References

[Ad] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.[AhS] L. Ahlfors, L. Sario, Riemann Surfaces, Princeton University Press, Princeton, 1960.[AnV] A. Andreotti, E. Vesentini, Carlemann estimates for the Laplace–Beltrami equation on

complex manifolds, Publ. Math. IHÉS 25 (1965), 81–130.[BehS] H. Behnke, K. Stein, Entwicklung analytischer Funktionen auf Riemannschen Flächen,

Math. Ann. 120 (1949), 430–461.[BerG] C. Berenstein, R. Gay, Complex Variables. An Introduction, Graduate Texts in Mathe-

matics, 125, Springer, New York, 1991.[Bis] E. Bishop, Subalgebras of functions on a Riemann surface, Pac. J. Math. 8 (1958), 29–

50.[C] C. Carathéodory, Über die gegenseitige Beziehung der Ränder bei der konformen Ab-

bildung des Inneren einer Jordanschen Kurve auf einen Kreis, Math. Ann. 73 (1913),no. 2, 305–320.

[De1] J.-P. Demailly, Estimations L2 pour l’opérateur ∂ d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète, Ann. Sci. Ec. Norm. Super. 15(1982), 457–511.

[De2] J.-P. Demailly, Cohomology of q-convex spaces in top degrees, Math. Z. 204 (1990),283–295.

[De3] J.-P. Demailly, Complex Analytic and Differential Geometry, online book.[Fa] M. Farber, Topology of Closed One-Forms, Mathematical Surveys and Monographs,

108, American Mathematical Society, Providence, 2004.[FarK] H. Farkas, I. Kra, Riemann Surfaces, Graduate Texts in Mathematics, 71, Springer, New

York, 1980.[Fl] H. Florack, Reguläre und meromorphe Funktionen auf nicht geschlossenen Rie-

mannschen Flächen, Schr. Math. Inst. Univ. Münster, no. 1, 1948.[Fol] G. Folland, Real Analysis: Modern Techniques and Applications, second ed., Wiley,

New York, 1999.[For] O. Forster, Lectures on Riemann Surfaces, Graduate Texts in Mathematics, 81, Springer,

Berlin, 1981.[Ga] D. Gardner, The Mergelyan–Bishop theorem, preprint.[GiT] D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,

second ed., Springer, Berlin, 1983.[GreW] R. E. Greene, H. Wu, Embedding of open Riemannian manifolds by harmonic functions,

Ann. Inst. Fourier (Grenoble) 25 (1975), 215–235.[GriH] P. Griffiths, J. Harris, Principles of Algebraic Geometry, Pure and Applied Mathematics,

Wiley-Interscience, New York, 1978.[GueNs] J. Guenot, R. Narasimhan, Introduction à la théorie des surfaces de Riemann, Enseign.

Math. (2) 21 (1975), nos. 2–4, 123–328.

T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones,DOI 10.1007/978-0-8176-4693-6, © Springer Science+Business Media, LLC 2011

545

546 References

[GuiP] V. Guillemin, A. Pollack, Differential Topology, Prentice-Hall, Englewood Cliffs, 1974.[HarR] F. Hartogs, A. Rosenthal, Über Folgen analytischer Funktionen, Math. Ann. 104 (1931),

no. 1, 606–610.[Hat] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2001.[Hö] L. Hörmander, An Introduction to Complex Analysis in Several Variables, third edition,

North-Holland, Amsterdam, 1990.[Hu] J. H. Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dy-

namics, Vol. 1: Teichmüller Theory, Matrix Editions, Ithaca, 2006.[JP] M. Jarnicki, P. Pflug, Extension of Holomorphic Functions, de Gruyter Expositions in

Mathematics, 34, Walter de Gruyter, Berlin, 2000.[KaK] B. Kaup, L. Kaup, Holomorphic Functions of Several Variables: An Introduction to the

Fundamental Theory, with the assistance of Gottfried Barthel, trans. from the Germanby Michael Bridgland, de Gruyter Studies in Mathematics, 3, Walter de Gruyter, Berlin,1983.

[Ke] J. L. Kelley, General Topology, Graduate Texts in Mathematics, 27, Springer, New York,1975.

[KnR] H. Kneser, T. Radó, Aufgaben und Lösungen, Jahresber. Dtsch. Math.-Ver., 35 (1926),issue 1/4, 49, 123–124.

[KobN1] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. I, Wiley ClassicsLibrary, Wiley, New York, 1996.

[KobN2] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. II, Wiley ClassicsLibrary, Wiley, New York, 1996.

[Kod] L. K. Kodama, Boundary measures of analytic differentials and uniform approximationon a Riemann surface, Pac. J. Math. 15 (1965), 1261–1277.

[Koe1] P. Koebe, Über die Uniformisierung beliebiger analytischer Kurven. I, Nachr. Akad.Wiss. Göttingen (1907), 191–210 (see also Math. Ann. 67 (1909), no. 2, 145–224).

[Koe2] P. Koebe, Über die Uniformisierung beliebiger analytischer Kurven. II, Nachr. Akad.Wiss. Göttingen (1907), 633–669 (see also Math. Ann. 69 (1910), no. 1, 1–81).

[Koe3] P. Koebe, Über die Uniformisierung beliebiger analytischer Kurven. III, Nachr. Akad.Wiss. Göttingen (1908), 337–358 (see also Math. Ann. 72 (1912), no. 4, 437–516).

[Koe4] P. Koebe, Über die Uniformisierung beliebiger analytischer Kurven. IV, Nachr. Akad.Wiss. Göttingen (1909), 324–361 (see also Math. Ann. 75 (1914), no. 1, 42–129).

[Mal] B. Malgrange, Existence et approximation des solutions des équations aux dérivéespartielles et des équations de convolution, Ann. Inst. Fourier 6 (1956), 271–355.

[Mat] Y. Matsushima, Differentiable Manifolds, translated by E. T. Kobayashi, Marcel Dekker,New York, 1972.

[Me] S. N. Mergelyan, Uniform approximations of functions of a complex variable (in Rus-sian), Usp. Mat. Nauk 7 (1952), no. 2 (48), 31–122.

[Mi] J. W. Milnor, Topology from the Differentiable Viewpoint, based on notes by DavidW. Weaver, revised reprint of the 1965 original, Princeton Landmarks in Mathematics,Princeton University Press, Princeton, 1997.

[MKo] J. Morrow, K. Kodaira, Complex Manifolds, reprint of the 1971 edition with errata,Chelsea, Providence, 2006.

[Mu] J. R. Munkres, Topology: A First Course, Prentice-Hall, Englewood Cliffs, 1975.[NR] T. Napier, M. Ramachandran, Elementary construction of exhausting subsolutions of

elliptic operators, Enseign. Math. 50 (2004), 367–390.[Ns1] R. Narasimhan, Imbedding of open Riemann surfaces, Nachr. Akad. Gött. 7 (1960),

159–165.[Ns2] R. Narasimhan, Imbedding of holomorphically complete complex spaces, Am. J. Math.

82 (1960), 917–934.[Ns3] R. Narasimhan, Analysis on Real and Complex Manifolds, North-Holland, Amsterdam,

1968.[Ns4] R. Narasimhan, Compact Riemann Surfaces, Lectures in Mathematics ETH Zürich,

Birkhäuser, Basel, 1992.

References 547

[Ns5] R. Narasimhan, Complex Analysis in One Variable, second ed., Birkhäuser, Boston,2001.

[OT] W. Osgood, E. H. Taylor, Conformal transformations on the boundaries of their regionsof definitions, Trans. Am. Math. Soc. 14 (1913), no. 2, 277–298.

[P] H. Poincaré Sur l’uniformisation des fontions analytiques, Acta Math. 31 (1907), 1–64.[R] R. Remmert, From Riemann surfaces to complex spaces, in Matériaux pour l’histoire

des mathématiques au XXe siécle (Nice, 1996), 203–241, Séminaires et congrès, 3, So-ciété Mathématique de France, Paris, 1998.

[Ri] I. Richards, On the classification of noncompact surfaces, Trans. Am. Math. Soc. 106(1963), no. 2, 259–269.

[Rud1] W. Rudin, Real and Complex Analysis, third ed., McGraw-Hill, New York, 1987.[Rud2] W. Rudin, Functional Analysis, second ed., International Series in Pure and Applied

Mathematics, McGraw-Hill, New York, 1991.[Run] C. Runge, Zur Theorie der eindeutigen analytischen Funktionen, Acta Math. 6 (1885),

no. 1, 229–244.[Sim] R. R. Simha, The uniformisation theorem for planar Riemann surfaces, Arch. Math.

(Basel) 53, no. 6 (1989), 599–603.[Sk1] H. Skoda, Application des techniques L2 à la théorie dex idéaux d’une algèbre de fonc-

tions holomorphes avec poids, Ann. Sci. Ec. Norm. Super. 5, no. 4 (1972), 545–579.[Sk2] H. Skoda, Formulation hilbertienne du Nullstellensatz dans les algèbres de fonctions

holomorphes, in L’Analyse harmonique dans le domaine complexe, Lecture Notes inMathematics, 366, Springer, Berlin, 1973.

[Sk3] H. Skoda, Morphismes surjectifs et fibrés linéaires semi-positifs, in Séminaire P. Lelong-H. Skoda (Analyse), 1976–1977, Lecture Notes in Mathematics, 694, Springer, Berlin,1978.

[Sk4] H. Skoda, Morphismes surjectifs et fibrés vectoriels semi-positifs, Ann. Sci. Ec. Norm.Super. (4), 11, (1978), 577–611.

[Sk5] H. Skoda, Relèvement des sections globales dans les fibrés semi-positifs, in Séminaire P.Lelong-H. Skoda (Analyse), 1978–1979, Lecture Notes in Mathematics, 822, Springer,Berlin, 1980.

[Sp] G. Springer, Introduction to Riemann Surfaces, second ed., Chelsea, New York, 1981.[T] C. Thomassen, The Jordan–Schönflies theorem and the classification of surfaces, Am.

Math. Mon. 99 (1992), no. 2, 116–130.[V] D. Varolin, Riemann Surfaces by Way of Complex Analytic Geometry, Graduate Studies

in Mathematics, 125, American Mathematical Society, Providence, 2011.[Wa] F. Warner, Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in

Mathematics, 94, Springer, New York, 1983.[Wel] R. O. Wells, Differential Analysis on Complex Manifolds, Springer, Berlin, 1980.[Wey] H. Weyl, The Concept of a Riemann Surface, translated from the third German ed. by

Gerald R. MacLane, International Series in Mathematics, Addison-Wesley, Reading,1964.

[WZ] R. L. Wheeden, A. Zygmund, Measure and Integral: An Introduction to Real Analysis,Pure and Applied Mathematics, 43, Marcel Dekker, New York, 1977.

Notation Index

AAdistr, 394, 463α ∗ β , 478A∗, 394Aut(X), 194

BB1(M,A), 506, 520B1(M,A), 513B1

deR(M,F), 503B�

1 (M,A), 527B1

�(M,A), 527∂S, 415{·, ·}(E,h), 131[D], 121[ · ]Hr

deR(M,F), [ · ]deR, 503[ · ]Hq

Dol(X,E), [ · ]Dol, 126[·, ·], 278, 534[ · ]x0 , [ · ]π1(X,x0), 479B(x; r), 378

CC 0, 383, 416Ck , 382, 421cl(S), 378, 415cos z, 22cot z, 22Cq(M,A), 505, 527

Cq�(M,A), 527

csc z, 22C

∗, 6

DD, 421D(E), 104Dp,q , 45Dp,q(E), 111

Dr , 439Dr (E), 111d , 429, 433, 442∂ , 47∂ , 47, 125∂distr, 60, 139D, D′, D′′, 61, 136D′′

distr, 62, 139Deck(ϒ), 492degD, 124degE, 124�2, 526�ω , 82�(z0;R), 6�(z0; r,R), 6�∗(z0;R), 6div(), 123div(s), 119Div(X), 119

EE , 116, 421E(E), 104, 116E p,q , 45, 116E p,q (E), 111, 116E r , 116, 439E r (E), 111, 116exp(z), 21ez , 21

F, 177Fσ , 377

T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones,DOI 10.1007/978-0-8176-4693-6, © Springer Science+Business Media, LLC 2011

549

550 Notation Index

Gγ , 434γ −, 478 (U, E(E)), 104 (U, F ), 117 (U, M(E)), 104 (U, O(E)), 104Gδ , 377genus(X), 165GL(n,F), 219

HH, 198H1(M,A), 506, 520H 1(M,A), 516H : α ∼ β , 477H 1

�(M,A), 527H�

1 (M,A), 527Hr

deR(M), 503H 1

Dol,L2 ,H1Dol,L2∩E , 140

Hq

Dol(X),Hq

Dol(X,E), 126Hom(·, ·), 407, 518hq , 165h∗, 130h ⊗ h′, 130hX , 77h∗

X , 81

IιES

, 168

JJac(X), 307JF , 383, 450Jθ,p , 307

KKX , 40κs , 292

Lλω , 457�p,qT ∗X, 45�p,qT ∗X ⊗ E, 110�rT ∗M,�r(T ∗M)C, 437�r V , 409�θ , 307log z, 21Lp , Lp(X,μ), 379L

p

loc, 379〈·, ·〉L2

p,q (ϕ), 〈·, ·〉L2p,q (ω,ϕ), 〈·, ·〉L2

1(ϕ), 54

〈·, ·〉L2p,q (E,h), 〈·, ·〉L2

p,q (E,ω,h), 〈·, ·〉L21(E,h), 132

‖ · ‖L2p,q (ϕ), ‖ · ‖L2

p,q (ω,ϕ), ‖ · ‖L21(ϕ), 54

‖ · ‖L2p,q (E,h), ‖ · ‖L2

p,q (E,ω,h), ‖ · ‖L21(E,h), 132

L2p,q(ϕ),L2

p,q (ω,ϕ),L21(ϕ), 55

L2p,q(E,h), L2

p,q (E,ω,h), L21(E,h), 133

MM, 33, 115M(E), 104, 115multp�, 33

Nn(γ ; z0), 201

OO, 4, 30, 115OD(E), 122O(E), 104, 115�(E), 111, 115�X , 46, 115ordp f , 14, 34ordp s, 104

PP

1, 28P 1(M,A), 512⊥, 397�∗, 429, 433, 438, 509, 511, 522, 524�∗, 126, 429, 433, 480, 509, 510, 522, 524π1(X,x0), 479�T M,�(T M)C , 433�T ∗M,�(T ∗M)C , 433PSL(n,F), 219

QQD(E), 122

RrankG, 525RP

2, 529

SS, 378, 415sE(·, ·),SE(·, ·), 167, 168sec z, 22#, 177σ (ν), 527◦S, 378, 415sin z, 22SL(n,F), 219∗, ∗, 178

Notation Index 551

∗, ∗#, 179�, 416

TtA, 393tan z, 22�h, 136�ω , 73�ϕ , 64⊗, 409, 412T M, (T M)C, 429, 432T ∗M,(T ∗M)C, 429, 433Tθ , 307(T X)p,q , 39(T ∗X)p,q , 39

VVC, 409V ∗, 401, 407

W∧, 409Wk , 532

ZZ1(M,A), 505Z1(M,A), 513Z1

deR(M,F), 503Z1

�(M,A), 527zζ , 22

Subject Index

AAbel–Jacobi embedding theorem, 192, 307Abel’s theorem, 303Accumulation point, 415Almost complex

structure, 311integrable, 311

surface, 311Almost everywhere, 447Ample, 293

very, 293Approximation of the identity, 391Argument

function, 21principle, 52, 202, 203, 227, 335, 337, 338

Atlas, 419Ck , 420holomorphic, 26holomorphically equivalent, 26line bundle, 102

holomorphically equivalent, 102Automorphism, 31, 194

groupof �, 221of C, 216of H, 222of P

1, 218–220

BBanach space, 378Barrier, 344Basis for a topology, 416Behnke–Stein theorem, 89Bessel’s inequality, 404

Biholomorphic, 18, 31Biholomorphism, 18, 31

local, 18, 31Bilinear function, 409, 412

skew-symmetric, 409symmetric, 409

Bishop–Kodama localization theorem, 98Bishop–Narasimhan–Remmert embedding

theorem, 191, 279Bolzano–Weierstrass property, 423Borel set, 377Boundary

of a set, 415operator ∂ , 505, 527

CCanonical homology basis, 269Canonical line bundle, 40, 105

degree of, 169Cauchy integral formula, 6, 201, 202, 334Cauchy–Pompeiu integral formula, 6Cauchy–Riemann equation

homogeneous, 4inhomogeneous, 7, 25L2 solution

for line-bundle-valued forms of type(0,0), 145

for line-bundle-valued forms of type(1,0), 140

for scalar-valued forms of type (0,0),74

for scalar-valued forms of type (1,0),65

local solution, 7Cauchy’s theorem, 6, 201, 202, 334

T. Napier, M. Ramachandran, An Introduction to Riemann Surfaces, Cornerstones,DOI 10.1007/978-0-8176-4693-6, © Springer Science+Business Media, LLC 2011

553

554 Subject Index

Cech1-form, 513

exact, 513line integral of, 513

cohomology, 512, 526Chain rule, 384, 430Change of variables formula, 386Characteristic function, 376Chordal Kähler form, 73, 76, 137

C∞ approximation, 390, 535Ck

atlas, 420differential form, 439

line-bundle-valued, 111function, 382, 421map, 383, 421section, 103vector field, 435

Closeddifferential form

d , 443∂ , 47∂ , 47, 125

set, 378, 415sequentially, 423

Closure, 378, 415Coboundary operator δ, 527Cocycle relation, 90, 150, 154, 518Coefficients

of a differential form, 45, 439of a vector field, 43, 435

Cohomology, 516Cech, 512, 526de Rham, 503Dolbeault, 126Dolbeault L2, 140singular, 504, 527

Commutator, 534subgroup, 278

Compact, 416locally, 418relatively, 416

Complete orthonormalbasis, 403set, 403

Complex differentiability, 16, 17Complex manifold

1-dimensional, 26n-dimensional, 35

Complex torus, 28, 197

Complexification, 409Connected, 416

component, 416locally, 416

Connecting homomorphism, 127Connection

canonicalfor scalar-valued forms, 61in a line bundle, 135, 136

Continuous, 415differential form

line-bundle-valued, 111section, 103sequentially, 423

Continuous closed 1-form, 513Convolution, 391Coordinate transformation, 419

Ck , 420holomorphic, 26

Coordinateshomogeneous, 290local Ck , 420local holomorphic, 26

Cotangentbundle, 433

(0,1), 39(1,0), 39complexified, 433holomorphic, 39projections, 433real, 433

map, 430, 433space, 429

(0,1), 39(1,0), 39holomorphic, 39

Coveringbranched, 294–298map, 483space, 483

C∞, 483C∞ equivalent, 487equivalent, 487holomorphic, 192holomorphically equivalent, 192universal, 489–491

Criticalpoint, 435value, 435

Curvature, 64, 136form, 64, 136negative, 64, 137nonnegative, 64, 137nonpositive, 64, 137

Subject Index 555

Curvature (cont.)of a Kähler form, 73positive, 64, 137, 147, 160, 162zero, 64, 137

Cycle, 505

D∂-Cauchy integral formula, 6De Rham

cohomology, 503pairing, 509, 522theorem, 524, 525

Deck transformation, 492Defining

function, 119local, 120

section, 119Degree

of a divisor, 124of a line bundle, 124

Diffeomorphism, 383, 421Differential, 383, 429, 433Differential form, 439

line-bundle-valued, 111negative, 452negative part, 453on a Riemann surface, 45–52positive, 452positive part, 453sequence

infimum, 453limit inferior, 453limit superior, 453supremum, 453

Differential operator, 392, 462Dirichlet problem, 338–350Discrete, 415Disjoint union, 28, 417, 421Distributional

∂ , 60, 139D′′, 62, 139differential operator, 394, 463solution, 394, 463

Divisor, 119–124defining section associated to, 121effective, 120line bundle associated to, 121linearly equivalent, 120of a line bundle map, 123of a section, 119principal, 119solution of, 119, 152, 303weak solution of, 152, 303

Dolbeaultcohomology, 126

class, 126exact sequence, 128L2 cohomology, 140lemma, 50

Dominated convergence theorem, 378, 457Dual

basis, 407line bundle, 107space

algebraic, 407norm, 401

EEdge of a singular 2-simplex, 527Embedding

Abel–Jacobi, 307C∞, 467holomorphic

into Cn, 279–290

into Pn, 291

Entire function, 4Evenly covered, 483Exact differential form

d , 443Ck , 48, 125, 443∂ , 48∂ , 48, 125locally, 48, 125, 443

Exact sequenceDolbeault, 128of sheaves, 119

Exhaustionby sets, 427function, 77, 427

strictly subharmonic, 82Exponential function, 21, 22

real, 388Exterior derivative, 442Exterior product, 409

FFatou’s lemma, 378, 457Finite type, 277

topological, 277Finiteness theorem, 164Flat operator, 177Formal

adjoint, 394transpose, 393

Fourier coefficients, 404Friedrichs lemma, 390

strong version, 535

556 Subject Index

Fubini’s theorem, 380Fundamental estimate

for scalar-valued forms, 64in a line bundle, 137on an almost complex surface, 328

Fundamental group, 479Fundamental theorem of algebra, 35

GGap, 174

sequence, 174generic, 175hyperelliptic, 175

General linear group, 219Genus, 165, 169Germ, 115, 117, 428Goursat’s theorem, 17Gram–Schmidt orthonormalization process,

403Group action

free, 493properly discontinuous, 493quotient by, 493

HHahn–Banach theorem, 402Harmonic, 64, 339

1-form, 186Hartogs–Rosenthal theorem, 98Hausdorff, 416Hilbert space, 399Hodge

conjugate star, 178conjugate star flat operator, 179conjugate star sharp operator, 179decomposition

for ∂ , 181for scalar-valued forms, 185

star, 178Holomorphic

atlas, 26attachment, 35–38

of caps, 207–209of tubes, 36, 241–263

functionin C, 4on a Riemann surface, 30

map, 30into C

n, 279into P

n, 291into a complex torus, 306local representation, 32

1-form, 46line-bundle-valued, 111

removal of tubes, 243–248section, 103vector field, 43

Holomorphically compatiblelocal charts, 26local trivializations, 102

Homeomorphism, 416Homology, 506, 519

canonical basis for, 267–273singular, 278, 507, 511, 520, 527

Homotopy, 477Hurwitz’s theorem, 301Hyperelliptic

gap sequence, 175involution, 300point, 175Riemann surface, 174, 299

Hyperplane bundle, 105curvature, 137Hermitian metric in, 129

IIdentity theorem, 13, 31Immersion

C∞, 467holomorphic

into Cn, 279

into Pn, 291

into a complex torus, 306Inner product

Hermitian, 397real, 397

Integral, 376differentiation past, 384of a differential form, 454

Interior, 378, 415Inverse function theorem

C∞, 384, 465holomorphic, 18, 43

JJacobi variety, 307Jacobian determinant, 383, 450Jordan curve, 228

theorem, 333

KKähler

form, 51metric, 130

Kodaira embedding theorem, 191, 294Koebe uniformization, 212

Subject Index 557

LLattice, 28, 306

period, 307Laurent series, 15Law of cosines, 398Lifting, 483

holomorphic, 193theorem, 484

Limit point, 415Line bundle

associated to a divisor, 121dual, 107Hermitian, 128holomorphic, 103homomorphism, 107isomorphism, 107map, 107set-theoretic, 101tensor product, 107

Line integral, 461along a 1-chain, 506, 520along a continuous path, 499

Linear functional, 407bounded, 401

Liouville’s theorem, 35Lipschitz, 384Local chart, 419

Ck , 420holomorphic, 26

Logarithmic function, 21, 22real, 388

Loop, 417Lp

norm, 379space, 379

Lp

locdifferential form, 458

line-bundle-valued, 111function, 379section, 103

L2

inner productof functions, 399of line-bundle-valued forms, 132of scalar-valued forms, 54

normof a function, 399of a line-bundle-valued form, 132of a scalar-valued form, 54

spaceof functions, 399of line-bundle-valued forms, 132, 133of scalar-valued forms, 55

MManifold, 419

Ck , 420complex

of dimension 1, 26product, 420real analytic, 420smooth, 420

Maximum principle, 14, 31for harmonic functions, 339for subharmonic functions, 339

Mean value property, 11, 341Measurable

differential form, 448line-bundle-valued, 111

function, 375, 447map, 375, 447section, 103set, 375, 447vector field, 448

Measurecomplete, 375completion of, 376counting, 376for a nonnegative form, 457Lebesgue, 377

outer, 377positive, 375space, 375

Mergelyan–Bishop theorem, 97Meromorphic

function, 331-form, 46

existence of, 67line-bundle-valued, 111

section, 103Metric

Hermitian, 128dual, 130tensor product, 130

Kähler, 130Riemannian, 475

Mittag-Leffler theorem, 88, 97, 160for a line bundle, 148, 160

Möbiusband, 449transformation, 219

Mollification, 391Mollifier, 391Monotone convergence theorem, 378, 457Montel’s theorem, 10, 72Multiplicity, 33

558 Subject Index

NNeighborhood, 415Net, 423Nongap, 174Nonorientable, 450Nonseparating, 239Norm, 378

complete, 378for an inner product, 397

OOne-point compactification, 418Open

mapping theorem, 14, 31Riemann surface, 27set, 378, 415

Order, 14, 34, 46, 104of a pole, 34, 46of a zero, 14, 34, 46

Orientable, 450double cover, 371, 462topological surface, 364

Orientation, 450compatible, 450equivalent, 450in a vector space, 411induced on a boundary, 459preserving, 451

Oriented, 450positively, 450

Orthogonal, 397decomposition, 399projection, 399

Orthonormal, 397Osgood–Taylor–Carathéodory theorem, 206

PParallelogram law, 398Partition of unity, 427Path, 417

Ck , 422connected, 418

locally, 418piecewise Ck , 422

Path homotopy, 477Perron method, 338Planar, 211Poincaré

duality, 187lemma, 443

Poissonformula, 343kernel, 343

Polar coordinates, 387

Pole, 33of a meromorphic 1-form, 46of a meromorphic section, 103simple, 34, 46, 104

Potential, 443local, 443

Power function, 22Power series, 12, 13Primitive, 4Principal part, 202Product

path, 478rule, 430topology, 417

Projection map, 102Projective space, 290Projectivized special linear group, 219Proper map, 427Pullback, 429, 433, 438, 509, 511, 522, 524Pushforward, 429, 433, 480, 509, 510, 522,

524Pythagorean theorem, 398

Qq-boundary, 527q-chain, 505, 526q-coboundary, 527q-cochain, 527q-cocycle, 527Quotient

map, 417space, 417topology, 417

RRank, 525Real projective plane, 529Realification, 408Regular value, 435Regularity

first-order, 532for ∂ , 63, 324for ∂/∂z, 10

Regularization, 391Representation of a section, 103Residue, 53

theorem, 53, 201, 203, 227, 335, 338Reverse path, 478Riemann mapping theorem, 191

in the plane, 204Riemann sphere, 28Riemann surface, 26

open, 27Riemann–Hurwitz formula, 299Riemann–Roch formula, 165, 169

Subject Index 559

Riemann’s extension theorem, 11, 31Rouché’s theorem, 52, 202, 203, 227, 335, 338Runge

approximation theorem, 91for a line bundle, 151

approximation theorem with poles, 92, 97for a line bundle, 151

open setholomorphically, 96topologically, 77

SSard’s theorem, 263, 276, 331, 449Schönflies’ theorem, 333

proof of, 350–356Schwarz inequality, 398Second countable, 416Section, 103

Ck , 103continuous, 103defining

associated to a divisor, 121holomorphic, 103L

p

loc, 103measurable, 103meromorphic, 103of a sheaf, 117

Separable, 403Separating, 239Sequence

Cauchy, 378convergent, 378, 423divergent, 378, 423limit of, 378, 423

Serreduality, 169mapping, 168pairing, 168

Sharp operator, 177Sheaf, 115–119

constant, 117definition, 117isomorphism, 117mapping, 117of holomorphic 1-forms, 115of holomorphic sections, 115of meromorphic 1-forms, 115of meromorphic sections, 115skyline, 122skyscraper, 116

Sheet interchange, 300σ -algebra, 375Simple function, 376Simply connected, 480

Singular 2-simplex, 526Sobolev lemma, 541Sobolev space, 532Special linear group, 219Stalk, 115, 117, 428Standard 2-simplex, 526Stereographic projection, 28Stokes’ theorem, 460Stratification, 286Strictly subharmonic, 64Strictly superharmonic, 64Strong deformation retraction, 363Structure

Ck , 420complex analytic, 26holomorphic, 26

Subharmonic, 64, 339Submanifold, 422Submersion, 467Superharmonic, 64Support, 426Surface, 419

Ck , 420Riemann, 26smooth, 420

TTangent

bundle, 432(0,1), 39(1,0), 39complexified, 432holomorphic, 39projections, 433real, 432

map, 429, 433space, 429

(0,1), 39(1,0), 39complexified, 429holomorphic, 39real, 429

vector, 429(0,1), 39(1,0), 39complex, 429holomorphic, 39real, 429to a path, 434

Tautological bundle, 108Taylor’s formula, 385Tensor product, 409, 412

line bundle, 107Tietze extension theorem, 348

560 Subject Index

Topological hull, 77extended, 81

Topological space, 415Topology, 415

product, 417subspace, 415

Torsion, 226, 332, 371, 521free, 226, 332, 371, 521

Toruscomplex, 28, 306real, 30

Total space, 102Triangulation, 282, 286, 370Trigonometric functions, 22

real, 387Trivialization

global, 102local, 102

Type (p, q), 43, 110

UUpper half-plane, 198

VVanishing derivatives, 444Vanishing theorem, 145, 148, 163

Vector field, 435Volume form, 453

WWeak

compactness, 403convergence, 403solution, 394, 463

Wedge product, 409Weierstrass

gap, 174gap theorem, 174nongap, 174point, 176theorem, 151weight, 175

Weight function, 54Winding number, 201, 226Wronskian, 175

ZZero

of a holomorphic function, 14, 33of a meromorphic 1-form, 46of a meromorphic function, 34of a meromorphic section, 103simple, 14, 33, 34, 46, 104