐ�ÑChoice of ÒÓÄS

Ò�ÄS ÔÒ�ÄSz Ä

È�Õ ÄS ä Ö

This can be assured by replacing � ¾ ® by another special nearby point ×� ¾ ® â^¼ �Êwith a good transversality property as illustrated above. S

Theorem 5.20 is not yet the complete generalization of Theorem 5.15 tononpseudoconvex hypersurfaces, since m is assumed to be totally real at everypoint. If m has hyperbolic complex tangencies, it is not clear whether a sphere ß ¾together with a strongly pseudoconvex boundary ñ2ª ¾ s ß ¾ as in the above keylemma can be constructed. The recent Theorem 5.13 indicates that this is possibleif hyperbolic complex tangencies are holomorphically flat, an assumption whichwould be rather ad hoc for the removal of compact sets ¼s³&m .

In fact, assuming generally that?

is an arbitrary globally minimal hypersur-face, that a given surface ß)³ ? has arbitrary topology (not necessarily diffeo-morphic to an open µ -disc) and possesses complex tangencies, the reduction to thefilling Theorem 5.17 seems to be impossible. Indeed, Fornæss-Ma ([FM1995])constructed an unknotted nonfillable µ -sphere ß�³ ° \ having only two ellip-tic complex tangencies. To the authors’ knowledge, the possibility of filling byLevi-flat Ð -spheres some µ -spheres lying in a nonpseudoconvex hypersurface is adelicate open problem. In addition, for the higher codimensional generalizationof Theorem 1.2, the idea of global filling seems to be irrelevant at present times,because no analog of the filling Theorem 5.17 is known in dimension g%Ø Ð .5.22. Beyond this survey. In the research article [MP2006a] placed in direct con-tinuation to this survey, we consider surfaces ß having arbitrary topology and wegeneralize Theorem 5.20 to arbitrary codimension, localizing the removabilityarguments and using only small analytic discs.

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