Geometric aspects of the potential theory on

Riemannian manifolds

Stefano PigolaUniversità dell�Insubria, Como

December 6, 2013

A. Potential theory and the Riemann mapping theorem

The potential theory of a given space (Euclidean, Riemannian, metric etc.) canbe thought of as the study of the family of harmonic functions on a space. Ifon the one hand, this study is in�uenced by the geometry of the underlyingspace, on the other hand it gives information on its geometry.

Geometry Analysis� volume growth� curvature bounds� structure at in�nity� topology� etc.

!

� (half)bnded (sub)harm functs are const� Harnack inequalities� heat/Green kernel properties� etc.

Perhaps the origin and the systematic use of potential theory both in complexdomains and on curved spaces can be traced back to the study of the conformaltype of these objects.

Prototype of this investigation is the Riemann mapping theorem, one of thedeepest results in the geometric function theory of planar domains.

Th. 1 Let � R2 be a simply connected domain, 6= R2, z0 2 . Then,there exists a direct conformal di¤eomorphism f : ! D1 s.t. f (z0) = 0.

Def. 2 A smooth function f : 1 ! 2 is called conformal if

Jac (f) (x) = � (x)O (x) ;

where � : 1 ! R>0 and O : 1 ! O (2) are smooth functions. Equivalently

hdxf (v) ; dxf (w)i = �2 (x) hv; wi :

In case O (x) 2 SO (2), or equivalently, det Jac (f) > 0, we say that theconformal map is direct (orientation preserving).

The Riemann mapping theorem gives a complete classi�cation, up to conformalequivalences, of the simply connected planar domains: namely, they are eitherD1 or R2. The corresponding classes are distinct.

Th. 3 Plane R2 and disc D1 are not conformally di¤eomorphic.

A natural way to distinguish the conformal type of these domains is to look atthe family of their harmonic functions, i.e., their potential theory.

Def. 4 A (su¢ ciently smooth) function u : ! R is said to be:

(�) harmonic if �u = 0 where � is the Euclidean Laplacian

� =2Xi=1

@2

@x2i= divr

(�) subharmonic if �u � 0.

(�) superharmonic if �u � 0.

Lemma 5 The harmonicity of functions on planar domains is preserved byconformal maps. Actually, given f = (f1; f2) : � R2 ! ~ � R2 conformaland u : ~! R then sgn (�u) = sgn (� (u � f)).

Proof. Use the composition law for the Laplacian

�(u � f) =2Xi=1

Hess (u) (df (ei) ; df (ei)) + hru;�fi ;

where �f = (�f1;�f2), ru = ( @u@x1; @u@x2

), fe1; e2g o.n. basis of R2 and

Hess (u) = [ @2u@xi@xj

] so that tr (Hess (u)) = �u.

Now, f conformal ) �f = 0 and df (ei) = �O � ei. Since fO � e1; O � e2gis again an o.n. basis, from the above we conclude

�(u � f) (x) = �2 (x) �ujf(x) :

Therefore, to prove that R2 and D1 are distinct we can simply look at their(sub-, super-)harmonic functions.

Claim 6 On the unit disc D1 we can construct as many bounded harmonicfunctions as we want.

Proof. Solve the Dirichlet problem(�u = 0 on D1u = f on @D1

where f 2 C0�D1�.

In sharp contrast we have:

Claim 7 The plane R2 has no non-constant subharmonic functions boundedfrom above. Namely(

�u � 0, on R2supR2 u = u

� < +1 ) u � u�:

Proof (Chen, Rigoli-Setti). Without loss of generality (take u� u�) assumeu� = 0: Apply the divergence theorem to X = euru:Z

DRdivX =

ZDReu jruj2 +

ZDReu�u �

ZDReu jruj2

andZDRdivX =

Z@DR

eu hru; �i �Z@DR

eu jruj �qj@DRj�

sZ@DR

eu jruj2:

Therefore ZDReu jruj2 �

qj@DRj

sZ@DR

eu jruj2

Let

F (R) =ZDReu jruj2 :

We show F (R) � 0: Integration in polar coordinates and Fubini Th. gives

F 0 (R) =Z@DR

eu jruj2 :

The above reads

F (R) �qj@DRj

qF 0 (R);

i.e.F 0 (R)

F (R)2� 1

j@DRj:

Integrating on [R0; R] gives

1

F (R0)�Z RR0

dr

j@Drj' log R

R0! +1:

It follows that F (R0) = 0 for every R0, hence, u � const:

Rmk 8 The previous proof relies on the fact that � is in divergence-form andZ +1 dr

j@Drj= +1

so it does not work on Rn for n � 3. In fact Rn�3 has again no non-constant(semi-)bounded harmonic functions (Harnack) but it does have non-constant,bounded above subharmonic functions. For instance, take

u (x) = �CnZRn

f(y)

jx�yjn�2dy;

with 0 � f 2 C1c (Rn) : Obviously, u � 0 and it can be veri�ed that�u = f;hence �u � 0.

B. Potential theory on Riemannian manifolds

So far, we considered both R2 and D1 endowed with the Euclidean metric.However, from the geometric viewpoint, it is more natural to endow D1 withits Poincaré hyperbolic metric

h; iH2 =4�

1� jxj2�2 h; i :

This makes D1 a very di¤erent geometric space. Think for instance of the�straight lines� in the Euclidean and hyperbolic spaces. However, since weare in dimension 2, the previous discussion is not a¤ected by this �conformalchange� of the metric.

� By a Riemannian manifold we mean an m-dimensional di¤erentiable man-ifold M endowed with a family h; ip of scalar products on the tangentspaces TpM that vary smoothly with the point p 2M .

� The Riemannian metric h; i on M induces an intrinsic distance

d (x; y) = inf ` ( ) ;

where is a piecewise C1 path from x to y and ` denotes its length.

� Say that (M; h; i) is complete () the metric space (M;d) is complete.

� Completeness() divergent paths (those leaving any �xed cmpt set) have` = +1. This means: walking with constant speed along a path you willnever exit M .

� The metric balls and spheres of (M; h; i) are de�ned by

BR (p) = fd (x; p) < Rg@BR (p) = fd (x; p) = Rg :

� If we allow @M 6= ; then M will be oriented (transition functs havedet Jac('�'

�1� ) > 0). In particular, there exists

� = outward pointing unit normal to @M:

� The intrinsic distance d is de�ned in the same way.

� M with @M 6= ; is complete i¤ (M;d) is complete.

� intM =Mn@M . For a domain D �M , the interior boundary of D is:

@ID = @D \ intM:and the Neumann part of D is:

@ND = D \ @M:

� Smooth domain D if @D is C1 hypersurface � with @� = @D \ @M:

On a Riemannian manifold (M; h; i) we have a natural second order, elliptic,linear operator called the Laplace-Beltrami operator.

� In local coordinates�x1; :::; xm

�around a given point p 2M , the Laplace-

Beltrami operator expresses as

�u =1pdet g

mXi;j=1

@

@xi

�qdet g gij

@u

@xj

�

=mX

i;j=1

gij@2u

@xi@xj+ �rst order terms,

where

gij =�@

@xi;@

@xj

�, and [gij] = [gij]

�1

Clearly, if h; i is the Euclidean metric on R2 then gij = �ij and we recover theexpression of the Euclidean Laplacian.

� On (M; h; i), the gradient of a function and the divergence of a vector�eld X = �mi=1X

i @@xi

are de�ned by

ru =mX

i;j=1

gij@u

@xj@

@xi

divX =1pdet g

mXi=1

@

@xi

�qdet gXi

�:

In particular, the usual formula

�u = divru

holds.

� The notions of harmonicity, subharmonicity and superharmonicity can betransplanted from planar domains to general Riemannian manifolds up tousing the Laplace-Beltrami operator.

Now, for dimM = 2; if we perform a conformal change of the metric:

gh; i = �2 (x) h; ithen from the above local expressions we get

f�u (x) = 1

�2 (x)�u (x)

proving that (sub- super-)harmonicity of a function is a conformal invariant.This applies in particular to D1 and it is on the base of the classi�cation ofgeneral Riemann surfaces, i.e., 2-dimensional oriented Riemannian manifoldsendowed with a conformal class of metrics.

We have seen: R2 with its �at metric h; i does not support any non-constant,bounded above subharmonic function. This property has a precise name.

Def. 9 Say that the Riemannian manifold (M; h; i), @M = ;, is parabolic if(�u � 0, on MsupM u = u� < +1 ) u � u�:

A non-parabolic manifold is called hyperbolic. If @M 6= 0 we add the Neumanncondition

@�u � 0, on @M:

This de�nition resembles the usual maximum principle/boundary point Lemmain compact spaces. From this viewpoint, parabolicity is a kind of compactness.We shall come back to this remark.

Similarly to planar domains, applying the divergence theorem exactly as we didin the Euclidean plane we have

Th. 10 Let (M; h; i) be a complete(!) Riemannian manifold with moderatearea growth: Z +1 dr���@IBr��� = +1:Then M is parabolic.

What is���@IBR���? De�ne the Riemannian measure on M

dv =qdet g � dx:

It induces an (m� 1)-dimensional Hausdor¤ measure dHm�1. Then

jBRj =ZBRdv,

���@IBR��� = Z@IBR

dHm�1 coarea=d

dRjBR \ intM j :

Rmk 11 The following chain of implications hold

jBRj . R2 )R

jBRj=2 L1 (+1)) 1���@IBR��� =2 L1 (+1) :

Rmk 12 Suppose @N = ; and M � N is smooth and open. Then, N canhave large volume growth whereas M is thin, hence with a moderate volumegrowth as a manifold with boundary.

It is already visible in the def. that parabolic manifolds share some importantproperties of compact manifolds. Using this viewpoint one has a natural wayto extend to non-compact settings geometric results that are known in thecompact realm.

C. Parabolicity vs Compactness I: maximum principle

For a compact smooth domain in a Riemannian manifold (M; h; i) the usualweak maximum principle states that

�u � 0, in ) supu = max

@u:

When @M 6= ;, we add the Neumann condition

@�u � 0, on @N

and, by the �boundary point Lemma�we conclude

supu = max

@Iu:

If is not compact, in general, this principle fails to hold.

Ex. 13 Take M = R2, = f(x; y) : y > 0g, u (x; y) = y. Then �u = 0,

sup@

u = 0 < supu = +1;

therefore some boundedness assumption has to be required.

Ex. 14 TakeM = H2 realized as the Poincaré disc D1 and let = D1nD1=2.Solve the Dirichlet problem8><>:

�R2u = 0 in u = 0 on @D1=2u = 1 on @D1 (the 1 in H2)

Then 0 � u (x) = log (2 jxj) = log 2 < 1. By conformal invariance �u = 0 in �M but

sup@M

u = 0 < 1 = supu:

Note that M is hyperbolic.

Th. 15 (Ahlfors maximum principle) M is parabolic , 8 � M with@I 6= ; it holds 8><>:

�u � 0; on @�u � 0, on @Nsup u < +1

) supu = sup

@Iu:

Moreover, ifM parabolic and =M , then the Neumann condition is avoided.

Proof (case @M = ;). Assume Ahlfors max. princ. Let �u � 0 andsupM u = u� < +1. Suppose u non-constant. Take " = fu > u� � "g 6=; for 0 < " << 1. Since u � u� � " on @", by Ahlfors u � u� � " on ",contradiction.

Assume M parabolic. Take � M and �u � 0 on , sup u < +1: Bycontradiction, sup u > sup@ u + ". De�ne v = max fu; sup u� "=2gon and v = sup u � "=2 on Mn. Then �v � 0, supM v < +1 andv 6� const: Contradiction.

Ahlfors maximum principle: as simple as deep. Deepness is measured by itsconsequences both in Geometry and in Analysis/Stochastic process.

� Geometry. We shall see an example in a moment. A-priori height estinatesfor CMC graphs over non-compact domains are obtained easily.

� Analysis/Stochastic process. Connection with other notions of parabol-icity, namely, Dirichlet parabolicty. This is systematically used by minimalsurface guys. In particular

volume growth) Neumann parab) Dirichlet parab.

The link between volume and Dirichlet parabolicity was never observedbefore. Back to Geometry. We get new deterministic and simple proofs ofresults proved by probabilistic methods. Example: Neel Theorem. Everyminimal graph over a domain of R2 is Dirichlet prarabolic.

C.1. Geometric application: height estimates for CMC graphs

Take � ,! R3 a surface with the induced metric h; i = h; iR3���T�.

Suppose � is oriented by a unit normal map N : �! S2. Then, 8v; w 2 Tp�

�p (v; w) = h�dxN (v); wi

is a symmetric bilinear form: the 2nd fundamental form of �.

Its eigenvalues k1 (p) ; k2 (p) are the principal curvatures.

The mean curvature of � with respect to N is

H (p) =1

2tr��p =

1

2(k1 (p) + k2 (p)) :

Special case: � = GraphD (u). In local coordinates (x1; x2) induced bythe parametrization ' (x) = (x; u (x)) : D � R2 ! R3, if we choose thedownward pointing Gauss map

N =(ru;�1)q1 + jruj2

it holds

formulas local coords � = GraphD (u)j

gij =D@i'; @j'

Ej gij = �ij + @iu@ju

�ij =D@2ij';N

Ej �ij =

@ijuq1+jruj2

H = 12gij�ij j H = �12 div

0@ ruq1+jruj2

1Athe latter non-linear, div-form operator is called the mean-curvature operator.

This formulas extend in an obvious way to graphs over a general manifold(M; h; i). Take M as the domain for a graphical surface in M � R:

� = GraphM (u) = f(x; u (x)) : x 2Mg :

� is oriented by the downward pointing normal �eld

N =(ru;�1)q1+jruj2

:

The 2nd fundamental form of � in M � R is

� (v; w) = � 1q1+jruj2

Hess (u) (v; w)

and its mean curvature (with respect to N ) is

mH = � div

0@ ruq1+jruj2

1A :

Suppose now that � is a surface in R3 with constant mean curvature H (x) �H with respect to a (downward) Gauss map N and with boundary on thehorizontal coordinate plane

@� �nx3 = 0

o:

De�ne the height function h : �! R from the coordinate plane by

h (p) = hp; e3i :

If � = GraphD (u) then we have the coordinate expression

h (x) = u (x) :

Claim. There is a relation between the maximal height of � and the constantmean curvature H.

� Let � = S2R \nx3 � 0

o, the upper half-sphere of radius R > 0. Then:

(i) � = GraphD2R

�qR2 � jxj2

�, (ii) H � 1=R and

max� h = R =1H .

� Let � = C2R\nx3 � 0

o, the upper half-cylinder of radius R > 0. Then:

(i) � = Graph[�R;R]�R�pR2 � t2

�, (ii) H � 1=2R and

max� h = R =12H .

� Let � = Or1;r2 \nx3 � 0

o, the upper half-unduloid with rays 0 < r1 �

r2. Or1;r2 is a CMC surface of revolution that �interpolates� betweenthe sphere and the cylinder. Then: (i) � = GraphD (u); (ii) H �1= (r1 + r2) and

max� h = r2 <1H .

Height estimates = a-priori estimates of the maximum & minimum height of� in terms of H (and of the geometry of the surface if needed).

Th. 16 (E. Heinz, J. Serrin, A. Ros & H. Rosenberg) Let

� = GraphD (u)

be a graph with constant mean curvature H > 0 with respect to the downwardGauss map. Assume that @� �

nx3 = 0

o. Then

0 � h (p) � 1

H

Rmk 17 Recall that R2, hece every domain of R2, has moderate volumegrowth!

� E. Heinz, J. Serrin (�70). � � R2�R compact surface. Proof: non-lineararguments based on the analysis of the mean curvature operator on R2.

� H. Rosenberg (�90). � � R2 � R is compact. Proof: linear argumentsbased on the maximum principle for the intrinsic Laplacian on the surface�.

� J. Spruck (2007). � � M � R is compact but M is a general manifold.Proof: interior gradient estimates for the mean curvature operator on M .The a-priori estimate is qualitative rather than qunatitative.

� A. Ros & H. Rosenberg (2010). � � R2 � R may be non-compact.Proof: convergence theory of CMC surfaces + the homogeneity of thedomain space R2.

� D. Impera & S.P. & A. G. Setti (2013). � �M�R may be non-compactprovided M has moderate volume growth, RicM � 0, and � � M �[�T; T ]. Proof: potential theory.

Proof. Quite simple. Starting point:

� (Elbert-Nelli-Rosenberg 2007) If � = GraphDu has constant mean cur-vature on D � Rm, m � 5, then sup juj < +1:

� (Li-Wang 2001, Impera-P.-Setti 2013). If � = GraphMu satis�es

supMjH (x)u (x)j < +1;

then, the volume of �-balls is controlled by that of M -balls���B�R��� � CjBMR j;R > 1.

Now (Rosenberg strategy �90) let

�(x) = \N (x) e3 ) cos� = � 1q1 + jruj2

< 0:

A (geometric) computation shows that

��u = 2H cos� mean curvature eq

and

�� cos�+k�k2 cos�=0 stability eq.

Let

� (x) = H � u (x) + cos� (x) , on �:

Then

��� =�2H2 � k�k2

�cos� � 0, on �:

Moreover, Erbert-Nelli-Rosenberg ) supD juj < +1. Therefore

sup�� < +1:

Since @� �nx3 = 0

othen

� � cos� � 0, on @�:By volume growth estimates, � is parabolic. Therefore

H sup�u� 1 � sup

��Ahlfors= sup

@�� � 0:

This givessup�u � 1

H:

Similarly

��u = 2H cos� � 0, on � & u = 0, on @� Ahlfor) u � 0 on �:This completes the proof.

D. Parabolicity vs Compactness II: divergence theorem

Let (M; h; i) be compact and X a smooth vector �eld. Then the usual diver-gence theorem states Z

MdivX =

Z@MhX; �i :

Parabolic manifolds share this global property. To see this we need the followingcharacterization which involves the concept of capacity.

Th. 18 M is parabolic() 8K ��M ,

cap (K) = infZMjr'j2 dv = 0;

the in�mum being taken with respect to all ' 2 C1c such that ' � 1 on K.

Interpretation: every K �� M has a small mass from the viewpoint ofharmonic functions.

The next result is known as the Kelvin-Nevanlinna-Royden criterion (KNR forshort). If @M = ;, It is due to T. Lyons and D. Sullivan but the most generalcondition was obtained by D. Valtorta and G. Veronelli. This version is fromImpera-P.-Setti.

Th. 19 M is parabolic() 8X 2 L2 (M) vector �eld s.t. (divX)� 2L1 (M) and hX; �i 2 L1 (@M) it holdsZ

MdivX =

Z@MhX; �i :

In particular, if @M = ;, ZMdivX = 0:

Interpretation: from the viewpoint of X, the �boundary at in�nity� of M isnegligible (or X has zero �boundary values�). Therefore, a global version ofStokes theorem holds.

Proof (of ) and @M = ;). Let j ��M be s.t. j %M . Since

cap (1) = 0;

we can choose 0 � 'j 2 C1c (j) s.t.

'j = 1 on 1, and r'j L2 ! 0:

Apply Stokes theorem

0 =ZMdiv

�X'j

�=ZM'j divX +

ZM

DX;r'j

E:

To conclude, note that����ZM

DX;r'j

E���� � kXkL2 kr'kL2 ! 0

and ZM'j divX !

ZMdivX:

D.2. Geometric Applications: slice-type theorems

The phenomenon we are going to describe was never observed before.

Th. 20 (slice-type th) Let M be complete and vol (M) < +1. Assume� = GraphM (u) has mean curvature H � 0. Then � is minimal, i.e.H � 0. If � is contained in a slab M � [��; �], then � =M � ftg.Proof. We have to show that u � const: Consider the vector �eld

X = ruq1+jruj2

:

Note that ZMjXj2 dv � vol (M) < +1

and

divX = �mH � 0:

Since vol (M) < +1)M parabolic, then

0 � �ZMmH dv =

ZMdivX dv

KNR= 0) H � 0:

Now take

Y = u ruq1+jruj2

:

Then ZMjY j2 dv � sup

Mjuj vol (M) < +1

and

div Y =jruj2q1+jruj2

�mH =jruj2q1+jruj2

� 0:

Therefore

0 �ZM

jruj2q1+jruj2

�ZMdiv Y

KNR= 0) u � const: �