Geometric Analysis (Cambridge Studies in Advanced Mathematics) [1 ed.] 052137331X, 9781107020641, 1107020646

Table of contents :
Cover
CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 134
CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS
Title
Copyright
Dedication
Contents
Geometric Analysis
Preface
1 First and second variational formulas for area
2 Volume comparison theorem
3 Bochner–Weitzenböck formulas
4 Laplacian comparison theorem
5 Poincaré inequality and the first eigenvalue
6 Gradient estimate and Harnack inequality
7 Mean value inequality
8 Reilly's formula and applications
9 Isoperimetric inequalities and Sobolev inequalities
10 The heat equation
11 Properties and estimates of the heat kernel
12 Gradient estimate and Harnack inequality for the heat equation
13 Upper and lower bounds for the heat kernel
14 Sobolev inequality, Poincaré inequality and parabolic mean value inequality
15 Uniqueness and the maximum principle for the heat equation
16 Large time behavior of the heat kernel
17 Green's function
18 Measured Neumann Poincaré inequality and measured Sobolev inequality
19 Parabolic Harnack inequality and regularity theory
20 Parabolicity
21 Harmonic functions and ends
22 Manifolds with positive spectrum
23 Manifolds with Ricci curvature bounded from below
24 Manifolds with finite volume
25 Stability of minimal hypersurfaces in a 3-manifold
26 Stability of minimal hypersurface in a higher dimensional manifold
27 Linear growth harmonic functions
28 Polynomial growth harmonic functions
29 Lq harmonic functions
30 Mean value constant, Liouville property, and minimal submanifolds
31 Massive sets
32 The structure of harmonic maps into a Cartan–Hadamard manifold
Appendix A Computation of warped product metrics
Appendix B Polynomial growth harmonic functions on Euclidean space
References
Index

Citation preview

C A M B R I D G E S T U D I E S I N A DVA N C E D M AT H E M AT I C S 1 3 4 Editorial Board ´ S , W. F U LTO N , A . K ATO K , F. K I RWA N , B. BOLLOBA P. S A R NA K , B . S I M O N , B . TOTA RO

GEOMETRIC ANALYSIS The aim of this graduate-level text is to equip the reader with the basic tools and techniques needed for research in various areas of geometric analysis. Throughout, the main theme is to present the interaction of partial differential equations (PDE) and differential geometry. More specifically, emphasis is placed on how the behavior of the solutions of a PDE is affected by the geometry of the underlying manifold, and vice versa. For efficiency, the author mainly restricts himself to the linear theory, and only a rudimentary background in Riemannian geometry and partial differential equations is assumed. Originating from the author’s own lectures, this book is an ideal introduction for graduate students, as well as a useful reference for experts in the field. Peter Li is Chancellor’s Professor at the University of California, Irvine.

CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS Editorial Board: B. Bollob´as, W. Fulton, A. Katok, F. Kirwan, P. Sarnak, B. Simon, B. Totaro All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit: http//www.cambridge.org/mathematics. Already published 86 J. J. Duistermaat & J. A. C. Kolk Multidimensional real analysis, I 87 J. J. Duistermaat & J. A. C. Kolk Multidimensional real analysis, II 88 M. C. Golumbic & A. N. Trenk Tolerance graphs 90 L. H. Harper Global methods for combinatorial isoperimetric problems 91 I. Moerdijk & J. Mrˇcun Introduction to foliations and Lie groupoids 92 J. Koll´ar, K. E. Smith & A. Corti Rational and nearly rational varieties 93 D. Applebaum L´evy processes and stochastic calculus (1st Edition) 94 B. Conrad Modular forms and the Ramanujan conjecture 95 M. Schechter An introduction to nonlinear analysis 96 R. Carter Lie algebras of finite and affine type 97 H. L. Montgomery & R. C. Vaughan Multiplicative number theory, I 98 I. Chavel Riemannian geometry (2nd Edition) 99 D. Goldfeld Automorphic forms and L-functions for the group GL(n,R) 100 M. B. Marcus & J. Rosen Markov processes, Gaussian processes, and local times 101 P. Gille & T. Szamuely Central simple algebras and Galois cohomology 102 J. Bertoin Random fragmentation and coagulation processes 103 E. Frenkel Langlands correspondence for loop groups 104 A. Ambrosetti & A. Malchiodi Nonlinear analysis and semilinear elliptic problems 105 T. Tao & V. H. Vu Additive combinatorics 106 E. B. Davies Linear operators and their spectra 107 K. Kodaira Complex analysis 108 T. Ceccherini-Silberstein, F. Scarabotti & F. Tolli Harmonic analysis on finite groups 109 H. Geiges An introduction to contact topology 110 J. Faraut Analysis on Lie groups: An introduction 111 E. Park Complex topological K-theory 112 D. W. Stroock Partial differential equations for probabilists 113 A. Kirillov, Jr An introduction to Lie groups and Lie algebras 114 F. Gesztesy et al. Soliton equations and their algebro-geometric solutions, II 115 E. de Faria & W. de Melo Mathematical tools for one-dimensional dynamics 116 D. Applebaum L´evy processes and stochastic calculus (2nd Edition) 117 T. Szamuely Galois groups and fundamental groups 118 G. W. Anderson, A Guionnet & O. Zeitouni An introduction to random matrices 119 C. Perez-Garcia & W. H. Schikhof Locally convex spaces over non-Archimedean valued fields 120 P. K. Friz & N. B. Victoir Multidimensional stochastic processes as rough paths 121 T. Ceccherini-Silberstein, F. Scarabotti & F. Tolli Representation theory of the symmetric groups 122 S. Kalikow & R. McCutcheon An outline of ergodic theory 123 G. F. Lawler & V. Limic Random walk: A modern introduction 124 K Lux & H. Pahlings Representations of groups 125 K. S. Kedlaya p-adic differential equations 126 R. Beals & R. Wong Special functions 127 E. de Faria & W. de Melo Mathematical aspects of quantum field theory 128 A. Terras Zeta functions of graphs 129 D. Goldfeld & J. Hundley Automorphic representations and L-functions for the general linear group, I 130 D. Goldfeld & J. Hundley Automorphic representations and L-functions for the general linear group, II 131 D. A. Craven The theory of fusion systems 132 J. V¨aa¨ n¨anen Models and games 133 G. Malle & D. Testerman Linear algebraic groups and finite groups of Lie type

Geometric Analysis PETER LI University of California, Irvine

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo, Delhi, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9781107020641 c Peter Li 2012  This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2012 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library Library of Congress Cataloguing-in-Publication Data Li, Peter, 1952– Geometric analysis / Peter Li, University of California, Irvine. pages cm. – (Cambridge studies in advanced mathematics ; 134) ISBN 978-1-107-02064-1 (Hardback) 1. Geometric analysis. I. Title. QA360.L53 2012 515 .1–dc23 2011051365 ISBN 978-1-107-02064-1 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

I would like to dedicate this book to my wife, Glenna, for her love and unwavering support.

Contents

Preface

page ix

1

First and second variational formulas for area

1

2

Volume comparison theorem

10

3

Bochner–Weitzenb¨ock formulas

19

4

Laplacian comparison theorem

32

5

Poincar´e inequality and the first eigenvalue

40

6

Gradient estimate and Harnack inequality

57

7

Mean value inequality

68

8

Reilly’s formula and applications

77

9

Isoperimetric inequalities and Sobolev inequalities

86

10

The heat equation

96

11

Properties and estimates of the heat kernel

109

12

Gradient estimate and Harnack inequality for the heat equation

122

13

Upper and lower bounds for the heat kernel

134

14

Sobolev inequality, Poincar´e inequality and parabolic mean value inequality

149

vii

viii 15

Contents Uniqueness and the maximum principle for the heat equation

169

16

Large time behavior of the heat kernel

177

17

Green’s function

189

18

Measured Neumann Poincar´e inequality and measured Sobolev inequality

203

19

Parabolic Harnack inequality and regularity theory

216

20

Parabolicity

241

21

Harmonic functions and ends

256

22

Manifolds with positive spectrum

267

23

Manifolds with Ricci curvature bounded from below

284

24

Manifolds with finite volume

299

25

Stability of minimal hypersurfaces in a 3-manifold

306

26

Stability of minimal hypersurfaces in a higher dimensional manifold

315

27

Linear growth harmonic functions

326

28

Polynomial growth harmonic functions

340

29

Lq harmonic functions

349

30

Mean value constant, Liouville property, and minimal submanifolds

361

31

Massive sets

370

32

The structure of harmonic maps into a Cartan–Hadamard manifold

381

Appendix A

Computation of warped product metrics

Polynomial growth harmonic functions on Euclidean space References Index

392

Appendix B

395 399 404

Preface

The main goal of this book is to present the basic tools that are necessary for research in geometric analysis. Though the main theme centers around linear theory, i.e., the Laplace equation, the heat equation, and eigenvalues for the Laplacian, the methods of dealing with these problems are quite often useful in the study of nonlinear partial differential equations that arise in geometry. A small portion of this book originated from a series of lectures given by the author at a Geometry Summer Program in 1990 at the Mathematical Sciences Research Institute in Berkeley. The lecture notes were revised and expanded when the author taught a regular course in geometric analysis. During the author’s visit to the Global Analysis Research Institute at Seoul National University, he was encouraged to submit these notes, though still in a rather crude form, for publication in their lecture notes series [L6]. The part of this book that concerns harmonic functions originated from the author’s lecture notes for a series of courses he gave on the subject at the University of California, Irvine. A part of this material was also used in a series of lectures the author gave at the XIV Escola de Geometria Diferencial in Brazil during the summer of 2006. These notes [L9] were printed for distribution to the participants of the program. As well as updating the Korean lecture notes with more recent developments and combining with them the harmonic function notes, the author has also inserted a treatment on the heat equation. The result takes the form of an introduction to the subject of geometric analysis on the one hand, with some application to geometric problems via linear theory on the other. Due to the vast literature in geometric analysis, it is prudent not to make any attempt to try to discuss nonlinear theory. The interested reader is encouraged to consult the excellent book of Schoen and Yau [SY2] in this direction.

ix

x

Preface

The aim of this book is to address entry-level geometric analysts by introducing the basic techniques in the most economical way. The theorems discussed are chosen sometimes for their fundamental usefulness and sometimes for the purpose of demonstrating various techniques. In many cases, they do not represent the best possible or most current results. The book is roughly divided into three main parts. The first part (Chapters 1–9) contains basic background material, including reviews of various topics that may be found in a standard Riemannian geometry book. It also provides a quick glimpse of a powerful technique, namely the maximal principle method, in obtaining estimates on a manifold. The second part (Chapters 10–19) gives an outline of the theory of the heat equation that forms a basis for further study of nonlinear geometric flows. It also established various estimates for nonnegative solutions of the heat equation. As a consequence, estimates for the constants that appear in the Sobolev inequality, the Poincar´e inequality, and the mean value inequality in terms of geometric quantities are established. Chapters 18 and 19 are quite technical and contain a presentation of Moser’s argument for the parabolic Harnack inequality on a manifold. Moreover, the dependency on the background geometry is explicitly stated. The last part (Chapters 20–32) of the book is primarily on harmonic functions and various applications to other geometric problems, such as minimal surfaces, harmonic maps, and the geometric structure of certain manifolds. The author would like to express his gratitude to Ovidiu Munteanu, Lei Ni, and Jiaping Wang for their suggestions on how to improve this book. He is particularly in debt to Munteanu for his detailed proof-reading of the draft. Acknowledgement is also due to the author’s graduate students Lihan Wang and Fei He, who were extremely helpful in pointing out necessary corrections to the manuscript. The preparation of this manuscript was partially supported by NSF grant DMS-0801988.

1 First and second variational formulas for area

In this chapter, we will derive the first and second variational formulas for the area of a submanifold. This will be useful in our later discussion on the volume and Laplacian comparison theorems. It will also be used in our studies of the stability issues of minimal submanifolds. Let M be a Riemannian manifold of dimension m with metric denoted by ds 2 . In terms of local coordinates {x 1 , . . . , xm } the metric is written in the form ds 2 = gi j d xi d x j , where we are adopting the convention that repeated indices are being summed over. If X and Y are tangent vectors at a point p ∈ M, we will also denote their inner product by ds 2 (X, Y ) = X, Y . If we let S(T M) be the set of smooth vector fields on M, then the Riemannian connection ∇ : S(T M) × S(T M) → S(T M) satisfies the following properties: (1) (2) (3) (4)

∇( f1 X 1 + f 2 X 2 ) Y = f 1 ∇ X 1 Y + f 2 ∇ X 2 Y ; ∇ X ( f 1 Y1 + f 2 Y2 ) = X ( f 1 )Y1 + f 1 ∇ X Y1 + X ( f 2 )Y2 + f 2 ∇ X Y2 ; X Y1 , Y2  = ∇ X Y1 , Y2  + Y1 , ∇ X Y2 ; and ∇ X Y − ∇Y X = [X, Y ], for all X, Y ∈ S(T M),

for all X, X 1 , X 2 , Y, Y1 , Y2 ∈ S(T M) and for all f 1 , f 2 ∈ C ∞ (M). Property (3) says that ∇ is compatible with the Riemannian metric, while property (4) means that ∇ is torsion free. Moreover, the Riemannian connection is the 1

2

Geometric Analysis

only connection satisfying the above properties. The curvature tensor of the Riemannian metric is then given by R X Y Z = ∇ X ∇Y Z − ∇Y ∇ X Z − ∇[X,Y ] Z , for X, Y, Z ∈ S(T M), and it satisfies the properties: (1) R X Y Z = −RY X Z ; (2) R X Y Z + RY Z X + R Z X Y = 0; and (3) R X Y Z , W  = R Z W X, Y , for all X, Y, Z , W ∈ S(T M). The sectional curvature of the 2-plane section spanned by a pair of orthonormal vectors X and Y is defined by K (X, Y ) = R X Y Y, X . If we take {e1 , . . . , em } to be an orthonormal basis of the tangent space of M, then the Ricci curvature is defined to be the symmetric 2-tensor given by Ri j =

m 

Rei ,ek ek , e j .

k=1

Observe that the diagonal elements of the Ricci curvature are given by  K (ei , ek ). Rii = k =i

Let N be an n-dimensional submanifold in M with n < m. The Riemannian 2 defined on M when restricted to N induces a Riemannian metric metric ds M 2 ds N on N . One can easily check that for vector fields X, Y ∈ S(T N ), if we define ∇ Xt Y = (∇ X Y )t to be the tangential component of ∇ X Y to N , then ∇ t is the Riemannian connection of N with respect to ds N2 . The normal component of ∇ yields the negative of the second fundamental form of N . In particular, one defines the second fundamental form by − → I I (X, Y ) = −(∇ X Y )n , and checks that it is tensorial with respect to X, Y ∈ S(T N ). Taking the trace − → of the bilinear form I I over the tangent space of N yields the mean curvature vector, given by − → − → tr( I I ) = H .

1 First and second variational formulas for area

3

Let us now consider a one-parameter family of deformations of N given by Nt = φ(N , t) for t ∈ (−, ) with N0 = N . Let {x1 , . . . , xn } be a coordinate system around a point p ∈ N . We can consider {x 1 , . . . , x n , t} to be a coordinate system of N × (−, ) near the point ( p, 0). Let us denote ei = dφ (∂/∂ xi ) for i = 1, . . . , n and T = dφ (∂/∂t) . The induced metric on Nt from M is then given by gi j = ei , e j . We may further assume that {x 1 , . . . , xn } form a normal coordinate system at p ∈ N . Hence gi j ( p, 0) = δi j and ∇ei e j ( p, 0) = 0. Let us define d At to be the area element of Nt with respect to the induced metric. For t sufficiently close to 0, we can write d At = J (x, t) d A0 . With respect to the normal coordinate system {x1 , . . . , xn }, the function J (x, t) is given by √ g(x, t) J (x, t) = √ g(x, 0) with g(x, t) = det(gi j (x, t)). To compute the first variation for the area of N , we compute J  ( p, t) = (∂ J /∂t)( p, t). By the assumption that gi j ( p, 0) = δi j , we have J  ( p, 0) = 12 g  ( p, 0). However, g = det(gi j ) =

n 

g1 j c1 j ,

j=1

where ci j are the cofactors of gi j . Therefore 

g ( p, 0) =

n 

g1 j ( p, 0) c1 j ( p, 0) +

j=1

=

n 

g1 j ( p, 0) c1 j ( p, 0)

j=1

  ( p, 0) + c11 ( p, 0). g11

By induction on the dimension, we conclude that g  ( p, 0) = other hand,

n

 i = 1 gii . On the

gii = T ei , ei  = 2∇T ei , ei  = 2∇ei T, ei , because {x1 , . . . , xn , t} form a coordinate system for N × (−, ). Let us point out that the quantity n  i=1

∇ei T, ei 

4

Geometric Analysis

is now well defined under an orthonormal change of basis and hence is globally defined. If we write T = T t + T n , where T t is the tangential component of T on N and T n is its normal component, then n n n    ∇ei T, ei  = ∇ei T t , ei  + ∇ei T n , ei  i=1

i=1

i=1

= div(T ) + t

n 

ei T , ei  − n

i=1

n 

T n , ∇ei ei 

i=1

− → = div(T t ) + T n , H , − → where H is the mean curvature vector of N . Hence the first variation for the volume form at the point ( p, 0) is given by    d − →  d At |( p,0) = divT t + T n , H d A0  . ( p,0) dt However, the right-hand side is intrinsically defined independent of the choice of coordinates and hence this formula is valid at any arbitrary point. If T is a compactly supported variational vector field on N , then using the divergence theorem the first variation of the area of N is given by    d − → A(Nt ) T n, H . = dt t=0

N

This shows that the mean curvature of N is identically 0 if and only if N is a critical point of the area functional. Definition 1.1 An immersed submanifold N → M is said to be minimal if its − → mean curvature vector vanishes identically, i.e., H ≡ 0. When N is a curve in M that is parametrized by arc-length with unit tangent vector e, then the first variational formula for length can be written as  l l d  t  L = T , e 0 − T n , ∇e e dt t=0 0 =

T, e|l0

−

l

T, ∇e e.

0

We will now proceed to derive the second variational formula for area. Let φ : N × (−, ) × (−, ) −→ M be a two-parameter family of variations of N . Using similar notation, we write dφ(∂/∂ xi ) = ei for i = 1, . . . , n, and denote the variational vector fields by dφ(∂/∂t) = T and dφ(∂/∂s) = S.

1 First and second variational formulas for area

5

In terms of a general coordinate system, the first partial derivative of J can be written as n  ∂J g i j ∇ei T, e j  J (x, t, s), (x, t, s) = ∂t i, j=1

where (g i j ) denotes the inverse matrix of (gi j ). Differentiating this with respect to s and evaluating at ( p, 0, 0) we have n    ∂2 J S g i j ∇ei T, e j J = ∂s∂t i, j=1

=

n 

(Sg i j )∇ei T, e j J +

i, j=1

+

n 



g i j S∇ei T, e j  J

i, j=1

n 

g i j ∇ei T, e j S(J )

i, j=1

=

n 

(Sg i j )∇ei T, e j  +

i, j=1

+

n 

S∇ei T, ei 

i=1

n 

⎞

⎛ n  ∇ei T, ei  ⎝ ∇e j S, e j ⎠ .

i=1

j=1

However, differentiating the formula n 

n

k =1

(Sg ik )gk j = −

k=1

g ik gk j = δi j , we obtain

n 

g ik (Sgk j ),

k=1

hence Sg = − ij

n 

g ik (Sgkl )gl j

k,l=1

= −Sgi j = −Sei , e j  = −∇ S ei , e j  − ∇ S e j , ei  = −∇ei S, e j  − ∇e j S, ei .

(1.1)

6

Geometric Analysis

The first term on the right-hand side of (1.1) now becomes n 

n 

(Sg )∇ei T, e j  = − ij

i, j=1

∇ei S, e j ∇ei T, e j 

i, j=1 n 

−

∇e j S, ei ∇ei T, e j .

i, j=1

The second term on the right-hand side of (1.1) can be written as n 

S∇ei T, ei  =

i=1

n 

∇ S ∇ei T, ei  +

i=1

=

n 

n 

∇ei T, ∇ S ei 

i=1

R Sei T, ei  +

i=1

n 

∇ei ∇ S T, ei  +

i=1

n 

∇ei T, ∇ei S,

i=1

where the term R Sei T, ei  on the right-hand side denotes the curvature tensor of M. Therefore, we have n n   ∂2 J =− ∇ei S, e j ∇ei T, e j  − ∇e j S, ei ∇ei T, e j  ∂s∂t i, j=1

+

n 

i, j=1

R Sei T, ei  +

i=1

+

n 

n 

∇ei ∇ S T, ei  +

i=1

n 

∇ei T, ∇ei S

i=1

⎞

⎛ n  ∇ei T, ei  ⎝ ∇e j S, e j ⎠ .

i=1

(1.2)

j=1

We will now consider some special cases that will simplify (1.2). Let us first assume that N is a curve parametrized by arc-length in M with unit tangent vector given by e, then the second variational formula for the length is given by  l ∂ 2 L  {−∇e S, e∇e T, e + R Se T, e} = ∂s∂t (s,t)=(0,0) 0 l {∇e ∇ S T, e + ∇e T, ∇e S} . + 0

1 First and second variational formulas for area

7

If we further assumed that N is a geodesic satisfying the geodesic equation ∇e e ≡ 0, then we have  l ∂ 2 L  {−(eS, e)(eT, e) + R Se T, e} = ∂s∂t (s,t)=(0,0) 0

l

+

{e∇ S T, e + ∇e T, ∇e S}

0



l

=

 ∇e T, ∇e S + R Se T, e − (eS, e)(eT, e)



0

+ ∇ S T, e|l0 . The second special case is when N is a general n-dimensional manifold and then if the two variational vector fields are the same and are normal to N , (1.2) becomes  n n n    ∂ 2 J  2 = − ∇ T, e  − ∇ T, e ∇ T, e  + RT ei T, ei  e j e i e j i j i ∂t 2 t=0 i, j=1

+

n 

i, j=1

∇ei ∇T T, ei  +

i=1

=−

n  i, j=1

n 

i=1

|∇ei T | + 2

i=1 n 

∇ei T, e j 2 −

n 

2 ∇ei T, ei 

i=1

∇e j T, ei ∇ei T, e j  −

i, j=1

n 

Rei T T, ei 

i=1

n   − − →  →2 |∇ei T |2 + T, H . + div(∇T T )t + (∇T T )n , H +

(1.3)

i=1

On the other hand, if {en+1 , . . . , em } denotes an orthonormal set of vectors normal to N in M, then n n n m     2 ∇ei T, ∇ei T  = ∇ei T, e j  + ∇ei T, eν 2 . i=1

i, j=1

i=1 ν=n+1

Also  − →  ∇ei T, e j  = T, I I i j = ∇e j T, ei ,

8

Geometric Analysis

− → where I I i j denotes the second fundamental form with value in the normal bundle of N . Hence, (1.3) becomes  n  − ∂ 2 J  → 2  T, I I = − − Rei T T, ei  + div(∇T T )t i j ∂t 2 t=0 i, j

i=1

m n   − − →   →2 + (∇T T )n , H + ∇ei T, eν 2 + T, H . i=1 ν=n+1

Therefore, the second variational formula for area in terms of compactly supported normal variations is given by ⎫ ⎧  ⎨  n     2  d2 − → →⎬ n −  T, I I A(N ) = − R T, e  + (∇ T ) , H − t  ij ei T i T ⎭ dt 2 N ⎩ t=0 i, j

+

⎧ ⎨ n m  N

⎩

i=1 ν=n+1

i=1

⎫ 2 ⎬ − → ∇ei T, eν 2 + T, H . ⎭ 

Definition 1.2 A minimally immersed submanifold N → M is said to be stable if the second variation for area with respect to all compactly supported normal variations is nonnegative. This means that the stability inequality    n n m  − → 2 T, I I i j − Rei T T, ei  + ∇ei T, eν 2 0≤− N i, j

N i=1

N i=1 ν=n+1

is valid for any compactly supported normal vector field T. If we further restrict N to be an orientable codimension-1 minimal submanifold of an orientable manifold M, we can write any normal variation in the form T = ψem , where ψ is a differentiable function on N and em is a unit normal vector field to N . Then the second variational formula can be written as ⎧ ⎫  ⎨  n ⎬ 2   d2 − → 2  T, I I A(N ) = − R(T, T ) + ∇ T, e  − t i j e m i  ⎭ dt 2 N ⎩ t=0 i, j

=

i=1

 N

 −ψ 2 h i2j − ψ 2 R(em , em ) + |∇ψ|2 ,

− → where I I i j = h i j em with h i j being the component of the second fundamental form and R(T, T ) denotes the Ricci curvature of M in the direction of T. Here we have also used the fact that

1 First and second variational formulas for area

9

∇ei T, em  = ψ∇ei em , em  + ei (ψ)em , em  = ei (ψ). In particular, the stability inequality in this case is given by 2 2 2 |∇ψ| ≥ ψ hi j + ψ 2 R(em , em ). N

N

(1.4)

N

The last special case is again to assume that N is an oriented hypersurface in an oriented manifold M and we restrict the variation to be given by hypersurfaces which are a constant distant from N . The variational vector field is then given by em with ∇em em ≡ 0. This situation is particularly useful for the purpose of controlling the growth of the volume of geodesic balls of radius r. − → In this case, if we write H = H em , the first variational formula for the area element becomes ∂J (x, 0) = H (x) J (x, 0), (1.5) ∂t and the second variational formula can be written as m−1  ∂2 J (x, 0) = − h i2j (x) J (x, 0) 2 ∂t i, j=1

− R(em , em )(x) J (x, 0) + H 2 (x) J (x, 0).

(1.6)

2 Volume comparison theorem

In this chapter, we will develop a volume comparison theorem originally proved by Bishop (see [BC]). Let p ∈ M be a point in a complete Riemannian manifold of dimension m. In terms of polar normal coordinates at p, we can write the volume element as J (θ, r )dr ∧ dθ, where dθ is the area element of the unit (m − 1)-sphere. The Gauss lemma asserts that the area element of submanifold ∂ B p (r ), which is the boundary of the geodesic ball of radius r , is given by J (θ, r )dθ. By the first and second variational formulas (1.5) and (1.6), if x = (θ, r ) is not in the cut-locus of p, we have J  (θ, r ) =

∂J (θ, r ) ∂r

= H (θ, r ) J (θ, r )

(2.1)

and J  (θ, r ) =

∂2 J (θ, r ) ∂r 2

=−

m−1 

h i2j (θ, r ) J (θ, r ) − Rrr (θ, r ) J (θ, r ) + H 2 (θ, r ) J (θ, r ),

i, j=1

(2.2) where Rrr = R(∂/∂r , ∂/∂r ), H (θ, r ), and (h i j (θ, r )) denote the Ricci curvature in the radial direction, the mean curvature and the second fundamental form of ∂ B p (r ) at the point x = (θ, r ) with respect to the unit normal vector ∂/∂r , respectively. 10

2

Volume comparison theorem

11

Using the inequalities m−1 

h i2j ≥

i, j=1

m−1 

h ii2

i=1



m−1 

≥ =

2 h ii

i=1

m−1 H2 m−1

(2.3)

and (2.1), we can estimate (2.2) by m−2 2 H J − Rrr J m−1 m − 2  2 −1 = (J ) J − Rrr J. m−1

J  ≤

(2.4)

Since any smooth metric is locally Euclidean, we have the initial conditions J (θ, r ) ∼ r m−1 and J  (θ, r ) ∼ (m − 1)r m−2 as r → 0. Let us point out that if M is a simply connected constant curvature space form with constant sectional curvature K , then all the above inequalities become equalities. In particular (2.4) becomes J  =

m − 2  2 −1 (J ) J − (m − 1)K J. m−1

Theorem 2.1 (Bishop [BC]) Let M be a complete Riemannian manifold of dimension m, and p be a fixed point of M. Let us assume that the Ricci curvature tensor of M at any point x is bounded below by (m − 1)K (r ( p, x)) for some function K depending only on the distance from p. If J (θ, r ) dθ is the area element of ∂ B p (r ) as defined above and J¯(r ) is the solution of the ordinary differential equation m − 2 ¯ 2 ¯−1 J¯ = ( J ) J − (m − 1)K J¯ m−1

12

Geometric Analysis

with initial conditions J¯(r ) ∼ r m−1 and J¯ (r ) ∼ (m − 1)r m−2 , as r → 0, then within the cut-locus of p the function J (θ, r )/ J¯(r ) is a nonincreasing function of r. Also, if H¯ (r ) = J¯ / J¯ , then H (θ, r ) ≤ H¯ (r ) whenever (θ, r ) is within the cut-locus of p. In particular, if K is a constant, then J¯dθ corresponds to the area element of the sphere of radius r in the simply connected space form of constant curvature K . Proof

By setting f = J 1/(m−1) , (2.1) and (2.4) can be written as f =

1 H f m−1

and f  ≤

−1 Rrr f m−1

≤ −K f. The initial conditions become f (θ, 0) = 0 and f  (θ, 0) = 1. Let f¯ = J¯1/(m−1) be the corresponding function defined using J¯. The function f¯ satisfies f¯ = −K f¯, f¯(0) = 0, and f¯ (0) = 1. Observe that when K is a constant, the√function f¯ > 0 for all values of r ∈ (0, ∞) when K ≤ 0, and for r ∈ (0, π/ K ) when K > 0. In general, f¯ > 0

2

Volume comparison theorem

13

on an interval (0, a) for some a > 0. At those values of r we can define F(θ, r ) =

f (θ, r ) . f¯(r )

We have F  = f¯−2 ( f  f¯ − f f¯ ) and F  = f¯−1 f  − 2 f¯−2 f  f¯ − f¯−2 f f¯ + 2 f¯−3 f ( f¯ )2 ≤ −2 f¯−1 f¯ F  , hence ( f¯2 F  ) = f¯2 (F  + 2 f¯−1 f¯ F  ) ≤ 0. Integrating from  to r yields F  (r ) ≤ F  () f¯2 () f¯−2 (r ) = ( f¯() f  () − f () f¯ ()) f¯−2 (r ). Letting  → 0, the initial conditions of f and f¯ imply that F  (r ) ≤ 0. In particular, f¯ f  − f¯ f ≤ 0, implying H (θ, r ) ≤ H¯ (r ). Moreover, that F is a nonincreasing function of r implies that J (θ, r )/ J¯(r ) is also a nonincreasing function of r.  By computing the area element and the mean curvature of the constant curvature space form explicitly, we have the following corollary. Corollary 2.2 Under the assumption of Theorem 2.1, if K is a constant, then √  ⎧ √ ⎪ K cot Kr for K > 0, (m − 1) ⎨ H ≤ (m − 1)r −1 ⎪

√  for K = 0, √ ⎩ for K < 0, (m − 1) −K coth −K r

14

Geometric Analysis

and J (θ, r ) J¯(r ) is a nonincreasing function of r, where ⎧ m−1 √  m−1 ⎪ √1 sin Kr ⎪ ⎪ K ⎨ m−1 J¯(r ) = r ⎪   ⎪ ⎪ ⎩ √ 1 m−1 sinhm−1 √−K r  −K

for

K > 0,

for

K = 0,

for

K < 0.

Let us take this opportunity to point out that this estimate implies that when √ K > 0, there must be a cut-point along any geodesic which has length π/ K . In particular, this proves Myers’ theorem. Corollary 2.3 (Myers) Let M be an m-dimensional complete Riemannian manifold with Ricci curvature bounded from below by Ri j ≥ (m − 1)K for some constant K > 0. Then M must be compact with diameter d bounded from above by π d≤ √ . K Corollary 2.4 Let M be an m-dimensional complete Riemannian manifold with Ricci curvature bounded from below by a constant (m − 1)K . Suppose M¯ is an m-dimensional simply connected space form with constant sectional curvature K . Let A p (r ) be the area of the boundary of the geodesic ball ¯ ) be the area of the boundary of ∂ B p (r ) centered at p ∈ M of radius r and A(r ¯ ) of radius r in M. ¯ Then for 0 ≤ r1 ≤ r2 < ∞, we have a geodesic ball ∂ B(r ¯ 2 ) ≥ A p (r2 ) A(r ¯ 1 ). A p (r1 ) A(r

(2.5)

¯ ), respectively, then If we let V p (r ) and V¯ (r ) be the volumes of B p (r ) and B(r for 0 ≤ r1 ≤ r2 , r3 ≤ r4 < ∞ we have







 V p (r2 ) − V p (r1 ) V¯ (r4 ) − V¯ (r3 ) ≥ V p (r4 ) − V p (r3 ) V¯ (r2 ) − V¯ (r1 ) . (2.6) Proof Let us define C(r ) to be a subset of the unit tangent sphere S p (M) at p such that for all θ ∈ C(r ) the geodesic given by γ (s) = exp p (sθ ) is

2

Volume comparison theorem

15

minimizing up to s = r. Clearly for r1 ≤ r2 we have C(r2 ) ⊂ C(r1 ). By Theorem 2.1, we have J (θ, r1 ) J¯(r2 ) ≥ J (θ, r2 ) J¯(r1 ) for θ ∈ C(r2 ). Integrating over C(r2 ) yields C(r2 )

J (θ, r1 ) dθ J¯(r2 ) ≥

C(r2 )

J (θ, r2 ) dθ J¯(r1 )

= A p (r2 ) J¯(r1 ). On the other hand, A p (r1 ) = ≥

C(r1 )

C(r2 )

J (θ, r1 ) dθ J (θ, r1 ) dθ.

Taking this together with the fact that ¯ ) = αm−1 J¯(r ) A(r with αm−1 being the area of the unit (m − 1)-sphere, we conclude (2.5). To see (2.6), we first assume that r1 ≤ r2 ≤ r3 ≤ r4 and we simply integrate the inequality ¯ 2 ) ≥ A p (t2 ) A(t ¯ 1) A p (t1 ) A(t over r1 ≤ t1 ≤ r2 and r3 ≤ t2 ≤ r4 . For the case when r1 ≤ r3 ≤ r2 ≤ r4 , we write



 V p (r2 ) − V p (r1 ) V¯ (r4 ) − V¯ (r3 ) 



= V p (r3 ) − V p (r1 ) V¯ (r2 ) − V¯ (r3 ) 



+ V p (r3 ) − V p (r1 ) V¯ (r4 ) − V¯ (r2 )



 + V p (r2 ) − V p (r3 ) V¯ (r2 ) − V¯ (r3 )



 + V p (r2 ) − V p (r3 ) V¯ (r4 ) − V¯ (r2 )

16

Geometric Analysis 



≥ V p (r2 ) − V p (r3 ) V¯ (r3 ) − V¯ (r1 ) 



+ V p (r4 ) − V p (r2 ) V¯ (r3 ) − V¯ (r1 )



 + V p (r2 ) − V p (r3 ) V¯ (r2 ) − V¯ (r3 ) 



+ V p (r4 ) − V p (r2 ) V¯ (r2 ) − V¯ (r3 ) 



= V p (r4 ) − V p (r3 ) V¯ (r2 ) − V¯ (r1 ) .



Let us point out that equality in (2.6) holds if and only if C(r1 ) = C(r4 ) and J (θ, r ) = J¯(r ) for all 0 ≤ r ≤ r4 and θ ∈ C(r1 ). In particular, if r1 = 0, then J (θ, r ) = J¯(r ) for all r ≤ r4 and θ ∈ S p (M). This implies that B p (r4 ) is ¯ 4 ). isometric to B(r Theorem 2.5 Let M be an m-dimensional complete Riemannian manifold with nonnegative Ricci curvature. Then the volume growth of M must satisfy the following estimates: (1) (Bishop) If αm−1 is the area of the unit (m − 1)-sphere, then V p (ρ) ≤

αm−1 m ρ m

for all p ∈ M and ρ ≥ 0. (2) (Yau [Y2]) For all p ∈ M, there exists a constant C(m) > 0 depending only on m, such that V p (ρ) ≥ C V p (1) ρ for all ρ > 2. Proof

Applying (2.6) to r1 = 0 = r3 and r4 = r, we have V p (r2 ) V¯ (r ) ≥ V p (r ) V¯ (r2 ).

Observing that lim

r2 →0

V p (r2 ) = 1, V¯ (r2 )

the upper bound follows. To prove the lower bound, let x ∈ ∂ B p (1 + ρ). Then (2.6) and the curvature assumption imply that Vx (2 + ρ) − Vx (ρ) ≤ Vx (ρ)

(2 + ρ)m − ρ m . ρm

(2.7)

2

Volume comparison theorem

17

However, since the distance between p and x is r ( p, x) = 1 + ρ, we have B p (1) ⊂ (Bx (2 + ρ) \ Bx (ρ)), hence V p (1) ≤ Vx (2 + ρ) − Vx (ρ).

(2.8)

Since Bx (ρ) ⊂ B p (1 + 2ρ), we have Vx (ρ) ≤ V p (1 + 2ρ), therefore combining this with (2.7) and (2.8), we conclude that V p (1) ≤ V p (1 + 2ρ)

(2 + ρ)m − ρ m . ρm

The lower bounded follows by observing that (2 + ρ)m − ρ m = C ρ −1 ρm for 12 ρ → ∞.



We would like to remark that if we assume that for a sufficiently small  > 0 the Ricci curvature has a lower bound of the form Ri j (x) ≥ −(1 + r ( p, x))−2 , then one can show that M must have infinite volume. On the other hand, if the Ricci curvature is bounded from below by Ri j (x) ≥ −C0 (1 + r ( p, x))−2−δ for some constants C 0 , δ > 0, then the upper bound is also valid and is of the form V p (r ) ≤ Cr m , where C(C0 , δ, m) > 0 is a constant depending on C0 , δ and m. It is also a good exercise to show that if a complete manifold has Ricci curvature bounded from below by Ri j ≥ r ( p, x)−2 for some constant  > 14 and for all r > 1, then M must be compact. The next theorem shows that when the upper bound of the diameter given by Myers’ theorem is achieved, then the manifold must be isometrically a sphere.

18

Geometric Analysis

Theorem 2.6 (Cheng [Cg1]) Let M be a complete m-dimensional Riemannian manifold with Ricci curvature bounded from below by Ri j ≥ (m − 1)K for some constant K > 0. If the diameter d of M satisfies π d= √ , K √ then M is isometric to the standard sphere of radius 1/ K . Proof By scaling, we may assume that K = 1. Let p and q be a pair of points in M which realize the diameter. The volume comparison theorem implies that   ¯ V (d) d V p (d) ≤ V p . 2 V¯ (d/2) √ The assumption that d = π/ K implies that V¯ (d) = 2, V¯ (d/2) hence

Similarly, we have

  d V p (d) ≤ 2V p . 2   d . Vq (d) ≤ 2Vq 2

However, by the triangle inequality and the fact that d = r ( p, q), we have B p (d/2) ∩ Bq (d/2) = ∅. Therefore, 2V (M) = V p (d) + Vq (d)      d d ≤ 2 Vp + Vq 2 2 ≤ 2V (M), where V (M) denotes the volume of M. This implies that the inequalities in the volume comparison theorem are in fact equalities. Hence by the remark following Corollary 2.4 M must be the standard sphere. 

3 Bochner–Weitzenb¨ock formulas

Applying the Bochner–Weitzenb¨ock formulas, sometimes referred to as the Bochner technique, is one of the most important techniques in the theory of geometric analysis. There are many formulas which can be derived for various situations. In this chapter, we will only derive the formula for differential forms so as to illustrate the flavor of this technique. For convenience sake, we will also introduce the moving frame notation. Let {e1 , . . . , em } be a locally defined orthonormal frame field of the tangent bundle. Let us denote the dual coframe field by {ω1 , . . . , ωm }. They have the property that ωi (e j ) = δi j . The connection 1-forms ωi j are given by exterior differentiation of the ωi s, and are uniquely defined by Cartan’s first structural equations dωi = ωi j ∧ ω j , where ωi j + ω ji = 0. Cartan’s second structural equations yield the curvature tensor dωi j = ωik ∧ ωk j + i j , with i j = 12 Ri jkl ωl ∧ ωk . Now let us consider the case that N is an n-dimensional submanifold of M. Let {e1 , . . . , em } be an adapted orthonormal frame field of M such that {e1 , . . . , en } are orthonormal to N . We will now adopt the indexing convention

19

20

Geometric Analysis

that 1 ≤ i, j, k ≤ n and n + 1 ≤ ν, μ ≤ m. The second fundamental form of N is given by ωνi = h iνj ω j . Relating the two notations, we have the formulas ωi j (ek ) = ∇ek ei , e j , Ri jkl = Rei e j el , ek , and − → h iνj = I I (ei , e j ), eν . The sectional curvature of the 2-plane section spanned by ei and e j is given by Ri ji j and the Ricci curvature is given by Ri j =

m 

Rik jk .

k=1

Let f ∈ C ∞ (M) be a smooth function defined on M. Its exterior derivative is given by d f = f i ωi .

(3.1)

The second covariant derivative of f can be defined by f i j ω j = d f i + f j ω ji .

(3.2)

Exterior differentiating (3.1), and applying the first structural equations, we have 0 = d fi ∧ ωi + f i dωi = d fi ∧ ωi + f i ωi j ∧ ω j = (d fi + f j ω ji ) ∧ ωi = f i j ω j ∧ ωi . This implies that f i j − f ji = 0 for all i and j. The symmetric 2-tensor given by f i j ω j ⊗ ωi is called the Hessian of f. Taking the trace of the Hessian, we define the Laplacian of f by  f = f ii .

3 Bochner–Weitzenb¨ock formulas

21

The third covariant derivative of f is defined by f i jk ωk = d f i j + f k j ωki + f ik ωk j . Exterior differentiation of (3.2) gives d fi j ∧ ω j + f i j dω j = d f j ∧ ω ji + f j dω ji . However, the first and second structural equations imply that 0 = −d fi j ∧ ω j − f i j dω j + d f j ∧ ω ji + f j dω ji = −d f i j ∧ ω j − f i j ω jk ∧ ωk + d f j ∧ ω ji + f j ω jk ∧ ωki + f j  ji = −(d f i j + f ik ωk j ) ∧ ω j + (d f j + f k ωk j ) ∧ ω ji + f j  ji = −(d f i j + f ik ωk j ) ∧ ω j + f jk ωk ∧ ω ji + f j  ji = −(d f i j + f ik ωk j + f k j ωki ) ∧ ω j + f j  ji = − f i jk ωk ∧ ω j +

1 2

f j R jikl ωl ∧ ωk .

This yields the commutation formula f i jk − fik j =

1 2 fl

(Rli jk − Rlik j )

= fl Rli jk . Contracting the indices k and i by setting k = i and summing over 1 ≤ i ≤ m, we have the Ricci identity f i ji − f ii j = fl Rl j . For p ≤ m, we will now take the convention on the indices so that 1 ≤ i, j, k, l ≤ m, 1 ≤ α, β, γ ≤ p, and 1 ≤ a, b, c, d ≤ p − 1. Let ω ∈  p (M) be an exterior p-form defined on M. Then in terms of the basis, we can write ω = ai1 ... i p ωi p ∧ · · · ∧ ωi1 , where the summation is being performed over the multi-index I = (i 1 , . . . , i p ). With this understanding, we can write ω = aI ωI . Exterior differentiation yields dω = da I ∧ ω I + a I dω I = da I ∧ ω I + a I (−1) p−α ωi p ∧ · · · ∧ dωiα ∧ · · · ∧ ωi1

22

Geometric Analysis = da I ∧ ω I + a I ωiα jα ∧ ωi p ∧ · · · ∧ ω jα ∧ · · · ∧ ωi1 = (da I + ai1 ... jα ... i p ω jα iα ) ∧ ω I .

One defines the covariant derivatives ai1 ,... i p , j by m 



ai1 ... i p , j ω j = dai1 ... i p +

ai1 ... jα ... i p ω jα iα

1≤α≤ p jα

j=1

for each multi-index I = (i 1 , . . . , i p ). Similarly, for ( p − 1)-forms, we have ai1 ... i p−1 , j ω j = dai1 ... i p−1 + ai1 ... ja ... i p−1 ω ja ia . Exterior differentiation yields dai1 ... i p−1 , j ∧ ω j + ai1 ... i p−1 ,k ωk j ∧ ω j = dai1 ... ja ... i p−1 ∧ ω ja ia + ai1 ... ja ... i p−1 ω ja k ∧ ωkia + 12 ai1 ... ja ... i p−1 R ja ia kl ωl ∧ ωk . The left-hand side becomes dai1 ... i p−1 , j ∧ ω j + ai1 ... i p−1 ,k ωk j ∧ ω j = ai1 ... i p−1 , jk ωk ∧ ω j − ai1 ... ka ... i p−1 , j ωka ia ∧ ω j = ai1 ... i p−1 , jk ωk ∧ ω j + dai1 ... ka ... i p−1 ∧ ωka ia +



ai1 ... jb ... ka ... i p−1 ω jb ib ∧ ωka ia

b =a

+ ai1 ... ja ... i p−1 ω ja ka ∧ ωka ia . Equating this to the right-hand side gives ai1 ... i p−1 , jk ωk ∧ ω j +



ai1 ... jb ... ka ... i p−1 ω jb ib ∧ ωka ia

b =a

= 12 R ja ia lk ai1 ... ja ... i p−1 ωk ∧ ωl .

3 Bochner–Weitzenb¨ock formulas

23

We now claim that the second term on the left-hand side is identically 0. Indeed, since  ai1 ... jb ... ka ... i p−1 ω jb ib ∧ ωka ia b =a

=



ai1 ... jb ... ka ... i p−1 ω jb ib ∧ ωka ia +

ba

the claim follows by interchanging the roles of ka and jb in the second term. Hence ai1 ... i p−1 , jk ωk ∧ ω j = 12 R ja ia lk ai1 ... ja ... i p−1 ωk ∧ ωl , implying that ai1 ... i p−1 ,lk − ai1 ... i p−1 ,kl = R ja ia lk ai1 ... ja ... i p−1 . Similarly, for p-forms, we also have ai1 ... i p , jk − ai1 ... i p ,k j = Rlα iα jk ai1 ... lα ... i p .

(3.3)

Let us now compute the Laplacian ω = − dδω − δdω for p-forms. First we have dω = a I, j ω j ∧ ω I  =



sgn(σ ) aσ ωi p+1 ∧ . . . ∧ ωi1 ,

i 1 0, K = 0, K 0, ⎪ ⎨ H¯ (r ) = (m − 1)r −1 for K = 0, ⎪ √ √ ⎩ (m − 1) −K coth( −K r ) for K < 0. To show that r has the desired estimate in the sense of distribution, it suffices to show that for any nonnegative compactly supported smooth function φ we have (φ) r ≤ φ H¯ (r ). M

M

In terms of polar normal coordinates at p, we can write ∞ ¯ φ H (r ) = φ H¯ (r ) J (θ, r ) dθ dr. M

0

C(r )

34

Geometric Analysis

On the other hand, for each θ ∈ S p (M) if we let R(θ ) be the maximum value of r > 0 such that the geodesic γ (s) = exp p (sθ ) minimizes up to s = r, then by Fubini’s theorem we can write ∞ R(θ) φ H¯ (r ) J (θ, r ) dθ dr = φ H¯ (r ) J (θ, r ) dr dθ. C(r )

0

S p (M) 0

However, for r < R(θ ), we have H¯ (r ) J (θ, r ) ≥ H (θ, r ) J (θ, r ) = Therefore, ¯ φ H (r ) ≥ M

S p (M) 0



=−



S p (M) 0

=−

R(θ)

M

≥−

∂φ + ∂r

φ

R(θ)



∂ J (θ, r ). ∂r

∂J dr dθ ∂r ∂φ J dr dθ + ∂r

S p (M)



R(θ)

S p (M)

[φ J ]0

dθ

φ(θ, R(θ )) J (θ, R(θ )) dθ

∇φ, ∇r , M

where we have used the facts that both φ and J are nonnegative and J (θ, 0) = 0. On the other hand, since r is Lipschitz, we conclude that (φ)r, − ∇φ, ∇r  = M

which proves the theorem.

M



We are now ready to prove a structural theorem for manifolds with nonnegative Ricci curvature. Let us first define the notions of a line and a ray in a Riemannian manifold. Definition 4.2 A line is a normal geodesic γ : (−∞, ∞) −→ M such that any of its finite segments γ |ab is a minimizing geodesic. Definition 4.3 A ray is a half-line γ+ : [0, ∞) −→ M, which is a normal minimizing geodesic. Theorem 4.4 (Cheeger–Gromoll [CG2]) Let M be a complete, m-dimensional, manifold with nonnegative Ricci curvature. If there exists a

4 Laplacian comparison theorem

35

line in M, then M is isometric to R × N , the product of a real line and an (m − 1)-dimensional manifold N with nonnegative Ricci curvature. Proof Let γ+ : [0, ∞) −→ M be a ray in M. One defines the Buseman function β+ with respect to γ+ by β+ (x) = lim (t − r (γ+ (t), x)). t→∞

We observe that β+ is a Lipschitz function with Lipschitz constant 1. Moreover, by the Laplacian comparison theorem, β+ (x) ≥ − lim (m − 1)r (γ+ (t), x)−1 t→∞

=0 in the sense of distribution. If γ is a line, then γ+ (t) = γ (t) and γ− (t) = γ (−t) for t ≥ 0 are rays. The corresponding Buseman functions β+ and β− are subharmonic in the sense of distribution. In particular, β− + β+ is also subharmonic on M. On the other hand, since γ is a line, the triangle inequality implies that 2t = r (γ (−t), γ (t)) ≤ r (γ (−t), x) + r (γ (t), x). Hence t − r (γ− (t), x) + t − r (γ+ (t), x) ≤ 0, and by taking the limit as t → ∞ we have β− (x) + β+ (x) ≤ 0. Moreover, it is also clear that β− (x) + β+ (x) = 0 for all x on γ . However, by the strong maximum principle, since the subharmonic function β− + β+ has an interior maximum, it must be identically constant. In particular, both β− and β+ are harmonic and β− ≡ −β+ . By regularity theory, β+ is a smooth harmonic function with |∇β+ | ≤ 1, and |∇β+ | ≡ 1 on the geodesic γ . For simplicity, let us write β = β+ . The Bochner formula gives |∇β|2 = 2βi2j + 2Ri j βi β j + 2∇β, ∇β ≥ 0.

(4.1)

36

Geometric Analysis

Hence by the fact that |∇β|2 achieves its maximum in the interior of M, the maximum principle for subharmonic functions again implies that |∇β|2 ≡ 1 on M. Substituting this into (4.1) yields βi j ≡ 0, and ∇β is a parallel vector field on M. This implies that M must split, which proves the theorem.  Corollary 4.5 Let M be a complete m-dimensional Riemannian manifold with nonnegative Ricci curvature. If M has at least two ends, then there exists a compact (m − 1)-dimensional manifold N of nonnegative Ricci curvature such that M = R × N. Proof The assumption that M has at least two ends implies that there exists a compact set D ⊂ M such that M \ D has at least two unbounded components. ∞ and {y }∞ , Hence there are two unbounded sequences of points {x i }i=1 i i=1 such that the minimal geodesics γi joining xi to yi must intersect D. By the compactness of D, if pi ∈ γi ∩ D and vi = γi ( pi ), then by passing through a subsequence we have pi → p and vi → v for some p ∈ D and v ∈ T p M. We now claim that the geodesic γ : (−∞, ∞) −→ M given by the initial conditions γ (0) = p and γ  (0) = v is a line. To see this, let us consider an arbitrary segment γ |[a,b] of γ . By the continuity of the initial conditions of the second order ordinary differential equation, we know that γi |[a,b] → γ |[a,b] because ( pi , vi ) → ( p, v). However, by the assumption, γi |[a,b] are minimizing geodesics, hence γ |[a,b] is also minimizing, and γ is a line. Theorem 4.4 now implies that M = R × N . The compactness of N follows from the assumption that M has at least two ends.  Another application of the Laplacian comparison theorem is the eigenvalue comparison theorem of Cheng. Theorem 4.6 (Cheng [Cg1]) Let M be a compact Riemannian manifold of dimension m and ∂ M be the boundary of M. Assume that the Ricci curvature of M is bounded by Ri j ≥ (m − 1)K for some constant K . Let us consider M¯ to be the simply connected space form with constant curvature K . Let μ1 (M) be the first nonzero eigenvalue of ¯ the Dirichlet Laplacian on M and i be the inscribe radius of M. If B(i) is a

4 Laplacian comparison theorem

37

¯ is its first Dirichlet eigenvalue, geodesic ball in M¯ with radius i and μ1 ( B(i)) then ¯ μ1 (M) ≤ μ1 ( B(i)). When ∂ M = ∅, let λ1 (M) be the first nonzero eigenvalue of the Laplacian ¯ and d be the diameter of M. If B(d/2) is a geodesic ball in M¯ with radius d/2  ¯ and μ1 B (d/2) is its first Dirichlet eigenvalue, then    d λ1 (M) ≤ μ1 B¯ . 2 Proof Let us first consider the case when M has boundary. Let B p (i) be an inscribed ball in M. By the monotonicity of eigenvalues, it suffices to show that ¯ μ1 (B p (i)) ≤ μ1 ( B(i)). ¯ Let u¯ be the first Dirichlet eigenfunction on B(i). By the uniqueness of u, ¯ we ¯ may assume that u¯ ≥ 0. If we let p¯ ∈ B(i) be its center and r¯ be the distance function to p, ¯ then u(¯ ¯ r ) must be a function of r¯ alone. By the facts that u¯ ≥ 0 and u(i) ¯ = 0, the strong maximum principle implies that (∂ u/∂ ¯ r¯ )(i) < 0. If there is some value of r¯ < i such that ∂ u/∂ ¯ r¯ > 0, then this would imply that u¯ has a interior local minimum. However, this violates the strong maximum principle. Hence, u¯  = ∂ u/∂ ¯ r¯ ≤ 0. ¯ ), where r denotes Let us define a Lipschitz function on B p (i) by u(r ) = u(r the distance function to p. Clearly, u satisfies the Dirichlet boundary condition. Computing the Laplacian of u gives u = u¯  r + u¯  |∇r |2 ≥ u¯  r + u¯  . ¯ then ¯ be the Laplacian on M, On the other hand, if we let  ¯ ¯ u¯ u¯ =  −μ1 ( B(i)) ¯ r + u¯  . = u¯  ¯ By Theorem 4.1 and the fact that u¯  ≤ 0, we conclude that ¯ u u ≥ −μ1 ( B(i))

38

Geometric Analysis

in the sense of distribution. Hence, by the Rayleigh principle for eigenvalues, we conclude that ! 2 B p (i) |∇u| μ1 (B p (i)) ≤ ! 2 B p (i) u ! − B p (i) uu = ! 2 B p (i) u ¯ ≤ μ1 ( B(i)). To prove the upper bound for the case when M has no boundary, we consider the disjoint balls B p (d/2) and Bq (d/2) centered at a pair of points p and q which realize the diameter. By the above estimate,       d d ≤ μ1 B¯ μ1 B p 2 2 and       d d ≤ μ1 B¯ . μ1 Bq 2 2 We now claim that λ1 (M) ≤ max{μ1 (M1 ), μ1 (M2 )} for any disjoint pair of open subsets M1 , M2 ⊂ M. This will establish the theorem by setting M1 = B p (d/2) and M2 = Bq (d/2). To prove the claim, let u 1 and u 2 be nonnegative first Dirichlet eigenfunctions on M1!and M2 respectively. By multiplying u i by a constant, we may assume that Mi u i = 1 for i = 1, 2. Let us define a Lipschitz function on M by ⎧ ⎨ u1 u = −u 2 ⎩ 0 Clearly,

! M

on on on

M1 , M2 , M \ (M1 ∪ M2 ).

u = 0, hence after applying the Rayleigh principle, we have







λ1 (M) M1

u 21

+ M2

u 22

= λ1 (M)

u2 M

≤

|∇u|2

M

=

|∇u 1 | +

|∇u 2 |2

2

M1

M2

4 Laplacian comparison theorem

39

= μ1 (M1 ) M1

u 21

+ μ1 (M2 ) 

M2

≤ max{μ1 (M1 ), μ1 (M2 )} M1

This establishes the claim.

u 22

u 21



+ M2

u 22

. 

5 Poincar´e inequality and the first eigenvalue

In this chapter, we will obtain lower estimates for the first eigenvalue of the Laplacian on a compact manifold. For the moment, we will primarily be concerned with the case when M has no boundary. The following lower bound was proved by Lichnerowicz [Lz], while Obata [O] considered the case when the estimate is achieved. Theorem 5.1 (Lichnerowicz and Obata) Let M be an m-dimensional compact manifold without boundary. Suppose that the Ricci curvature of M is bounded from below by Ri j ≥ (m − 1)K for some constant K > 0, then the first nonzero eigenvalue of the Laplacian on M must satisfy λ1 ≥ m K . Moreover,√equality holds if and only if M is isometric to a standard sphere of radius 1/ K . Proof

Let u be a nonconstant eigenfunction satisfying u = −λu,

with λ > 0. Consider the smooth function Q = |∇u|2 + 40

λ 2 u m

5 Poincar´e inequality and the first eigenvalue

41

defined on M. Computing its Laplacian   2λ uu i Q = 2u j u ji + m i = 2u 2ji + 2u j u jii +

2λ 2 2λ u + uu ii m i m

2λ 2λ |∇u|2 + u(u), m m where we have used the Ricci identity and the convention that summation is performed on repeated indices. On the other hand, = 2u 2ji + 2Ri j u i u j + 2u j (u) j +

m 

u 2ji ≥

i, j=1

m 

u ii2

i=1

m ≥ =

i=1 u ii

2

m (u)2 m

λ2 u 2 . m Hence, by the assumption on the Ricci curvature, we have =

2λ2 u 2 2λ 2λ2 u 2 + 2(m − 1)K |∇u|2 − 2λ|∇u|2 + |∇u|2 − m m m   λ = 2(m − 1) K − (5.1) |∇u|2 . m

Q ≥

If λ ≤ m K , then Q is a subharmonic function. By the compactness of M and the maximum principle, Q must be identically constant and all the above inequalities are equalities. In particular, the right-hand side of (5.1) must be identically 0. Hence λ = m K because u is nonconstant. Moreover, λ 2 λ u = |u|2∞ , m m where |u|∞ = sup M |u|. If we normalize u such that |u|∞ = 1, and observe that at the maximum and minimum points of u its gradient must vanish, then we conclude that max u = 1 = − min u and √ |∇u| = K. √ 2 1−u |∇u|2 +

42

Geometric Analysis

Integrating this along a minimal geodesic γ joining the points where u = 1 and u = − 1, we have √ |∇u| √ d K ≥ 1 − u2 γ 1 du ≥ √ 1 − u2 −1 = π, where d denotes the diameter of M. However, Cheng’s theorem (Theorem 2.6) implies that M must be the standard sphere.  We will now give a sharp lower bound for the first eigenvalue on manifolds with nonnegative Ricci curvature. The estimate of Lichnerowicz becomes trivial in this case, since the Ricci curvature does not have a positive lower bound. However, one could still estimate the first eigenvalue in terms of the diameter of M alone. Let λ1 be the least nontrivial eigenvalue of a compact manifold and let φ be the corresponding eigenfunction. By multiplying φ by a constant it is possible to arrange that a − 1 = inf φ, M

a + 1 = sup φ, M

where 0 ≤ a(φ) < 1 is the median of φ. Lemma 5.2 (Li–Yau [LY1]) Suppose M m is a compact manifold without boundary and its Ricci curvature is nonnegative. Then the first nontrivial eigenvalue satisfies λ1 ≥

π2 , (1 + a)d 2

where d is the diameter of M. Proof

Setting λ = λ1 and u = φ − a, the eigenvalue equation becomes u = −λ(u + a).

Let P = |∇u|2 + cu 2 , where c = λ(1 + a). Suppose x0 ∈ M is the point where P achieves its maximum. If |∇u(x 0 )| = 0 we may rotate the frame so that u 1 (x0 ) = |∇u(x0 )|. Differentiating in the ei direction yields 1 2 Pi

= u j u ji + cuu i .

5 Poincar´e inequality and the first eigenvalue

43

At the point x0 , this yields 0 = u 1 (u 11 + cu) and u ji u ji ≥ u 211 = c2 u 2 .

(5.2)

Covariant differentiating with respect to ei again, using the commutation formula (5.2), the definition of P, and evaluating at x 0 , we have 0 ≥ 12 P = u ji u ji + u j u jii + cu i2 + cuu ≥ c2 u 2 + u j (u) j + R ji u j u i + cu i2 − cλu(u + a) ≥ c2 u 2 − λu i2 + cu i2 − cλu(u + a)   = (c − λ) u i2 + cu 2 − acλu ≥ aλP(x0 ) − acλ. Hence, for all x ∈ M

  |∇u(x)|2 ≤ λ(1 + a) 1 − u(x)2 .

(5.3)

Note that (5.3) is trivially satisfied if ∇u(x0 ) = 0, hence the assumption ∇u(x0 ) = 0 is not necessary. Let γ be the shortest geodesic from the minimizing point of u to the maximizing point. The length of γ is at most d. Integrating the gradient estimate (5.3) along this segment with respect to arc-length, we obtain 1 " " |∇u|ds du d λ(1 + a) ≥ λ(1 + a) ds ≥ √ ≥ √ = π.  2 1−u 1 − u2 −1 γ γ In view of Lemma 5.2 and known examples, Li and Yau conjectured that the sharp estimate λ1 ≥

π2 d2

should hold for compact manifolds with nonnegative Ricci curvature. In fact, if the first eigenspace has multiplicity greater than 1, this was verified in [L3]. This conjecture was finally proved by Zhong and Yang by applying the

44

Geometric Analysis

maximum principle to a judicious choice of test function. We will now present this argument. Lemma 5.3 The function  " 2 arcsin(u) + u 1 − u 2 − u π

z(u) = defined on [−1, 1] satisfies

z˙ u + z¨ (1 − u 2 ) + u = 0,

(5.4)

z˙ 2 − 2z z¨ + z˙ ≥ 0,

(5.5)

2z − z˙ u + 1 ≥ 0,

(5.6)

(1 − u 2 ) ≥ 2|z|.

(5.7)

and

Proof

Differentiating the definition of z(u) yields z˙ =

4" 1 − u2 − 1 π

and z¨ =

−4u . √ π 1 − u2

Clearly (5.4) is satisfied. To see (5.5), we have z˙ − 2z z¨ + z˙ = 2

4

√ π 1 − u2

#

$  4 " 1 − u 2 + u arcsin(u) − (1 + u 2 ) . π

Since the right-hand side is an even function, it suffices to check that  4 " 1 − u 2 + u arcsin(u) − (1 + u 2 ) ≥ 0 π on [0, 1]. Computing its derivative # $  d 4 " 4 2 2 1 − u + u arcsin(u) − (1 + u ) = arcsin(u) − 2u, du π π

5 Poincar´e inequality and the first eigenvalue

45

which is nonpositive on [0, 1]. Hence  4 " 1 − u 2 + u arcsin(u) − (1 + u 2 ) π & % "  4 1 − u 2 + u arcsin(u) − (1 + u 2 ) ≥ π u=1 = 0. Inequality (5.6) follows easily because 2z − z˙ u + 1 =

4 arcsin(u) + 1 − u, π

which is obviously nonnegative. To see (5.7), we will consider the cases −1 ≤ u ≤ 0 and 0 ≤ u ≤ 1 separately. It is clearly that the inequality is valid at −1, 0, and 1. Let us set  " 4 f (u) = 1 − u 2 − arcsin(u) + u 1 − u 2 + 2u. π Then  4 " f˙ = −2u − 2 1 − u 2 + 2, π f¨ = −2 +

8u , √ π 1 − u2

and ... f =

8 3

π(1 − u 2 ) 2

.

When −1 ≤ u ≤ 0, we have ...f¨ ≤ 0, hence f (u) ≥ min{ f (−1), f (0)} = 0. For the case 0 ≤ u ≤ 1, we have f ≥ 0, hence f˙ ≤ max{ f˙(0), f˙(1)} # $ 8 = max 2 − , 0 π = 0. Therefore f (u) ≥ f (1) proving (1 − u 2 ) ≥ 2z on [−1, 1]. Note that (1 − u 2 ) ≥ 2|z| follows from the previous inequality because z is an odd function  while (1 − u 2 ) is even.

46

Geometric Analysis

Lemma 5.4 Suppose M is a compact manifold without boundary whose Ricci curvature is nonnegative. Assume that a nontrivial eigenfunction φ corresponding to the eigenvalue λ is normalized so that for 0 ≤ a < 1, a + 1 = sup φ and a − 1 = inf M φ. By setting u = φ − a, its gradient must satisfy the estimate

 |∇u|2 ≤ λ 1 − u 2 + 2aλz(u),

(5.8)

where z(u) =

 " 2 arcsin(u) + u 1 − u 2 − u. π

(5.9)

Proof We will first prove an estimate similar to (5.8) for u = (φ − a), where 0 <  < 1. The lemma will follow by letting  → 1. By the definition of u, we have u = −λ(u + a) with − ≤ u ≤ . By (5.3) we may assume a > 0. Consider the function Q = |∇u|2 − c(1 − u 2 ) − 2aλz(u), and because of (5.3) and (5.7) we can choose c large enough so that sup M Q = 0. The lemma follows if c ≤ λ, for a sequence of  → 1, hence we may assume that c > λ. Let the maximum point of Q be x0 . We claim that |∇u(x 0 )| > 0 since otherwise ∇u(x0 ) = 0 and 0 = Q(x0 ) = −c(1 − u 2 )(x 0 ) − 2aλz(x0 ) ≤ −(c − aλ)(1 −  2 ) by (5.7), which is a contradiction. Differentiating in the ei direction gives 1 2 Qi

= u j u ji + cuu i − aλ˙z u i .

(5.10)

At x0 , we can rotate the frame so u 1 (x0 ) = |∇u(x0 )| and using Q i = 0 we have u ji u ji ≥ u 211 = (cu − aλ˙z )2 .

(5.11)

5 Poincar´e inequality and the first eigenvalue

47

Differentiating again, using the commutation formula, Q(x 0 ) = 0, (5.7), (5.10), and (5.11), we get 0 ≥ 12 Q(x0 ) = u ji u ji + u j u jii + cu i2 + cuu − aλ¨z u i2 − aλ˙z u = u ji u ji + u j (u) j + R ji u j u i + (c − aλ¨z ) u i2 + (cu − aλ˙z ) u  ( '  ≥ (cu − aλ˙z )2 + (c − λ − aλ¨z ) c 1 − u 2 + 2aλz − λ (cu − aλ˙z ) (u + a)     = −acλ (1 − u 2 )¨z + u z˙ + u + a 2 λ2 −2z z¨ + z˙ 2 +  z˙ + aλ(c − λ) {−u z˙ + 2z + 1} + (c − λ)(c − aλ). However by (5.4), (5.5), and (5.6), we conclude that 0 ≥ acλ(1 − )u − a 2 λ2 (1 − )˙z + (c − λ)(c − aλ)   4 ≥ −acλ(1 − ) − a 2 λ2 (1 − ) − 1 + (c − λ)(c − aλ) π ≥ −(c + λ)λ(1 − ) + (c − λ)2 . This implies that # c≤λ

2 + (1 − ) +

√

(1 − )(9 − )

2

$ .

Clearly, when  → 1 this yields the desired estimate.



Theorem 5.5 (Zhong–Yang [ZY]) Suppose M is a compact manifold without boundary whose Ricci curvature is nonnegative. Let a ≥ 0 be the median of a normalized first eigenfunction with a + 1 = sup φ and a − 1 = inf φ, and let d be the diameter. Then the first nontrivial eigenvalue satisfies d 2 λ1 ≥ π 2 +

4 6 π − 1 a 2 ≥ π 2 (1 + 0.02a 2 ). π 2

Proof Arguing with u = φ − a as before, let γ be the shortest geodesic from a minimum point of u to a maximum point, with length at most d. Integrating

48

Geometric Analysis

the gradient estimate (5.8) along this segment with respect to arc-length and using the oddness of the function, we have 1/2

dλ

≥λ

1/2 γ

ds

≥

"

γ

≥

|∇u|ds 1 − u 2 + 2az(u)

1#

√

0

≥ 0

1

1

1

1 − u 2 + 2az

$

+√ du 1 − u 2 − 2az

# $ 1 3a 2 z 2 2+ du √ 1 − u2 1 − u2

≥ π + 3a

1

2 0

=π+

z

2

√ 1 − u2

4 3a 2  π . − 1 π2 2



This technique also applies to manifolds with boundary. Let M n be a compact manifold with smooth boundary whose Ricci curvature is nonnegative. Suppose that the second fundamental form of ∂ M is nonnegative. Then the first nontrivial eigenvalue of the Laplacian with Neumann boundary conditions also satisfies inequality (5.8). The proof is the same as that of Lemma 5.4 except that the possibility that the maximum of the function Q may occur at the boundary must be considered. In fact, the boundary convexity assumption implies that the maximum of Q cannot occur on the boundary as indicated in the proof of Corollary 5.8. The next theorem gives an estimate of the first eigenvalue for the general compact Riemannian manifold without boundary. The estimate depends on the lower bound of the Ricci curvature, the upper bound of the diameter, and the dimension of M alone. An estimate of this form was first conjectured by Yau in [Y3]. It was first proved by the author [L1] for manifolds with nonnegative Ricci curvature with an improvement by Li and Yau [LY1] as stated in Lemma 5.2. The general case was also proved by Li and Yau in [LY1] and will be presented in Theorem 5.7. The following lemma will be useful for the upcoming theorem and also for various estimates in the next chapter.

5 Poincar´e inequality and the first eigenvalue

49

Lemma 5.6 Let M be a complete m-dimensional Riemannian manifold. Suppose the Ricci curvature of M is bounded from below by Ri j ≥ −(m − 1)R for some constant R ≥ 0. Let u be a function defined on M satisfying the equation u = −λu, and if we define Q = |∇ log(a + u)|2 , then m |∇ Q|2 Q −1 2(m − 1)

Q −

+ ∇v, ∇ Q Q −1  +



2(m − 2) 2 λu Q− m−1 m−1 a+u

4 λu 2λa − − 2(m − 1)R m−1a+u a+u

 Q+

 ≥

2 Q2 m−1

2 m−1



λu a+u

2 .

Proof If we set v = log(a + u), then a direct computation shows that v satisfies the equation v = −|∇v|2 − λ +

λa . a+u

Using the Bochner formula, we compute Q = 2vi2j + 2Ri j vi v j + 2∇v, ∇v ≥ 2vi2j − 2(m − 1)R Q − 2∇v, ∇ Q −

2λa |∇v|2 . (a + u)

(5.12)

Choosing an orthonormal frame {e1 , . . . , em } at a point so that |∇v| e1 = ∇v, we can write

50

Geometric Analysis

|∇|∇v| | = 4 2 2

m m   j=1

= 4v12

2 vi vi j

i=1

m 

v12 j

j=1

= 4|∇v|2

m 

v12 j .

(5.13)

j=1

On the other hand, vi2j

2 ≥ v11 +2

m 

2 v1α +

α=2

≥

2 v11

+2

m  α=2

m

m 

2 v1α

α=2 vαα

+

m 

2 v1α +

(v − v11 )2 m−1

2 v1α +

1 m−1

α=2 2 = v11 +2

m  α=2

≥

2

m−1

α=2 2 = v11 +2

2 vαα

 |∇v|2 +

λu + v11 a+u

2

  m m  2 1 λu 2 |∇v|2 + v1 j + m−1 m−1 a+u j=1

+

2v11 m−1

 |∇v|2 +

λu a+u

 .

However, using the identity 2v1 v11 = e1 (|∇v|2 ) = ∇|∇v|2 , ∇v |∇v|−1 , we conclude that 2v11 = |∇v|−2 ∇|∇v|2 , ∇v.

(5.14)

5 Poincar´e inequality and the first eigenvalue

51

Substituting the above identity into (5.14), we obtain   m m  2 1 λu 2 2 2 |∇v| + vi j ≥ v1 j + m−1 m−1 a+u j=1

1 + ∇|∇v|2 , ∇v |∇v|−2 m−1



λu |∇v| + a+u 2

 .

Combining the above inequality with (5.12) and (5.13) yields m |∇ Q|2 Q −1 − 2(m − 1)R Q Q ≥ 2(m − 1)   2λu 2(m − 2) 2 −1 + ∇v, ∇ Q Q − Q + Q2 (m − 1)(a + u) m−1 m−1 +

λu 2 4 Q+ m−1 a+u m−1



λu a+u

2 −

2λa Q. a+u

(5.15) 

Theorem 5.7 (Li–Yau [LY1]) Let M be a compact m-dimensional Riemannian manifold without boundary. Suppose that the Ricci curvature of M is bounded from below by Ri j ≥ −(m − 1)R for some constant R ≥ 0, and d denotes the diameter of M. Then there exist constants C1 (m), C 2 (m) > 0 depending on m alone, such that the first nonzero eigenvalue of M satisfies √ C1 λ1 ≥ 2 exp(−C2 d R). d Proof Let u be a nonconstant eigenfunction satisfying u = −λu. By the fact that





−λ

u= M

u = 0, M

u must change sign. Hence we may normalize u to satisfy min u = − 1 and max u ≤ 1. Let us consider the function v = log(a + u) for some constant a > 1.

52

Geometric Analysis

If x0 ∈ M is a point where the function Q = |∇v|2 achieves its maximum, then Lemma 5.6 and the maximum principle assert that   λu 2λa 2 4 2 Q + − − 2(m − 1)R Q 0≥ m−1 m−1a+u a+u  λu 2 a+u   2 4λ 2(m + 1) λa ≥ Q2 + − − 2(m − 1)R Q, m−1 m−1 m−1 a+u 2 + m−1



which implies that Q(x) ≤ Q(x 0 ) ≤ (m − 1)2 R +

(m + 1)aλ (a − 1)

for all x ∈ M. Integrating Q 1/2 = |∇ log(a + u)| along a minimal geodesic γ joining the points at which u = − 1 and u = max u, we have     a + max u a ≤ log log a−1 a−1 |∇ log(a + u)| ≤ γ

) ≤d

(m + 1)aλ + (m − 1)2 R a−1

for all a > 1. Setting t = (a − 1)/a, we have 

 1 1 2 2 (m + 1)λ ≥ t log − (m − 1) R t d2 for all 0 < t < 1. Maximizing the right-hand side as a function of t by setting " t = exp(−1 − 1 + (m − 1)2 Rd 2 ), we obtain the estimate " " 2 2 Rd 2 ) exp(−1 − 1 + (m − 1)2 Rd 2 ) λ≥ (1 + 1 + (m − 1) (m + 1)d 2 √ C1 exp(−C2 d R) 2 d as claimed. ≥



5 Poincar´e inequality and the first eigenvalue

53

We would like to point out that when M is a compact manifold with boundary, there are corresponding estimates for the first Dirichlet eigenvalue and the first nonzero Neumann eigenvalue using the maximum principle. In fact, Reilly [R] proved the Lichnerowicz–Obata result for the Dirichlet eigenvalue on manifolds whose boundary has nonnegative mean curvature with respect to the outward normal vector. In 1990, Escobar [E] established the Lichnerowicz– Obata result for the first nonzero Neumann eigenvalue on manifolds whose boundary is convex with respect to the second fundamental form. There are estimates analogous to that of Theorem 5.7 for both the Dirichlet and Neumann eigenvalues on manifolds with boundary. In general, the estimate for the Dirichlet eigenvalue [LY1] depends also on the lower bound of the mean curvature of the boundary with respect to the outward normal, and the estimate for the Neumann eigenvalue depends also on the lower bound of the second fundamental form of the boundary and the -ball condition (see [Cn]). However, when the boundary is convex, the Neumann eigenvalue has an estimate similar to manifolds without boundary. Corollary 5.8 (Li–Yau [LY1]) Let M be a compact m-dimensional Riemannian manifold whose boundary is convex in the sense that the second fundamental form is nonnegative with respect to the outward pointing normal vector. Suppose that the Ricci curvature of M is bounded from below by Ri j ≥ −(m − 1)R for some constant R ≥ 0, and d denotes the diameter of M. Then there exist constants C1 (m), C 2 (m) > 0 depending on m alone, such that the first nonzero Neumann eigenvalue of M satisfies √ C1 λ1 ≥ 2 exp(−C2 d R). d Proof In view of the proof of Theorem 5.7, it suffices to show that the maximum value for the functional Q does not occur on the boundary of M. Supposing the contrary that the maximum point for Q is x0 ∈ ∂ M, let us denote the outward pointing unit normal vector by em , and assume that {e1 , . . . , em−1 } are orthonormal tangent vectors to ∂ M. Since Q satisfies the differential inequality (5.15), the strong maximum principle implies that em (Q)(x 0 ) > 0. Using the Neumann boundary condition on u, we conclude that em v = 0. Moreover, since the second covariant derivative of v is defined by

 v ji = ei e j − ∇ei e j v,

54

Geometric Analysis

we have em (Q) = 2

m 

(ei v)(em ei v)

i=1

=2

m−1 

(eα v)(vαm + ∇em eα v)

α=1

=2

m−1 

 (eα v) vmα + ∇em eα v

α=1

=2

m−1 

 (eα v) eα em v − ∇eα em v + ∇em eα v .

α=1

Using em v = 0 again and the fact that the second fundamental form is defined by h αβ = ∇eα em , eβ , we have em (Q) = −2

m−1 

(eα v)h αβ (eβ v) + 2

α,β=1

≤2

m−1 

m−1 

(eα v)∇em eα , eβ (eβ v)

α,β=1

(eα v)∇em eα , eβ (eβ v).

α,β=1

On the other hand, since ∇em eα , eβ  = −∇em eβ , eα , we conclude that 2

m−1 

(eα v)∇em eα , eβ (eβ v) = −2

α,β=1

m−1 

(eα v)∇em eβ , eα (eβ v).

α,β=1

Hence em (Q) ≤ 0, which is a contradiction.



5 Poincar´e inequality and the first eigenvalue

55

When M is a complete manifold, it is often useful to have a lower bound of the first eigenvalue for the Dirichlet Laplacian on a geodesic ball. This is provided by the next theorem. Theorem 5.9 (Li–Schoen [LS]) Let M be a complete manifold of dimension m. Let p ∈ M be a fixed point such that B p (2ρ) ∩ ∂ M = ∅ for 2ρ ≤ d. Assume that the Ricci curvature on B p (2ρ) satisfies Ri j ≥ −(m − 1)R for some constant R ≥ 0. For any α ≥ 1, there exist constants C1 (α), C2 (m, α) > 0, such that for any compactly supported function f on B p (ρ) √ |∇ f |α ≥ C1 ρ −α exp(−C2 (1 + ρ R)) | f |α . B p (ρ)

B p (ρ)

In particular, the first Dirichlet eigenvalue of B p (ρ) satisfies √ μ1 ≥ C1 ρ −2 exp(−C2 (1 + ρ R)). Proof Let q ∈ ∂ B p (2ρ). By the triangle inequality B p (ρ) ⊂ (Bq (3ρ) \ Bq (ρ)). Theorem 4.1 implies that √ √ r ≤ (m − 1) R coth(r R) √ ≤ (m − 1)(r −1 + R) for r (x) = r (q, x). For k > m − 2, we have r −k = −kr −k−1 r + k(k + 1)r −k−2 √ ≥ −k(m − 1)r −k−1 (r −1 + R) + k(k + 1)r −k−2 √ = kr −k−1 ((k + 2 − m)r −1 − (m − 1) R) √ ≥ kr −k−1 ((k + 2 − m)(3ρ)−1 − (m − 1) R) √ on B p (ρ). Choosing k = m − 1 + 3(m − 1)ρ R this becomes r −k ≥ kr −k−1 (3ρ)−1 ≥ k(3ρ)−k−2 on B p (ρ).

(5.16)

56

Geometric Analysis

Let f be a nonnegative function supported on B p (ρ). Multiplying (5.16) with f and integrating over B p (ρ) yields k(3ρ)−k−2 f ≤ f r −k B p (ρ)

B p (ρ)

=−

B p (ρ)

≤k

B p (ρ)

≤ kρ −k−1 implying

B p (ρ)

|∇ f | ≥ C1 ρ

−1

∇ f, ∇r −k 

|∇ f |r −k−1 B p (ρ)

√

|∇ f |,

exp(−C2 (1 + ρ R))

B p (ρ)

f.

The case when α = 1 is shown by simply applying the above inequality to | f |. For α > 1, we set f = |g|α . Then we have

1/α

(α−1)/α α

B p (ρ)

|∇g|α

≥α

B p (ρ)

|g|

B p (ρ) α−1

|g|α

|∇g| =

B p (ρ)

|∇g α |

√ ≥ C1 ρ −1 exp(−C2 (1 + ρ R)) which implies the desired inequality.

B p (ρ)

|g|α , 

6 Gradient estimate and Harnack inequality

In this chapter we discuss an important estimate that is essential to the study of harmonic functions as well as many elliptic and parabolic problems. In 1975, Yau [Y1] developed a maximum principle method to prove that complete manifolds with nonnegative Ricci curvature must have a Liouville property. His argument was later localized in his paper with Cheng [CgY] and resulted in a gradient estimate for a rather general class of elliptic equations. In 1979, the maximum principle method was used by Li [L1] in proving eigenvalue estimates for compact manifolds. This method (presented in Chapter 5) was then refined and used by many authors ([LY1], [ZY], etc.) for obtaining sharp eigenvalue estimates. In 1986, Li and Yau [LY2] used a similar philosophy to prove a parabolic version of the gradient estimate for the parabolic Schr¨odinger equation. This method has since been used by Hamilton, Chow, Cao, and many other authors to yield estimates for various nonlinear parabolic equations. In [LW6] Li and Wang realized that Yau’s gradient estimate is sharp and equality is achieved on a manifold with negative Ricci curvature. The sharpness of this was somewhat surprising since the parabolic gradient estimate is sharp on a manifold with nonnegative Ricci curvature. In the following treatment, we will present a proof for the sharp gradient estimate for positive functions satisfying the equation  f = −λ f, where λ ≥ 0 is a constant. The argument will give both the local and global estimates. Various immediate consequences of the gradient estimate will also be derived.

57

58

Geometric Analysis

Theorem 6.1 Let M m be a complete Riemannian manifold of dimension m. Assume that the geodesic ball B p (2ρ) ∩ ∂ M = ∅. Suppose that the Ricci curvature on B p (2ρ) is bounded from below by Ri j ≥ −(m − 1)R for some constant R ≥ 0. If u is a positive function defined on B p (2ρ) ⊂ M satisfying u = −λ u for some constant λ ≥ 0, then there exists a constant C depending on m such that (4(m − 1)2 + 2)R |∇u|2 (x) ≤ + C((1 +  −1 )ρ −2 + λ) u2 4 − 2 for all x ∈ B p (ρ) and for any  < 2. Moreover, if ∂ M = ∅ with Ri j ≥ −(m − 1)R everywhere and u is defined on M, then |∇u|2 (m − 1)2 R −λ+ (x) ≤ 2 u2

*

(m − 1)4 R 2 − (m − 1)2 λ R 4

and λ≤ Proof

If we set v = log u and Q = |∇v|2 , then Lemma 5.6 asserts that

Q − ≥

(m − 1)2 R . 4

m |∇ Q|2 Q −1 + ∇v, ∇ Q Q −1 2(m − 1)

2 Q2 + m−1



4λ − 2(m − 1)R m−1



 Q+

2(m − 2) 2λ Q− m−1 m−1

2λ2 . m−1



6

Gradient estimate and Harnack inequality

59

Let φ be a nonnegative cutoff function and G = φ Q, then we have G = (φ) Q + 2∇φ, ∇ Q + φ Q ≥

m φ G + 2φ −1 ∇φ, ∇G − 2|∇φ|2 φ −2 G + |∇G|2 G −1 φ 2(m − 1) m + |∇φ|2 φ −2 G 2(m − 1)   2λ m −1 −1 2(m − 2) φ ∇φ, ∇G − ∇v, ∇G Q Q− − (m − 1) m−1 m−1   2(m − 2) 2λ + ∇v, ∇φ Q− m−1 m−1   2 4λ 2λ2 + φ −1 G 2 + − 2(m − 1)R G + φ. (6.1) m−1 m−1 m−1

Using the inequality 1

1

|∇v, ∇φ| ≤ |∇φ| φ − 2 G 2 , (6.1) can be written as G ≥

3m − 4 m − 2 −1 φ G+ φ ∇φ, ∇G − |∇φ|2 φ −2 G φ m−1 2(m − 1)   4λ m + |∇G|2 G −1 + − 2(m − 1)R G 2(m − 1) m−1 2(m − 2) ∇v, ∇G m−1 2λ 2(m − 2) |∇φ| φ −3/2 G 3/2 + ∇v, ∇G − m−1 (m − 1)Q

−

−

2λ |∇φ| φ −1/2 G 1/2 m−1

+

2 2λ2 φ −1 G 2 + φ. m−1 m−1

However, at the maximum point x 0 of G, the maximum principle asserts that G(x0 ) ≤ 0 and ∇G(x0 ) = 0.

60

Geometric Analysis

Hence at x0 , we have 0 ≥ (m − 1)(φ) G −

  3m − 4 |∇φ|2 φ −1 G + 4λ − 2(m − 1)2 R φ G 2

− 2(m − 2)|∇φ| φ −1/2 G 3/2 − 2λ |∇φ| φ 1/2 G 1/2 + 2G 2 + 2λ2 φ 2 . (6.2) Let us choose φ(x) = φ(r (x)) to be a function of the distance r to the fixed point p with the property that B p (ρ),

φ = 1 on φ = 0 on

M \ B p (2ρ),

1

−Cρ −1 φ 2 ≤ φ  ≤ 0 on

B p (2ρ)\ B p (ρ),

and |φ  | ≤ C ρ −2

B p (2ρ)\ B p (ρ).

on

Then the Laplacian comparison theorem asserts that φ = φ  r + φ  √ ≥ −C 1 (ρ −1 R + ρ −2 ), and also |∇φ|2 φ −1 ≤ C2 ρ −2 . Hence (6.2) yields

√ 0 ≥ −(C3 ρ −1 R + C4 ρ −2 ) G + (4 λ φ − 2(m − 1)2 R φ) G 3

1

− C5 ρ −1 G 2 − C 6 λ ρ −1 G 2 + 2G 2 + 2λ2 φ 2 .

(6.3)

We observe that since x0 is the maximum point of G and φ = 1 on B p (ρ), φ(x0 ) |∇v|2 (x0 ) ≥ sup |∇v|2 (x). B p (ρ)

Using the fact that φ(x0 ) |∇v|2 (x0 ) ≤ φ(x0 ) sup |∇v|2 (x), B p (2ρ)

we conclude that σ (ρ) ≤ φ(x0 ) ≤ 1,

6

Gradient estimate and Harnack inequality

61

where σ (ρ) denotes σ (ρ) =

sup B p (ρ) |∇v|2 (x) sup B p (2ρ) |∇v|2 (x)

.

Applying this estimate to (6.3), we have √ 0 ≥ −(C3 ρ −1 R + C 4 ρ −2 − 4 λ σ (ρ) + 2(m − 1)2 R) G − C 5 ρ −1 G 3/2 − C 6 λ ρ −1 G 1/2 + 2G 2 + 2λ2 σ 2 (ρ).

(6.4)

On the other hand, Schwarz inequality asserts that

and

−C5 ρ −1 G 3/2 ≥ − G 2 −

C 52 −1 −2  ρ G, 4

−C6 λ ρ −1 G 1/2 ≥ − λ2 −

C62 −1 −2  ρ G, 4

√ −C3 ρ −1 R ≥ − R − C7  −1 ρ −2

for  > 0. Hence combining this with (6.4) we obtain 0 ≥ −(C 8 (1 +  −1 ) ρ −2 − 4 λ σ (ρ) + (2(m − 1)2 + )R) G + (2 − )G 2 + 2λ2 σ 2 (ρ) −  λ2 . In particular, if  < 2, then for any x ∈ B p (ρ) we conclude that |∇v|2 (x) ≤ G(x 0 ) " B + B 2 − 4(2 − ) λ2 (2σ 2 (ρ) − ) , ≤ 4 − 2

(6.5)

where B = C8 (1 +  −1 ) ρ −2 − 4 λ σ (ρ) + (2(m − 1)2 + )R. Since 0 ≤ σ (ρ) ≤ 1, we conclude that (4(m − 1)2 + 2)R |∇u|2 (x) ≤ + C((1 +  −1 )ρ −2 + λ) u2 4 − 2 for all x ∈ B p (ρ) and for all  < 2. If u is defined on M, then it follows that |∇v|2 is a bounded function. In particular, if we take ρ → ∞ in (6.5), then σ (ρ) → 1

62

Geometric Analysis

and (6.5) becomes |∇v|2 ≤

(2(m − 1)2 + )R − 4λ 4 − 2 " ((2(m − 1)2 + )R − 4λ)2 − 4(2 − )2 λ2 . + 4 − 2

Letting  → 0, we obtain (m − 1)2 R |∇v| ≤ −λ+ 2 2

*

(m − 1)4 R 2 − λ (m − 1)2 R. 4

This gives a contradiction if λ > (m − 1)2 R/4, and the second assertion follows.  The following Harnack type inequality is a direct consequence of the gradient estimate. Corollary 6.2 Let M m be a complete manifold with Ricci curvature bounded from below by Ri j ≥ −(m − 1)R for some constant R ≥ 0. If u is a positive function defined on the geodesic ball B p (2ρ) ⊂ M satisfying u = −λ u for some constant λ ≥ 0, then there exists constants C 9 , C10 > 0 depending on m such that u(x) ≤ u(y) C9 exp(C10 ρ

√

R + λ)

for all x, y ∈ B p (ρ/2). Proof Let γ be the shortest curve in B p (ρ) joining y to x, and clearly the length of γ is at most 2ρ. Integrating the quantity |∇ log u| along γ yields log u(x) − log u(y) ≤

γ

|∇ log u|.

(6.6)

6

Gradient estimate and Harnack inequality

63

On the other hand, applying the gradient estimate of Theorem 6.1, we obtain 

γ

1/2 (4(m − 1)2 + 2)R + C((1 +  −1 )ρ −2 + λ) 4 − 2 γ   √ C10 R + λ + C12 ρ −1 ≤

|∇ log u| ≤

γ

≤ C 10 ρ

√

R + λ + 2C12 .

The corollary follows by combining this inequality with (6.6).



An upper bound for the infimum of the spectrum that is a consequence of the comparison theorem of Cheng (Theorem 4.6) can also be recovered using the above estimate. Definition 6.3 Let M m be a complete Riemannian manifold without boundary. We denote the greatest lower bound for the L 2 -spectrum of the Laplacian  by μ1 (M). The notation μ1 (M) does not necessarily mean that it is an eigenvalue of , but is motivated by the characterization of μ1 (M) by μ1 (M) = lim μ1 (Di ) i→∞

for any compact exhaustion {Di } of M. Corollary 6.4 (Cheng [Cg1]) Let M m be a complete Riemannian manifold without boundary. Suppose the Ricci curvature of M is bounded from below by Ri j ≥ −(m − 1), then the greatest lower bound of the L 2 -spectrum of the Laplacian has an upper bound given by μ1 (M) ≤

(m − 1)2 . 4

To prove Cheng’s theorem, we need the following lemma, which was first proved by Fischer-Colbrie and Schoen [FCS] for general operators of the form L =  + V (x), where V (x) is a smooth potential function. We will provide the proof for the case V is a constant λ as an application of Theorem 6.1. However, it is important to point out that a Harnack inequality similar to Theorem 6.1 is valid in a more general setting of L =  + V (x) (see [CgY]), hence the argument provided below will yield the result of Fischer-Colbrie and Schoen.

64

Geometric Analysis

We would also like to point out that using the Nash–Moser Harnack inequality of Chapter 19 the potential function V is only required to be in some L p space. Lemma 6.5 (Fischer-Colbrie–Schoen) Let M m be a complete Riemannian manifold, and μ1 (M) be the greatest lower bound of . Then the value μ satisfies μ ≤ μ1 if and only if the operator L =  + μ admits a positive solution of the equation Lu = 0 on M. Proof Let us first assume that there exists a positive function u defined on M such that Lu = 0. For any smooth, compact, subdomain D ⊂ M, let f be the first Dirichlet eigenfunction for the operator L =  + μ, satisfying L f = −μ1 (D, L) f

on

D,

with boundary condition f =0

∂ D.

on

By multiplication by −1 if necessary, we may assume that f is positive in the interior of D. In particular, −μ1 (D, L) u f = u Lf − f Lu D

=

D

∂D

D

u fν .

The positivity of f on D and its boundary condition assert that f ν < 0, hence together with the positivity of u, we conclude that μ1 (D, L) > 0. Since D is arbitrary, this implies that μ1 (M, L) ≥ 0, hence μ1 (M) ≥ μ. Conversely, let us assume that μ1 (M) ≥ μ and μ1 (M, L) ≥ 0. In particular, by monotonicity μ1 (D, L) ≥ 0 for any compact subdomain D ⊂ M. Let us consider a compact exhaustion of M by a sequence of subdomains {Di } with ∞ D = M. On each D , since μ (D , L) > 0, one can find Di ⊂ Di+1 and ∪i=1 i i 1 i a positive solution to the Dirichlet problem L f i = 0 on

Di ,

6

Gradient estimate and Harnack inequality

65

with boundary condition fi = 1

on

∂ Di .

Note that f i must be nonnegative on Di since the subdomain defined by  = { f i < 0} will satisfy μ1 (, L) > 0, hence contradicting the fact that f i is an eigenfunction with eigenvalue 0 on . We claim that f i must be positive in the interior of Di . If not, then there exists x ∈ Di with fi (x) = 0 which is a local minimum and hence a critical point of f i . On the other hand, a regularity result (see [Cg2]) asserts that at x the function f i is asymptotically a spherical harmonic in Rm , contradicting the fact that f i has a local minimum at x. For a fixed point p ∈ M, after multiplication by a constant, we may assume that f i ( p) = 1. Of course, f i still satisfies L fi = 0 but with a different boundary condition. For any fixed ρ > 0, Corollary 6.2 asserts that 0 ≤ f i ≤ C on B p (2ρ), for some constant C > 0 independent of i. Hence a subsequence of { fi } converges uniformly on B p (2ρ). We also consider the nonnegative cutoff function φ supported on B p (2ρ) with the properties that φ = 1 on

B p (ρ),

M \ B p (2ρ),

φ = 0 on and |∇φ|2 ≤ C ρ −2

on

B p (2ρ)\ B p (ρ)

for some fixed constant C > 0 independent of ρ. For sufficiently large i, since B p (2ρ) ⊂ Di we have 0= φ 2 fi L fi M





=−

φ f i ∇φ, ∇ f i  + μ

M

≤− This implies



φ 2 |∇ f i |2 − 2 1 2

M







φ 2 |∇ f i |2 + 2 M

M



M

|∇φ|2 f i2 + μ

M

φ 2 fi2

φ 2 fi2 .

B p (ρ)

|∇ f i |2 ≤

φ 2 |∇ f i |2 M





≤4 M

|∇φ|2 f i2 + 2μ

M

≤ 4C (ρ −2 + μ) V p (2ρ),

φ 2 f i2

66

Geometric Analysis

hence we conclude that f i is bounded in H 1,2 (B p (ρ)). In particular, we choose a subsequence of f i converging uniformly on compact subsets and in H 1,2 to a nonnegative function f . In particular, f is a positive weak solution and by regularity is a strong solution to L f = 0.  Obviously, Corollary 6.4 follows by combining the second part of Theorem 6.1 and Lemma 6.5. Corollary 6.6 (Yau [Y1], Cheng [Cg3]) Let M m be a complete manifold with nonnegative Ricci curvature. There exists a constant C(m) > 0, such that for any harmonic function defined on M if we denote i(ρ) =

inf

x∈B p (ρ)

u(x),

then |∇u|(x) ≤ C ρ −1 (u(x) − i(2ρ)) for all x ∈ B p (ρ). In particular, M does not admit any nonconstant harmonic function satisfying the growth estimate lim inf r −1 (x) u(x) ≥ 0. x→∞

Proof Observe that u − i(2ρ) is a positive harmonic function defined on B p (2ρ). Applying Theorem 6.1 to this function and setting  = 1, we obtain |∇u|2 (x) ≤ 2Cρ −2 (u(x) − i(2ρ))2 for all x ∈ B p (ρ), proving the first part of the corollary. To prove the second part, we observe that the maximum principle asserts that i(ρ) is achieved at some point x ∈ ∂ B p (ρ). In particular, ρ −1 i(ρ) = r −1 (x) u(x) and the growth assumption on u implies that lim ρ −1 i(ρ) ≥ 0.

ρ→∞

Hence for any fixed x ∈ M, we apply the estimate |∇u|(x) ≤ lim C ρ −1 (u(x) − i(2ρ)) ρ→∞

≤0 and conclude that u must be a constant function.



The following example demonstrates the sharpness of Theorem 6.1 even on manifolds with negative curvature.

6

Gradient estimate and Harnack inequality

67

Example 6.7 Let M m = Hm be the hyperbolic m-space with constant curvature −1. Using the upper half-space model, Hm is given by Rm + = {(x 1 , x2 , . . . , xm ) | xm > 0} with metric   ds 2 = xm−2 d x12 + · · · + d x m2 . For x = (x1 , . . . , xm ) let us consider the function u(x) = xmα . Direct computation yields  m   ∂u 2 2 2 |∇u| = xm ∂ xi i=1

= α u2 2

and u = xm2

m  ∂ 2u i=1

∂ xi2

− (m − 2)x m

∂u ∂ xm

= α(α + 1 − m)u. In particular, for (m − 1)/2 ≤ α ≤ m − 1, we see that u = −λ u with λ = α(m − 1 − α). This implies that the gradient estimate of Theorem 6.1 is sharp since * (m − 1)2 (m − 1)4 −λ+ − (m − 1)2 λ = α 2 . 2 4

7 Mean value inequality

We will prove a version of the mean value inequality which is adapted to the theory of subharmonic functions on a Riemannian manifold. Let us begin by proving a theorem of Yau [Y2]. Lemma 7.1 (Yau) Let M be a complete Riemannian manifold. Suppose p ∈ M and ρ4 > 0 are such that the geodesic ball B p (ρ4 ) centered at p of radius ρ4 satisfies B p (ρ4 ) ∩ ∂ M = ∅. Let f be a nonnegative subharmonic function defined on B p (ρ4 ). Then for any constant α > 1 and for any 0 ≤ ρ1 < ρ2 < ρ3 < ρ4 , we have the estimates B p (ρ3 )\B p (ρ2 )

f

α−2

4 |∇ f | ≤ (α − 1)2 2

(ρ2 − ρ1 )−2

B p (ρ2 )\B p (ρ1 )

+(ρ4 − ρ3 )−2

fα

B p (ρ4 )\B p (ρ3 )

fα

and B p (ρ3 )

f α−2 |∇ f |2 ≤

4 (ρ4 − ρ3 )−2 (α − 1)2

B p (ρ4 )\B p (ρ3 )

f α.

In particular, if M has no boundary, then a nonconstant, nonnegative, L α subharmonic function does not exist.

68

7 Mean value inequality Proof

69

Let φ(r (x)) be a cutoff function defined by ⎧ 0 for r ≤ ρ1 , ⎪ ⎪ ⎪ ⎪ r − ρ ⎪ 1 ⎪ ⎪ for ρ1 ≤ r ≤ ρ2 , ⎪ ⎪ ⎪ ⎨ ρ2 − ρ1 φ(r ) = 1 for ρ2 ≤ r ≤ ρ3 , ⎪ ⎪ ⎪ ρ4 − r ⎪ ⎪ for ρ3 ≤ r ≤ ρ4 , ⎪ ⎪ ρ4 − ρ3 ⎪ ⎪ ⎪ ⎩ 0 for ρ4 ≤ r.

We now consider the integral 0≤ φ 2 f α−1  f B p (ρ4 )

= −(α − 1)



2 α−2

B p (ρ4 )

φ f

|∇ f | − 2 2

B p (ρ4 )

φ f α−1 ∇φ, ∇ f .

After applying the algebraic inequality       α−1 2 φf ∇φ, ∇ f   B p (ρ4 )  2 α−1 2 α−2 2 φ f |∇ f | + |∇φ|2 f α , ≤ 2 α − 1 B p (ρ4 ) B p (ρ4 ) we have α−1 α−1 α−2 2 f |∇ f | ≤ φ 2 f α−2 |∇ f |2 2 2 B p (ρ3 )\B p (ρ2 ) B p (ρ4 ) 2 ≤ |∇φ|2 f α α − 1 B p (ρ4 ) 2 ≤ fα (ρ2 − ρ1 )2 (α − 1) B p (ρ2 )\B p (ρ1 ) 2 + f α. (ρ4 − ρ3 )2 (α − 1) B p (ρ4 )\B p (ρ3 ) The second inequality follows similarly by taking φ to be ⎧ for r ≤ ρ3 , ⎪ ⎪1 ⎪ ⎨ ρ4 − r for ρ3 ≤ r ≤ ρ4 , φ(r ) = ρ4 − ρ3 ⎪ ⎪ ⎪ ⎩ 0 for ρ4 ≤ r.

70

Geometric Analysis

If M has no boundary, we simply set ρ3 = ρ, ρ4 = 2ρ and let ρ → ∞. The fact that f is L α implies that f α−2 |∇ f |2 = 0, M

hence |∇ f | ≡ 0, and f must be a constant function.



Theorem 7.2 (Li–Schoen [LS]) Let M be a complete Riemannian manifold of dimension m. Let p ∈ M be a fixed point such that the geodesic ball B p (4ρ) centered at p of radius 4ρ satisfies B p (4ρ) ∩ ∂ M = ∅. Suppose f is a nonnegative subharmonic function defined on B p (4ρ). Assume that the Ricci curvature on B p (4ρ) is bounded by Ri j ≥ − (m − 1)R for some constant R ≥ 0. Then there exist constants C 3 , C 4 (m) > 0 with C 4 depending only on m such that √ 2 −1 f 2. sup f ≤ C 3 (1 + exp(C4 ρ R))V p (4ρ) B p (4ρ)

x∈B p (ρ)

Proof Let h be a harmonic function on B p (2ρ) obtained by the solving the Dirichlet boundary problem h = 0

B p (2ρ)

on

and h= f

∂ B p (2ρ).

on

Since f is nonnegative, the maximum principle implies that h is positive on the ball B p (ρ) and f ≤h

on

B p (ρ).

The Harnack inequality (Corollary 6.2) implies that   √ sup h ≤ inf h exp(C(1 + ρ R)). B p (ρ)

B p (ρ)

Hence, in particular, we have sup f 2 ≤ sup h 2

B p (ρ)

B p (ρ)

√

≤ exp(2C(1 + ρ R))V p (ρ)

−1

B p (ρ)

h2.

(7.1)

7 Mean value inequality

71

We will now estimate the L 2 -norm of h in terms of the L 2 -norm of f . By the triangle inequality, we observe that h2 ≤ 2 (h − f )2 + 2 f2 B p (ρ)

B p (ρ)

B p (ρ)

≤2

B p (2ρ)

(h − f )2 + 2

B p (4ρ)

f 2.

(7.2)

However, since the function h − f vanishes on ∂ B p (2ρ), the Poincar´e inequality (Theorem 5.9) implies that √ (h − f )2 ≤ C 1 ρ 2 exp(C 2 (1 + ρ R)) |∇(h − f )|2 (7.3) B p (2ρ)

B p (2ρ)

for some constants C1 > 0 and C2 (m) > 0. Using the triangle inequality again, we have |∇(h − f )|2 ≤ 2 |∇h|2 + 2 |∇ f |2 . B p (2ρ)

B p (2ρ)

B p (2ρ)

The fact that a harmonic function has the least Dirichlet integral of all functions with the same boundary data asserts that |∇(h − f )|2 ≤ 4 |∇ f |2 . B p (2ρ)

B p (2ρ)

Now the argument in Lemma 7.1 implies that |∇ f |2 ≤ Cρ −2 B p (2ρ)

B p (4ρ)

f2

for some constant C > 0. Taking this together with (7.1), (7.2), (7.3), and the volume comparison (Corollary 2.4), the theorem follows.  Let us point out that the constant in the mean value inequality depends only on the lower bound of the Ricci curvature and the radius of the ball is essential in some of the geometric applications. In fact, it is well known that one can prove another version of the mean value inequality by using an iteration method of Moser. This will be presented in Chapter 18 and the difference of these two methods will also be pointed out later. We will now give an application of this mean value inequality to the study of the space of harmonic functions on a certain class of manifolds. This result can be viewed as a generalization of Yau’s Liouville theorem. Let us first prove a lemma that is useful in estimating the dimension of a linear space.

72

Geometric Analysis

Lemma 7.3 (Li [L2]) Let H be a finite dimensional space of L 2 functions defined over a set D. If V (D) denotes the volume of the set D, then there exists a function f 0 in H such that f 02 ≤ V (D) sup f 02 . dim H D

D

Proof Let f 1 , . . . , f k be an orthonormal basis for H with respect to the L 2 inner product. Let us consider the function F(x) =

k 

fi2 (x),

i=1

which is well defined under an orthonormal change of basis. Clearly F(x). dim H = D

Now let us consider the subspace H p of H consisting of functions that vanish at a point p ∈ D. The space is clearly of at most codimension 1, otherwise there are f 1 and f 2 in the complement of H p which are linearly independent. This implies that both f 1 ( p) = 0 and f 2 ( p) = 0. However, clearly the linearly combination f 1 ( p) f 2 − f 2 ( p) f 1 is a function in H p , which is a contradiction. This implies that by a change of orthonormal bases, there exist f 0 in the orthogonal complement of H p that has unit L 2 -norm, and satisfies the identity F( p) = f 02 ( p). Hence, in particular, if we choose p ∈ D such that F achieves its maximum, then dim H = F D

≤ V (D)F( p) = V (D) f 02 ( p) = V (D) sup f 02 . D

This proves the lemma.



7 Mean value inequality

73

Theorem 7.4 (Li–Tam [LT6]) Let M be an m-dimensional complete noncompact Riemannian manifold without boundary. Suppose that the Ricci curvature of M is nonnegative on M \ B p (1) for some unit geodesic ball centered at p ∈ M. Let us assume that the lower bound of the Ricci curvature on B p (1) is given by Ri j ≥ −(m − 1)R for some constant R ≥ 0. If we let H (M) be the space of functions spanned by the set of harmonic functions f that has the property that when restricted to each unbounded component of M \ D it is bounded either from above or from below for some compact subset D ⊂ M, then H (M) is of finite dimension. Moreover, there exists a constant C(m, R) > 0 depending only on m and R, such that dim H (M) ≤ C(m, R). Proof

By the definition of H (M), there exists R0 > 1 such that f =

k 

vi ,

i=1

where each vi is bounded from one side on each end of M \ B p (R0 ). Let E be an end of M \ B p (R0 ). If v is a harmonic function defined on M which is positive on E and if x is a point in E with r ( p, x) ≥ 2R0 , then by applying Theorem 6.1 to the ball Bx (r ( p, x)/2) and using the curvature assumption, there is a constant C > 0 independent of v, such that |∇v|(x) ≤ Cr −1 ( p, x)v(x).

(7.4)

Since all the vi are bounded on one side on E, there are constants a1 , . . . , ak and i = ± 1 such that the harmonic functions u i = ai + i vi are positive on E. Hence, by applying (7.4) to u 1 , . . . , u m , we can estimate the gradient of f by |∇ f |(x) ≤

k 

|∇vi |(x)

i=1

=

k 

|∇u i |(x)

i=1

≤ Cr −1 ( p, x)

k  i=1

u i (x).

(7.5)

74

Geometric Analysis

Using the fact that |∇ f | is a subharmonic function on M \ B p (1), the maximum principle implies that for any given δ > 0 |∇ f |(x) − sup |∇ f | ≤ Cδ

m 

∂E

u i (x) + 1

i=1

for all x ∈ E. Letting δ → 0, we conclude that sup |∇ f | ≤ sup |∇ f |. E

∂E

Since E is an arbitrary end of M \ B p (R0 ), we have sup

M\B p (R0 )

|∇ f | ≤ sup |∇ f |. ∂ B p (R0 )

(7.6)

In fact, sup |∇ f | ≤ sup |∇ f |

M\B p (1)

∂ B p (1)

after applying the maximum principle to the subharmonic function |∇ f | on the set M \ B p (1) and (7.6). In particular, this implies that sup |∇ f | ≤ sup |∇ f |. M

B p (1)

(7.7)

Let us now consider the codimension-1 subspace Hp (M) of H (M) defined by Hp (M) = { f ∈ H (M)| f ( p) = 0}. For any f ∈ Hp (M), the fundamental theorem of calculus implies that sup f 2 ≤ 16 sup |∇ f |2 .

B p (4)

B p (4)

Together with (7.7), we have sup f 2 ≤ 16 sup |∇ f |2 .

B p (4)

B p (1)

7 Mean value inequality

75

Applying the gradient estimate (Theorem 6.1) to the function f + sup B p (2) | f | yields sup f 2 ≤ 16 sup |∇ f |2

B p (4)

B p (1)

≤ C(R + 1) sup

B p (1)

2 f + sup | f | B p (2)

≤ C(R + 1) sup f 2 . B p (2)

However, using this and the mean value inequality (Theorem 7.2) when applied to the nonnegative subharmonic function | f |, we conclude that there exist constants C3 , C4 (m) > 0 such that √ f 2. (7.8) V p (4) sup f 2 ≤ C3 exp(C4 R) B p (4)

B p (4)

On the other hand, Lemma 7.3 implies that for any finite dimensional subspace H of Hp (M), there exists a function f 0 such that f 02 ≤ V p (4) sup f 02 . dim H B p (4)

B p (4)

Hence applying (7.8) to f 0 yields the estimate √ dim H ≤ C3 exp(C 4 R). Since this estimate holds for any finite dimensional subspace H, this implies that √ dim Hp (M) ≤ C 3 exp(C4 R). Therefore, √ dim H (M) ≤ C3 exp(C4 R) + 1 

as was to be proven.

Let us remark that if M has nonnegative Ricci curvature, then (7.7) can be written as sup |∇ f | ≤ |∇ f |( p). M

76

Geometric Analysis

However, since |∇ f | is a subharmonic function on M, the maximum principle implies that |∇ f | must be identically constant. If f is not a constant function, we can apply the Bochner formula to |∇ f | again, and conclude that ∇ f is a parallel vector field. This implies that M must split and that f is a linear growth harmonic function. In particular, f cannot be a positive harmonic function, hence we recover Yau’s theorem.

8 Reilly’s formula and applications

In this chapter we discuss some of the applications of the integral version of Bochner’s formula derived by Reilly [R]. This formula is particularly useful when studying embedded minimal surfaces and surfaces with constant mean curvature. We first point out some standard formulas concerning submanifolds in Rm+1 and Sm+1 . Lemma 8.1 Let {x1 , . . . , x m+1 } be rectangular coordinates of Rm+1 , and let us denote the position vector by X = (x1 , . . . , xm+1 ). If M is a subman− → − → ifold of Rm+1 with the induced metric and if I I and H denote the second fundamental form and the mean curvature vector of M, then − → Hess M X = − I I and − → M X = − H , where Hess M (X ) and  M (X ) are the Hessian of X and the Laplacian of X computed on M. Proof Let us first assume that M m ⊂ N n is a submanifold of an arbitrary m manifold N . Let us choose an adapted orthonormal frame with {ei }i=1 tangenn tial to M and {eν }ν=m+1 normal to M. Then the Hessian of M and the Hessian of N are related by 77

78

Geometric Analysis (Hess N f )i j = (ei e j − ∇ei e j ) f = (Hess M f )i j −

n 

∇ei e j , eν  f ν

ν=m+1

= (Hess M f )i j +

n  − →  I I i j , ν f ν

(8.1)

ν=m+1

for any 1 ≤ i, j ≤ m and for any function f defined on a neighborhood of M. For N = Rm+1 , we apply (8.1) to the position vector X = (x 1 , . . . , xm+1 ) and observe that HessRm+1 X = 0, hence concluding that − → (Hess M X )i j = − I I i j , em+1  X m+1 − → = − I I i j , em+1  em+1 − → = −I I i j , 

and the lemma follows.

Corollary 8.2 A submanifold M of Rm+1 is minimal if and only if the coordinate functions are harmonic. In particular, there are no compact minimal submanifolds in Rm+1 other than points. Lemma 8.3 Let M be an m-dimensional submanifold of the standard unit sphere Sn , then M is minimal if and only if all the coordinate functions of Sn ⊂ Rn+1 are eigenfunctions of M satisfying  M X = −m X. Proof By Lemma 8.1, and using the fact that the position vector X is also the unit normal vector on Sn , we have − → (HessSn X )i j = − I I i j = −δi j X. Applying (8.1) to the pair M ⊂ Sn yields − → (HessSn X )αβ = (Hess M X )αβ + ( I I M )αβ

8 Reilly’s formula and applications

79

for tangent vectors eα and eβ which are tangential to M. Taking the trace of both sides gives − → −m X =  M X + H M , 

which proves the lemma.

The following integral formula was established by Reilly [R] in his proof of Aleksandrov’s theorem. Theorem 8.4 (Reilly) Let D be a manifold of dimension m + 1 whose boundary is given by a smooth m-dimensional manifold M. Suppose f is a function defined on D satisfying f = g

on

D

and f =u then m m+1

on







g2 ≥ D

M,

M



+

H f ν2 + 2 m  M α,β=1

M

fν  M u

h αβ u α u β +

Ri j f i f j , D

where H and h αβ denote the mean curvature and the second fundamental form of M with respect to the outward unit normal ν,  M is the Laplacian on M, and Ri j is the Ricci curvature of D. Moreover, equality holds if and only if fi j =

g δi j m+1

on D. Proof

Let us consider the Bochner formula 1 2 |∇

f |2 = f i2j + f i f i j j = f i2j + f i ( f )i + Ri j f i f j = f i2j + ∇ f, ∇g + Ri j f i f j .

80

Geometric Analysis

Using the inequality m+1 

 fi2j ≥

i, j=1

=

m+1 i=1 f ii

2

m+1 g2 , m+1

we have 1 2 |∇

f |2 ≥

g2 + ∇ f, ∇g + Ri j f i f j . m+1

Integrating this over D yields 1 1 2 2 |∇ f | ≥ g + ∇ f, ∇g + Ri j f i f j . 2 m+1 D D D D

(8.2)

On the other hand, after integration by parts, the second term on the right-hand side becomes 2 ∇ f, ∇g = − g + g fν , D

D

M

where ν is the outward unit normal to M, hence (8.2) becomes m 2 2 1 |∇ f | ≥ − g + g f + Ri j f i f j . ν 2 m+1 D D M D

(8.3)

If we pick orthonormal frame {e1 , . . . , em+1 } near the boundary of D such that {e1 , . . . , em } are tangential to M, and ν = em+1 is the outward unit normal vector, then the divergence theorem implies that 1 2

|∇ f |2 = D

m+1 

(ei f )(em+1 ei f ).

M i=1

Using the boundary data of f , and choosing ∇em+1 em+1 = 0 at a point, we conclude that m+1  i=1

(ei f )(em+1 ei f ) = (em+1 f )(em+1 em+1 f ) + = (em+1 f )  f −

m 

(eα f )(em+1 eα f )

α=1

m 

f αα +

α=1

= f ν (g − H f ν −  M u) +

m 

(eα f )(em+1 eα f )

α=1 m 

(eα f )(em+1 eα f ), (8.4)

α=1

8 Reilly’s formula and applications

81

where  M is the Laplacian on M and H is the mean curvature of M with respect to the unit normal ν. However, em+1 eα f = eα em+1 f + ∇em+1 eα f − ∇eα em+1 f = eα em+1 f +

m 

∇em+1 eα , eβ  f β −

β=1

m 

∇eα em+1 , eβ  f β , (8.5)

β=1

because ∇em+1 eα , em+1  = −eα , ∇em+1 em+1  =0 and ∇eα em+1 , em+1  = 12 eα |em+1 |2 = 0. Using (8.4), (8.5), and the fact that ∇eα em+1 , eβ  = −em+1 , ∇eα eβ  = h αβ , we can write 1 |∇ f |2 2 D





= M

+

g fν − m 

α,β=1 M

H M

f ν2

− M

fν  M u +

∇em+1 eα , eβ  f α f β −

 m

(eα f )(eα em+1 f )

M α=1 m 

M α,β=1

h αβ u α u β .

On the other hand,

m 

∇em+1 eα , eβ  f α f β = −

M α,β=1

eα , ∇em+1 eβ  f α f β

M α,β=1

=−

m 

m 

∇em+1 eα , eβ  f α f β

M α,β=1

(8.6)

82

Geometric Analysis

implies that both sides are identically 0. Also integrating by parts yields  m (eα f )(eα em+1 f ) = − f ν  M u. M α=1

M

Combining this with (8.3) and (8.6), we have 2 − H fν − 2 fν  M u − M

≥−

M

m m+1



g2 + D

m  M α,β=1

h αβ u α u β

Ri j f i f j , D

which was to be proved. The equality case is clear from the Above argument.  Theorem 8.5 (Aleksandrov–Reilly) Any compact embedded hypersurface of constant mean curvature in Rm+1 is a standard sphere. Proof Let M m be a compact embedded hypersurface in Rm+1 with constant mean curvature H . By compactness, we claim that H > 0. Indeed, if p ∈ M is the maximum point for the function |X |2 on M, then the identity  M |X |2 = 2 M X, X  + 2|∇ X |2 and the maximum principle implies that 0 ≥  M X, X  − → = − H , X  at the point p. On the other hand, it is clear that X must be a multiple of the unit normal vector ν at p, hence 0 ≥ −H |X | and H ≥ 0. Since Corollary 8.2 asserts that H = 0, this confirms the claim that H > 0. After scaling, we may assume that H = m. The assumption that M is embedded implies that M must enclose a bounded domain D in Rm+1 . Let us now consider the solution f of the boundary value problem  f = −1

on

D

and f = 0 on

M = ∂ D.

8 Reilly’s formula and applications

83

Applying Theorem 8.4 to f, we have V (D) ≥ m+1

f ν2 .

M

(8.7)

Schwarz’s inequality now implies that 2



f ν2

A(M) M

≥ 

M

=

fν 2 f

D

= V 2 (D), where A(M) is the area of M. Therefore, taking this together with (8.7), we obtain the inequality A(M) ≥ (m + 1)V (D).

(8.8)

On the other hand, Lemma 8.1 asserts that  M X = −m ν, hence X, X 

0= D





=−

|∇ X | + 2

D

M

= −(m + 1)V (D) − = −(m + 1)V (D) +

X, X ν  1 m 1 m

X,  M X 

M

|∇ M X |2 M

= −(m + 1)V (D) + A(M), where we have used the fact that |∇ X |2 = m + 1 and |∇ M X |2 = m. This implies that (8.8) is in fact an equality, hence all the inequalities that were used to derive (8.8) are in fact equalities. In particular, fi j = −

δi j m+1

on

D,

(8.9)

and f m+1 must be identically constant on M. Let us denote the maximum point of f in D by p. Since p is a critical point, (8.9) implies that it is an isolated

84

Geometric Analysis

maximum. On the other hand, by choosing p to be the origin of Rm+1 , the function g=−

|X |2 2(m + 1)

has a maximum at p and its Hessian is given by δi j . m+1

gi j = −

In particular, this implies that f − g must be a constant function on D. Since f = 0 on M, we conclude M must be the sphere given by the level set of |X |2 . In particular, M must be the standard sphere since the mean curvature of M is m.  Theorem 8.6 (Choi–Wang [CW]) Let M m be a compact, embedded, oriented minimal hypersurface in a compact, oriented Riemannian manifold N m+1 . Suppose that the Ricci curvature of N is bounded from below by Ri j ≥ m R for some constant R > 0. Then the first nonzero eigenvalue of M has a lower bound given by λ1 (M) ≥

mR . 2

Proof The assumption that N has positive Ricci curvature implies that its first homology group H 1 (N , R) is trivial. By an exact sequence argument, we conclude that M divides N into two connected components N1 and N2 with ∂ N1 = M = ∂ N2 . Let D be one of the components to be chosen later. If u is the first nonconstant eigenfunction on M satisfying  M u = −λ1 (M)u, then let f be the solution of  f = 0 on

D

with boundary condition f =u

on

M.

Applying Theorem 8.4, we have u fν + h αβ u α u β + m R |∇ f |2 . 0 ≥ −2λ1 (M) M

M

D

8 Reilly’s formula and applications On the other hand,

85



2 M

u fν = 2

M

=

f fν

( f 2 ) D



=2

|∇ f |2 . D

Hence, we have





(2λ1 (M) − m R)

|∇ f | ≥ 2

D

M

h αβ u α u β .

(8.10)

Let us observe that the right-hand side is independent of the extended function f . If we choose a different component of N \ M to perform this computation, the second fundamental form will differ by a sign, hence we may choose a component, say N1 , so that h αβ u α u β ≥ 0. M

Hence taking this together with (8.10), we conclude that either λ1 (M) ≥ m R/2, or ∇ f = 0 on N1 . However, the latter is impossible because f has a boundary value u which is nonconstant. This proves the estimate. 

9 Isoperimetric inequalities and Sobolev inequalities

In this chapter, we will show that a class of isoperimetric inequalities that occurs in geometry is in fact equivalent to a class of Sobolev type inequalities. The relationship between these inequalities was exploited in the study of eigenvalues of the Laplacian as early as the 1920s by Faber [F] and Krahn [K]. The equivalence was first formally established by Federer and Fleming [FF] (also see [Bm]) in 1960. In 1970, Cheeger [C] observed that the same argument can apply to estimating the first eigenvalue of the Laplacian. We will first define the isoperimetric and Sobolev constants on a manifold. Let us assume that M is a compact Riemannian manifold with or without boundary ∂ M. Definition 9.1 If ∂ M = φ, we define the Dirichlet α-isoperimetric constant of M by I Dα (M) =

inf

⊂M ∂∩∂ M=∅

A(∂) 1

V () α

,

where the infimum is taken over all subdomains  ⊂ M with the property that ∂ is a hypersurface not intersecting ∂ M. Similarly, we define the Neumann α-isoperimetric constant of M. Definition 9.2 The Neumann α-isoperimetric constant of M is defined by I Nα (M) =

inf

∂1 =S=∂2 M=1 ∪S∪2

A(S) , min{V (1 ), V (2 )}1/α

where the infimum is taken over all hypersurfaces S dividing M into two parts, denoted 1 and 2 . Note that in this case there is no assumption on whether M has boundary or not. 86

9

Isoperimetric inequalities and Sobolev inequalities

87

Definition 9.3 If ∂ M = ∅, we define the Dirichlet α-Sobolev constant of M by ! M |∇ f | ! , S Dα (M) = inf α 1/α f ∈H1,1 (M) ( M |f| ) f |∂ M =0

where the infimum is taken over all functions f in the first Sobolev space with Dirichlet boundary condition. We also define the Neumann α-Sobolev constant of M. Definition 9.4 The Neumann α-Sobolev constant of M is defined by ! |∇ f | !M S Nα (M) = inf , f ∈H1,1 (M) (infk∈R M | f − k|α )1/α where the first infimum is taken over all functions f in the first Sobolev space, and the second infimum is taken over all real numbers k. Again, there is no assumption on whether M has boundary or not. Theorem 9.5 For any α > 0, we have I Dα (M) = S Dα (M). Proof To see that I Dα (M) ≤ S Dα (M), it suffices to show that for any Lipschitz function f defined on M with boundary condition f |∂ M ≡ 0 must satisfy 1/α  α |∇ f | ≥ I Dα (M) |f| . M

M

Without loss of generality, we may assume that f ≥ 0. Let us define Mt = {x ∈ M| f (x) > t} to be the sublevel set of f . By the coarea formula, ∞ |∇ f | = A(∂ Mt )dt M

0

≥ I Dα (M)



∞

V (Mt )1/α dt.

(9.1)

0

We now claim that for any s ≥ 0, we have the inequality α  s s V (Mt )1/α dt ≥α t α−1 V (Mt )dt. 0

0

This is obvious for the case s = 0. Differentiating both sides as functions of s, we obtain  s α  s α−1 d 1/α 1/α V (Mt ) dt =α V (Mt ) dt V (Ms )1/α (9.2) ds 0 0

88

Geometric Analysis

and d ds

 α

s

t

α−1

 V (Mt )dt

= αs α−1 V (Ms ).

(9.3)

0

!s Observing that 0 V (Mt )1/α dt ≥ sV (Ms )1/α , because Ms ⊂ Mt for t ≤ s, we conclude that (9.2) is greater than or equal to (9.3). Integrating from 0 to s yields the inequality as claimed. Applying this inequality to (9.1) yields M

 |∇ f | ≥ I Dα (M) α

∞

t α−1 V (Mt )dt

1/α .

0

However, the coarea formula implies that

∞

α



∞

d(t α ) dt 0 ∞ tα =

t α−1 V (Mt )dt =

0

t

∂ Mt

0



∞



∂ Ms

d As dsdt |∇ f |

d At dt |∇ f |

f α.

= M

This proves I Dα (M) ≤ S Dα (M). We will now prove that I Dα (M) ≥ S Dα (M). Let  be a subdomain of M with smooth boundary ∂ such that ∂ ∩ ∂ M = φ. We denote the -neighborhood of ∂ in  by N = {x ∈ |d(x, ∂) < }. Note that for  > 0 sufficiently small, the distance function d(·, ∂) to ∂ is a smooth function on N . Let us define the function f  (x) =

⎧ ⎨0 ⎩

1  d(x, ∂)

1

on on on

M \, N , \ N .

Clearly f  is a Lipschitz function defined on M with Dirichlet boundary condition. Moreover,  1 A(∂ Nt \∂)dt. |∇ f  | = M 0 

9

Isoperimetric inequalities and Sobolev inequalities

89

On the other hand, we have 1/α  α |∇ f  | ≥ S Dα (M) | f | M

M

1

≥ S Dα (M)V (\ N ) α . Hence 1 

0



A(∂ Nt \∂)dt ≥ S Dα (M)V (\ N )1/α ,

and letting  → 0 yields A(∂) ≥ S Dα (M)V ()1/α . Since  is arbitrary, this proves I Dα (M) ≥ S Dα (M).



Theorem 9.6 For any α > 0, we have min{1, 21−1/α } I Nα (M) ≤ S Nα (M) and S Nα (M) ≤ max{1, 21−1/α } I Nα (M). Proof that

Let f be a Lipschitz function defined on M and k ∈ R be chosen such M+ = {x ∈ M| f (x) − k > 0}

and M− = {x ∈ M| f (x) − k < 0} satisfy the conditions that V (M+ ) ≤ 12 V (M) and V (M− ) ≤ 12 V (M). To show that S Nα (M) ≥ min{1, 21−1/α } I Nα (M), it suffices to show that 1/α  1−1/α α |∇u| ≥ min{1, 2 } I Nα (M) |u| M

M

for u = f − k. Note that if Mt = {x ∈ M|u(x) > t},

90

Geometric Analysis

then for t > 0, we have V (Mt ) ≤ V (M+ ) ≤ 12 V (M). This implies that min{V (Mt ), V (M \ Mt ))} = V (Mt ), and A(∂ Mt ) ≥ I Nα (M)V (Mt )1/α . Therefore by the same argument as in the proof of Theorem 9.5, we have 

M+

|∇u| ≥ I Nα (M)

|u|

α

|u|

α

1/α .

M+

Similarly, we also obtain 

M−

|∇u| ≥ I Nα (M)

Hence

+

M

|∇u| ≥ I Nα (M)

|u|α

1/α .

M−



1/α

|u|α

+

M+

M−

 ≥ min{1, 2

1−1/α

1/α ,

} I Nα (M)

α

1/α

|u|

.

M

This proves S Nα (M) ≥ min{1, 21−1/α } I Nα (M). To prove that max{1, 21−1/α }I Nα (M) ≥ S Nα (M), we consider any hypersurface S dividing M into two components denoted by 1 and 2 . Let us assume that V (2 ) ≤ V (1 ). For  > 0 sufficiently small, let us define N = {x ∈ 2 | d(x, S) < } and the function

⎧ ⎪ 1 ⎪ ⎪ ⎨ 1 f  (x) = 1 − d(x, S) ⎪  ⎪ ⎪ ⎩ 0

on

1 ,

on

N ,

on

2 − N .

Let 0 ≤ k ≤ 1 be chosen such that | f  − k |α = inf | f  − k|α . M

k∈R M

9

Isoperimetric inequalities and Sobolev inequalities

91

By using a similar argument to that in the proof of Theorem 9.5, we have |∇ f  | = |∇ f  | M

N



≥ S Nα (M)  ≥ S Nα (M)

α

M

1

1/α

| f  − k |

| f  − k |α +

2\N

| f  − k |α

1/α

1/α ≥ S Nα (M) (1 − k )α V (1 ) + kα V (2 \ N )

1/α ≥ S Nα (M) (1 − k )α + kα V (2 \ N )1/α .

(9.4)

We now observe that (1 − k)α + k α ≥ 21−α for all 0 ≤ k ≤ 1 and α > 1, also (1 − k)α + k α ≥ 1 for all 0 ≤ k ≤ 1 and α ≤ 1. Hence by taking  → 0, the left-hand side of (9.4) tends to A(S), while the right-hand side of (9.4) is bounded from below by S Nα (M) min{1, 2(1−α) /α}V (2 )1/α . This establishes  the inequality max{1, 21−1/α }I Nα (M) ≥ S Nα (M). Let us point out that when the dimension of M is m and α > m/(m − 1), then by the fact that the volume of geodesic balls of radius r behaves like V (r ) ∼

αm−1 m r m

and the area of their boundary is asymptotic to A(r ) ∼ αm−1r m−1 , it is clear that I Dα (M) = 0 = I Nα (M). Hence it is only interesting to consider those α ≤ m/(m − 1). Corollary 9.7 (Cheeger [C]) Let M be a compact Riemannian manifold. If ∂ M = ∅, let μ1 (M) be the first Dirichlet eigenvalue on M and λ1 (M) be its first nonzero Neumann eigenvalue for the Laplacian. When ∂ M = ∅, we will denote the first nonzero eigenvalue of M by λ1 (M) also. Then μ1 (M) ≥

I D1 (M)2 4

λ1 (M) ≥

I N1 (M)2 . 4

and

92 Proof

Geometric Analysis By Theorem 9.5, to see that μ1 (M) ≥

I D1 (M)2 , 4

it suffices to show that any Lipschitz function f with Dirichlet boundary condition must satisfy S D1 (M)2 |∇ f |2 ≥ f 2. 4 M M Applying the definition of S D1 (M) to the function f 2 , we have |∇ f 2 | ≥ S D1 (M) f 2. M

(9.5)

M

On the other hand,

|∇ f 2 | = 2

M

| f ||∇ f | M



1/2 

≤2

1/2 |∇ f |2

f2 M

.

M

Hence, the desired inequality follows from this and (9.5). For the Neumann eigenvalue, we simply observe that if u is the first eigenfunction satisfying u = −λ1 (M)u, then u must change sign. If we let M+ = {x ∈ M|u(x) > 0} and M− = {x ∈ M|u(x) < 0}, then μ1 (M+ ) = λ1 (M) = μ1 (M− ). Let us assume that V (M+ ) ≤ V (M− ). In particular, this implies that I D1 (M+ ) ≥ I N1 (M). Hence by our previous argument, λ1 (M) = μ1 (M+ )

proving the corollary.

≥

I D1 (M+ )2 4

≥

I N1 (M)2 , 4 

9

Isoperimetric inequalities and Sobolev inequalities

93

Corollary 9.8 Let M be a compact Riemannian manifold with boundary. For any function f ∈ H 1,2 (M) and f |∂ M ≡ 0, we have (2−α)/α  2  2−α |∇ f |2 ≥ | f |2α/(2−α) . I Dα (M) 2 M M Proof By applying Theorem 9.5 and the definition of S Dα (M) to the function | f |2/(2−α) , we obtain 1/α  2/(2−α) 2α/(2−α) |∇| f | | ≥ I Dα (M) |f| . M

M

On the other hand, Schwarz’s inequality implies that 2 2/(2−α) |∇| f | |= | f |α/(2−α) |∇ f | 2−α M M  1/2  1/2 2 2α/(2−α) 2 |f| |∇ f | . ≤ 2−α M M 

This proves the corollary.

Corollary 9.9 Let M be a complete Riemannian manifold with or without boundary. There exist constants C1 , C2 > 0 depending only on α, such that |∇ f |2 ≥ C 1 S Nα (M)2 M

 ×

|f|

2α/(2−α)

(2−α)/α

−(2−2α)/α



− C2 V (M)

M

|f|

2

M

for all f ∈ H 1,2 (M). Proof

Let us first observe that a function g satisfies sgn(g) |g|α−1 = 0 M

if and only if





α

|g| = inf

k∈R M

M

|g − k|α .

In particular,  S Nα (M)

|g| M

α

1/α

≤

|∇g|. M

(9.6)

94

Geometric Analysis For any given f ∈ H 1,2 (M), let us choose k ∈ R such that sgn( f − k) | f − k|(2α−2)/(2−α) = 0.

(9.7)

M

Using g = sgn( f − k) | f − k|2/(2−α) , (9.6) implies that 2 2−α

M

| f − k|α/(2−α) |∇ f | ≥ S Nα (M)

1/α

 | f − k|2α/(2−α)

.

M

Applying Schwarz’s inequality as in the proof of Corollary 9.9 yields 

|∇ f | ≥ 2

M

2−α S Nα (M) 2

2  | f − k|

2α/(2−α)

M



≥

2 2−α S Nα (M) 2 (2−α)/α  × 21−α | f |2α/(2−α) − V (M) |k|2α/(2−α) 

≥

(2−α)/α

M

2−α S Nα (M) 2

2 

(2−α)(1−α)/α

 |f|

2

2α/(2−α)

(2−α)/α

M

 − V (M)(2−α)/α |k|2 .

(9.8)

By changing the sign of f if necessary, we may assume k ≥ 0. We can estimate k from above as follows. Let us define M+ = {x ∈ M | f (x) − k ≥ 0} and M− = {x ∈ M | f (x) − k < 0}. The condition (9.7) implies that (2α−2)/(2−α) ( f − k) = M+

However, since (2α−2)/(2−α) ( f − k) ≤ C3 M+

(k − f )(2α−2)/(2−α) . M−

M+

| f |(2α−2)/(2−α) − V (M+ ) k (2α−2)/(2−α)

9 and M−

Isoperimetric inequalities and Sobolev inequalities

(k − f )(2α−2)/(2−α) ≥ C 4 V (M− ) k (2α−2)/(2−α) −



95

| f |(2α−2)/(2−α) , M−

for some constants C3 , C4 > 0 depending on α only, we conclude that | f |(2α−2)/(2−α) − V (M+ ) k (2α−2)/(2−α) C3 M+

≥ C4 V (M− ) k (2α−2)/(2−α) − This implies that





| f |(2α−2)/(2−α) . M−

| f |(2α−2)/(2−α) ≥ V (M) k (2α−2)/(2−α)

C5 M

for some constant C5 depending only on α. Applying the H¨older inequality to the left-hand side, we obtain C6 | f |2 ≥ V (M) k 2 . M

Substituting the above inequality into (9.8) yields the corollary.



10 The heat equation

In this chapter, we will discuss the existence and some basic properties of the fundamental solution of the heat equation (heat kernel). The heat equation on M × (0, T ), for some constant 0 < T ≤ ∞, is given by   ∂ − f (x, t) = 0. ∂t Typically, we consider solving it by prescribing an initial datum f 0 on M, i.e., lim f (x, t) = f 0 (x).

t→0

The fundamental solution for the heat equation is a kernel function H (x, y, t) defined on M × M × (0, ∞) with the property that the function f (x, t) defined by f (x, t) = H (x, y, t) f 0 (y) dy M

solves the heat equation   ∂ f (x, t) = 0 − ∂t

on

M × (0, ∞)

(10.1)

with the initial condition lim f (x, t) = f 0 (x).

t→0

(10.2)

When M is a compact manifold with boundary, the two natural boundary conditions are the Dirichlet and Neumann boundary conditions. For the first case, the solution is required to satisfy f (x, t) = 0 96

on

∂ M × (0, ∞)

(10.3)

10

The heat equation

97

in addition to (10.1) and (10.2). In the case of Neumann boundary condition, the solution should satisfy ∂f (x, t) = 0 ∂ν

on

∂ M × (0, ∞),

(10.4)

where ν is the outward normal to ∂ M. No boundary condition is necessary when M is a compact manifold without boundary. Let us first consider the Dirichlet boundary condition. Elliptic theory asserts that there is a set of eigenvalues {μ1 < μ2 ≤ . . . ≤ μi ≤ . . . } with corresponding eigenfunctions {φi } satisfying φi = −μi φi , such that they form an orthonormal basis with respect to the L 2 -norm. In particular, any function f 0 ∈ L 2 (M) can be written in the form f 0 (x) =

∞ 

ai φi (x)

i=1

with

ai =

f 0 φi d x. M

Formally, the function given by ∞ 

f (x, t) =

e−μi t ai φi (x)

i=1

will satisfy (10.1) with boundary condition (10.3) and initial condition (10.2). This is equivalent to saying that H (x, y, t) f 0 (y) dy, f (x, t) = M

if we define the kernel by H (x, y, t) =

∞ 

e−μi t φi (x) φi (y).

i=1

To justify the above discussion, we first prove a theorem on H (x, y, t). The key estimates follow from that in [L2] where they were derived for the more general setting of differential forms. It is also important to point out that H (x, y, t) is obviously symmetric in the x and y variables.

98

Geometric Analysis

Theorem 10.1 Let M be a compact manifold with boundary. The kernel function H (x, y, t) =

∞ 

e−μi t φi (x) φi (y)

i=1

is well defined on M × M × (0, ∞). It is the unique kernel, such that for any f 0 ∈ L 2 (M), the function given by H (x, y, t) f 0 (y) dy f (x, t) = M

solves the heat equation   ∂ − f (x, t) = 0 ∂t

on

M × (0, ∞)

and f (x, t) = 0

on

∂ M × (0, ∞),

with lim f (x, t) = f 0 (x).

t→0

Moreover, H (x, y, t) is positive on (M \∂ M) × (M \∂ M) × (0, ∞) and it satisfies H (x, y, t) ≤ 1. M

Proof We begin by estimating the supremum norm of an eigenfunction. Let us recall that Sobolev inequality asserts that for m ≥ 3 there exists a constant CSD > 0 depending only on M, such that  (m−2)/m |∇ f |2 ≥ C SD | f |2m/(m−2) (10.5) M

M

for all f ∈ Hc1,2 (M). In fact, Corollary 9.8 implies that CSD ≥ (m − 2)/ 2 (2m − 2) I Dm/(m−1) (M) . When m = 2, the Sobolev inequality can be taken to be  2/ p |∇ f |2 ≥ CSD | f |p M

M

for all f ∈ Hc1,2 (M) and for any fixed 2 ≤ p < ∞. For our purpose, we do not need the precise value of the Sobolev constant CSD , beyond the fact that it only depends on the manifold M.

10

The heat equation

99

Let us assume that m ≥ 3 since the case when m = 2 can be done similarly. For any nonnegative function u ∈ Hc1,2 (M) and for any constant k ≥ 2, integrating by parts and applying the Sobolev inequality gives u k−1 u = −(k − 1) u k−2 |∇u|2 M

M

=− ≤− ≤−

4(k − 1) k2

|∇(u k/2 )|2 M

4(k − 1)C SD k2 2CSD k

 |u|km/(m−2)

(m−2)/m

M

 |u|km/(m−2)

(m−2)/m

.

(10.6)

M

In particular, if u = |φi |, then u satisfies u ≥ −μi u. Hence (10.6) asserts that |φi |k ≥ M

which can be rewritten as 

2CSD k μi

2CSD k μi

 |φi |km/(m−2)

(m−2)/m

,

M

1/k φi kβ ≤ φi k

for all k ≥ 2, with β = m/(m − 2) and φi k denoting the L k -norm of φi . Setting k = 2β j for j = 0, 1, 2, . . . , we have  φi 2β j+1 ≤

β j μi CSD

1/2β j φi 2β j .

Iterating this estimate and using φi 2 = 1, we conclude that φi 2β j+1 ≤

1/2β j   β μi =0

CSD

Letting j → ∞ and applying the fact that lim – φi  p = φi ∞ ,

p→∞



.

100

Geometric Analysis

we obtain m/4

φi ∞ ≤ C 1 μi

,

(10.7)

where C1 > 0 is a constant depending on CSD . We will now estimate μk using k. Let E be the k-dimensional vector space spanned by the first k eigenfunctions {φ1 , φ2 , . . . , φk }. We will show that for any u ∈ E, the estimate (m−1)/2

u∞ ≤ C 2 μk

V (M)(m−1)/m – u2

(10.8)

must hold for some constant C2 > 0 depending only on m and CSD . This estimate and Lemma 7.3, together imply μk ≥ C 3 k 1/(m−1) V (M)−2/m

(10.9)

for some constant C3 depending only on m and CSD . To prove (10.8), we observe that for a C ∞ function u, since |∇u|2 = |∇|u||2 , the identities (u 2 ) = 2u u + 2|∇u|2 and |u|2 = 2|u| |u| + 2|∇|u||2 imply that u u = |u| |u|. Hence applying (10.6) to |u|, the inequality can be written as |u|k−2 u u ≤ − M

2CSD k

 |u|km/(m−2)

(m−2)/m

.

(10.10)

M

We claim that for any u ∈ E and j ≥ 0, we have the estimate – u2β j+1 ≤

1/(2β j  μk V (M)2/m β 

 −1)

– u2 .

C

=0

(10.11)

To see this, for each j ≥ 0, let us choose h ∈ E, such that, h2β j+1 h2

= max u∈E

u2β j+1 u2

.

(10.12)

10

The heat equation

101

Setting u = h and k = 2β j in (10.10), we have  1/β CSD 2β j+1 |h| βj M j ≤− |h|2β −2 hh M



2β j+1

≤

|h|

1/(2β j +1) 

M

2β j

|h|

(2β j −1)/2β j

V (M)(β−1)/(2β

j +1)

.

M

(10.13) To estimate the first term on the right-hand side for h ∈ E, we write h=

k 

a φ ,

=1

hence h = −

k 

μ a φ .

=1

Observe that, for p ≥ 2 and 1 ≤ i ≤ k, the function F(μi ) = |h| p M

is convex as a function of μi because  k p  ∂2 F ∂ 2   = μ a φ      2  ∂μi2 M ∂μi  =1

 p−2  k    = p( p − 1) μ a φ  ai2 φi2   M =1

≥ 0. This implies that for a fixed 1 ≤ i ≤ k, F(μi ) is bounded above by either F(0) or F(μk ). In particular, we conclude that there exists a subset {α} of {1, 2, . . . , k} such that 2β j+1     2β j+1 |h| ≤ μk aα φα  . (10.14)   M M α

102

Geometric Analysis

Since the function



α aα

φα ∈ E, the extremal property of h asserts that

2β j+1

β j+1  −β j+1      2 2β j+1 2 aα φα  ≤ aα |h| h   M α M M α

≤

|h|2β

j+1

.

M

Combining this with (10.13) and (10.14), we have C βj

 |h|2β

j+1

1/β

2β j −1

≤ μk ||h||2β j+1 ||h||2β j

M

V (M)(β−1)/(2β

j +1)

,

hence  – h2β j+1 ≤

μk V (M)2/m β j C

1/(2β j −1) – h2β j .

(10.15)

For j = 0, the extremal property of h implies (10.11) for that case. We now assume that (10.11) is valid for j − 1. To show that (10.11) is valid for j, we consider h satisfying (10.12). The induction hypothesis together with (10.15) yields  – h2β j+1 ≤

μk V (M)2/m β j C

1/(2β j −1) – h2β j

1/(2β j −1) 1/(2β  −1)  j−1  μk V (M)2/m β  μk V (M)2/m β j ≤ – h2 C C =0

=

1/(2β j  μk V (M)2/m β  =0

C

 −1)

– h2 .

Again, the extremal property of h implies (10.11) for any j, upholding the claim. Letting j → ∞, we conclude the validity of inequality (10.8), hence also of (10.9). The convergence of the infinite series ∞  i=1

e−μi t φi (x) φi (y)

10

The heat equation

103

can now be seen by using (10.7) to get e−μi t φi (x) φi (y) ≤ e−μi t φi 2∞ ≤ C1 e−μi t μi

m/2

≤ C1 C(t) e−μi t/2 , where we have used the fact that e−xt x m/2 ≤ C(t) e−xt/2 for some constant C(t) > 0 depending only on t and for all 0 ≤ x < ∞. Applying (10.8), we conclude that ∞ 

e−μi t φi (x) φi (y) ≤ C 1 C(t)

i=1

∞ 

e−C4 i

1/(m−1)

t

,

i=1

which clearly converges uniformly on M × M × [a, ∞) for any a > 0. Hence the kernel H (x, y, t) =

∞ 

e−μi t φi (x) φi (y)

i=1

is well defined and satisfies the Dirichlet boundary condition in both x and y  for t > 0. Moreover, if f0 (x) = i∞= 1 ai φi (x) in L 2 (M), then f (x, t) =

H (x, y, t) f 0 (y) dy M

=

∞ 

e−μi t ai φi (x),

i=1

which obviously satisfies the initial condition lim f (x, t) = f 0 (x).

t→0

We also note that ∇φi , ∇φ j  = δi j μi M

104

Geometric Analysis

implies that the finite sum 2  k    −μi t e φi (x) ∇φi (y) dy   M i=1



k 

=

e−(μi +μ j ) t φi (x) φ j (x) ∇φi (y), ∇φ j (y) dy

M i, j=1

=

k 

e2μi t μi φi (x) φi (x)

i=1

converges as k → ∞ according to our previous argument. Hence the truncated  sum ik = 1 e−μi t φi (x) φi (y) converges to H (x, y, t) weakly in H 1,2 (M). In particular, H (x, y, t) is a weak solution of the heat equation since each truncated sum solves the heat equation. Regularity theory implies that H (x, y, t) is a smooth solution and also f (x, t) solves the Dirichlet heat equation (10.1) and (10.3), with initial condition f0 . ! The strong maximum principle asserts that f (x, t) = M H (x, y, t) f 0 (y) dy is positive on (M \∂ M) × (0, ∞) whenever f 0 ≥ 0 on M. This implies that H (x, y, t) is positive on (M \∂ M) × (M \∂ M) × (0, ∞). If f¯(x, t) is another solution of the heat equation with initial condition f 0 , then the maximum principle implies that the solution of the heat equation f (x, t) − f¯(x, t) must be identically 0 since it vanishes on (M × {0}) ! ∪ (∂ M × (0, ∞)). In particular, H (x, y, t) is unique. The property that M H (x, y, t) dy ≤ 1 follows from the fact that it solves the heat equation with initial condition f 0 = 1 and the maximum principle. The case of m = 2 can be proved in a similar manner using the twodimensional Sobolev inequality.  Note that the same argument can be used to prove the existence of the heat kernel for compact manifolds without boundary, or for compact manifolds with Neumann boundary condition. In either of these cases, the heat kernel is given by K (x, y, t) =

∞ 

e−λi t ψi (x) ψi (y),

i=0

where ψi is the (Neumann) eigenfunction with eigenvalue λi satisfying ψi = −λi ψi .

10

The heat equation

105

1

Since λ0 = 0 and ψ0 = V (M)− 2 , we only need to derive estimates on ψi and λi for i ≥ 1. To see this, we replace the Sobolev inequality by that from Corollary 9.9 setting α = m/(m − 1) when m ≥ 3, and α = 43 when m = 2. It takes the form  (m−2)/m |∇ f |2 ≥ C1 | f |2m/(m−2) − C2 | f |2 , M

M

M

where C 1 and C2 are constants depending only on M. Inequality (10.6) now takes the form (m−2)/m  2C1 2C2 k km/(m−2) −λi u ≤− u + uk . k k M M M with u = |ψ|, hence  ψi 2β j+1 ≤ and

β j λi + C 2 C1

1/2β j ψi 2β j

  m/4 ψi ∞ ≤ C3 λi + C 4 .

We can also estimate λi using a similar method and the existence of K (x, y, t) is established. The positivity of K (x, y, t) can be seen by applying the Hopf boundary point lemma asserting that if (x, t) ∈ ∂ M × (0, ∞) is a maximum point for a solution to the heat equation, f (x, t), then (∂ f /∂ν)(x, t) > 0. In particular, ! if f 0 ≥ 0, then f (x, t) = M K (x, y, t) f 0 (y) dy must have its maximum on M × {0} since ∂ f /∂ν = 0 on ∂ M × (0, ∞), hence f (x, t) ≥ 0. Again this implies that K (x, y, t) is positive on M × M × (0, ∞). The uniqueness of K ! also follows. In particular, M K (x, y, t) dy = 1 because the constant 1 also solves the Neumann heat equation with 1 as the initial condition. We now have the following version of Theorem 10.1 for the Neumann heat kernel. We would also like to point out that in a [WZ], Wang and Zhou found an elegant way to obtain the estimates for eigenfunctions using gradient estimates. In fact, their arguments also apply to eigenforms and yield an estimate on the first nonzero eigenvalues for p-forms. Theorem 10.2 The kernel function K (x, y, t) =

∞  i=0

e−λi t ψi (x) ψi (y)

106

Geometric Analysis

is well defined on M × M × (0, ∞). It is the unique kernel such that for any f 0 ∈ L 2 (M) the function given by f (x, t) = K (x, y, t) f 0 (y) dy M

solves the heat equation   ∂ f (x, t) = 0 − ∂t

on

M × (0, ∞)

and ∂f (x, t) = 0 ∂ν

on

∂ M × (0, ∞),

with lim f (x, t) = f 0 (x).

t→0

Moreover, K (x, y, t) is positive on M × M × (0, ∞) and it has the property that K (x, y, t) = 1. M

We will now establish the fact that the Dirichlet heat kernel has a minimal property among all heat kernels. Definition 10.3 Let M be a manifold with or without boundary. We say that H (x, y, t) is a heat kernel if it is positive, symmetric in the x and y variables, and satisfies the heat equation   ∂ − H (x, y, t) = 0 ∂t with the initial condition lim H (x, y, t) = δx (y),

t→0

where δx (y) denotes the point mass delta function at x. Note that if f (x, t) and g(x, t) are two solutions of the heat equation, then the maximum principle allows us to compare f (x, t) with g(x, t) by considering the difference f (x, t) − g(x, t). However, if K (x, y, t) and H (x, y, t) are heat kernels, it is difficult to compare their values at t = 0. The following proposition gives an effective way of comparing them and it is referred to as the Duhamel principle.

10

The heat equation

107

Proposition 10.4 Let M be a Riemannian manifold with boundary. If H (x, y, t) and K (x, y, t) are two heat kernels, then 



 ∂ H (x, y, t − s) K (y, z, s) K (x, z, t) − H (x, z, t) = ∂ν y 0 ∂M    ∂ H (x, y, t − s) K (x, y, s) dy ds − ∂ν y t

on M × M × (0, ∞), where ∂/∂ν y denotes the outward normal derivative with respect to the y variable. Proof Observe that since the two kernels H (x, y, t) and K (x, y, t) are delta functions at t = 0, it follows that

∂ ∂s

t

0

H (x, y, t − s) K (y, z, s) dy ds = K (x, z, t) − H (x, z, t). M

On the other hand, since they both satisfy the heat equation, we have 0

t

∂ ∂s

H (x, y, t − s) K (y, z, s) dy ds M

=−

0

+

M

t 0

=−

M

t 0

+

t

H (x, y, t − s)  y K (y, z, s) dy ds M

t ∂M

0

+

 y H (x, y, t − s) K (y, z, s) dy ds M

0

=−

 ∂ H (x, y, t − s) K (y, z, s) dy ds ∂(t − s)   ∂ H (x, y, t − s) K (y, z, s) dy ds ∂s

t 

t 0

∂M



 ∂ H (x, y, t − s) K (y, z, s) dy ds ∂ν y   ∂ H (x, y, t − s) K (y, z, s) dy ds. ∂ν y

108 Hence we conclude that

Geometric Analysis

 ∂ H (x, y, t − s) K (y, z, s) dy ds K (x, z, t) − H (x, z, t) = ∂ν y 0 ∂M  t  ∂ H (x, y, t − s) K (y, z, s) dy ds. − 0 ∂ M ∂ν y t



 Corollary 10.5 Let M be a manifold with boundary. The Dirichlet heat kernel H (x, y, t) is minimal among all heat kernels. In particular, the Neumann heat kernel K (x, y, t) dominates H (x, y, t) with K (x, y, t) > H (x, y, t) on M × M × (0, ∞). Proof According to the previous proposition, the boundary condition of H (x, y, t) asserts that  t  ∂ K (x, z, t) − H (x, z, t) = − H (x, y, t − s) K (x, y, s) dy ds. 0 ∂ M ∂ν y Using the facts that K (x, y, s) > 0, H (x, y, t − s) > 0 with y ∈ M \∂ M, and H (x, y, t − s) = 0 with y ∈ ∂ M, we conclude that the right-hand side is positive by the Hopf boundary lemma, this proves the inequality.  We also have the following monotonicity property for the Dirichlet heat kernel. Corollary 10.6 Let 1 and 2 be two compact subdomains of M with the property that 1 ⊂ 2 . Suppose H1 (x, y, t) and H2 (x, y, t) are their corresponding Dirichlet heat kernels. Then H1 (x, y, t) ≤ H2 (x, y, t) on 1 × 1 × (0, ∞).

11 Properties and estimates of the heat kernel

Recall that on Euclidean space Rm , the heat kernel is given by the formula  2  r (x, y) H (x, y, t) = (4πt)−m/2 exp − . 4t Since all manifolds are locally Euclidean, this formula is in fact an approximation of any heat kernel as t → 0. Theorem 11.1 Let M be a complete Riemannian manifold (with or without boundary). Suppose H (x, y, t) is a heat kernel. Then for any x ∈ M \ ∂ M, − m2

H (x, y, t) ∼ (4πt)



r 2 (x, y) exp − 4t



as t → 0 and r (x, y) → 0. Also, for fixed x, y ∈ M \ ∂ M with x = y, we have H (x, y, t) → 0 as t → 0. Proof

For any integer k > 0, let us define the function −m/2

G(x, y, t) = (4πt)



r 2 (x, y) exp − 4t

 k

u i (x, y) t i .

i=0

Taking the Laplacian with respect to the y-variable, we have  2 r G = (4πt)−m/2 exp − 4t

  k k k  1 r2  r r r i i − + 2 ui t − ∇r, ∇u i  t + (u i ) t i , × − 2t 2t t 4t i=0

i=0

i=0

109

110

Geometric Analysis

also

 k  2   k  m r m r2  ∂G −2 i i−1 exp − ui t + i ui t = (4π t) − + 2 . ∂t 4t 2t 4t i=0

i=1

Hence,  −

∂ ∂t



 2 r G = (4π t)−m/2 exp − 4t ⎛  k−1  k−1  (m − 1)  r r + ×⎝ − u i+1 t i − r ∇r, ∇u i+1  t i 2 2 i=−1

+

k  i=0

(u i ) t − i

k−1 

i=−1

(i + 1) u i+1 t i .

i=0

For a fixed x, let y be within the injectivity radius of x, we now choose u i (x, y) as functions of y to satisfy   r r m−1 (11.1) − u 0 + r ∇r, ∇u 0  = 0, 2 2 and then inductively   r r m−1 − u i+1 + r ∇r, ∇u i+1  + (i + 1) u i+1 = u i 2 2

(11.2)

for 0 ≤ i ≤ k − 1. To see that this can be done, we observe that r (x, y) = J −1 (y) (∂ J /∂r )(y), where J (y) is the area element of the sphere of radius r (x, y) at the point y. Equation (11.1) can be written as (m − 1) ∂u 0 1 ∂J + u0 − u 0 = 0, ∂r 2J ∂r 2r and u 0 = C r (m−1)/2 J −1/2 , with the constant C chosen to satisfy u 0 (x, y) = 1, is a solution of (11.1). Equation (11.2) can be written as ∂u i+1 1 ∂J m − 3 − 2i + u i+1 − u i+1 = r −1 u i , ∂r 2J ∂r 2r

(11.3)

11

Properties and estimates of the heat kernel

111

which can be solved by setting u i+1 = r

(m−3−2i)/2

J

−1/2



r

s (2i+1−m)/2 J 2 u i ds. 1

0

In particular, we obtain  2   r ∂ G = (4πt)−m/2 exp − u k t k . − ∂t 4t Note that these computations are valid when y ∈ / ∂ M is within the injectivity radius of x. Let us choose φ(y) to be a cutoff function satisfying # φ(y) =

1 0

on on

Bx (ρ), M \ Bx (2ρ),

where 2ρ is chosen to be less than the injectivity radius at x and the distance from x to ∂ M. For a fixed x ∈ / ∂ M, we now define F(x, y, t) = φ(y) G(x, y, t), which is supported on Bx (2ρ) and it is equal to G(x, y, t) on Bx (ρ). In particular, on Bx (2ρ) \ Bx (ρ),        − ∂ F(x, y, t)   ∂t    2   r −m/2 k  = φ (4π t) exp − u k t + 2∇φ, ∇G + G  φ  4t   2 ρ ≤ C2 t −m/2 exp − . (11.4) 4t Note that since all manifolds are locally Euclidean and because of (11.3), limt→0 G(x, y, t) = δx (y). In particular, ∂ H (z, y, t − s) F(x, z, s) dz ds F(x, y, t) − H (x, y, t) = 0 ∂s M t =− z H (z, y, t − s) F(x, z, s) dz ds

t

0

M



 ∂ F(x, z, s) dz ds ∂s 0 M   t ∂ =− F(x, z, s) dz ds. H (z, y, t − s)  − ∂s 0 M +

t

H (z, y, t − s)

112

Geometric Analysis

Hence using the fact that

! M

H (x, y, t) dy ≤ 1 and (11.4), we have

|H (x, y, t) − F(x, y, t)| ≤ C1



t

s 0



+ C2 ×

k−m/2 Bx (ρ)

t

0

H (z, y, s) dz ds

  ρ2 s −m/2 exp − 4s

Bx (2ρ)\Bx (ρ)

H (z, y, s) dz ds

≤ C3 t k+1−m/2 . If y ∈ / Bx (ρ), since F(x, y, t) = φ(y)G(x, y, t) and G(x, y, t) → 0 as t → 0, we conclude that H (x, y, t) → 0 as t → 0. On the other hand, if y ∈ Bx (ρ), then     2 k   (x, y) r   u i (x, y) t i  ≤ C 3 t k+1−m/2  H (x, y, t) − (4πt)−m/2 exp −   4t i=0



and the theorem follows because of (11.3).

Another important property of the heat kernel is the semi-group property. In ∞ −μ t fact, this follows from the definition of H (x, y, t) = i=1 e i φi (x) φi (y). 2 Using the fact that φi are orthonormal with respect to L , we have  H (x, z, t) H (z, y, s) dz = e−μi t e−μ j s φi (x) φ j (y) φi (z) φ j (z) dz M

M

i, j

=

∞ 

e−μi (t+s) φi (x) φi (y)

i=1

= H (x, y, t + s).

(11.5)

11

Properties and estimates of the heat kernel

113

In particular, H 2 (x, y, t) dy

H (x, x, 2t) =

(11.6)

M

and H (x, y, t + s) ≤ H 1/2 (x, x, 2t) H 1/2 (y, y, 2s).

(11.7)

This can be used to estimate the heat kernel from above. Theorem 11.2 Let M be a Riemannian manifold with boundary. The Dirichlet heat kernel is bounded from above by  H (x, y, t) ≤

2CSD m

−m/2

t −m/2 ,

for all x, y ∈ M and t > 0, where CSD > 0 is the Sobolev constant from (10.5). Moreover, the kth Dirichlet eigenvalue must satisfy the estimate 2e CSD μk ≥ m



k V (M)

2/m

and the kth Dirichlet eigenfunction must satisfy 

φk 2∞ Proof

2CSD ≤e m

−m/2

m/2

μk

||φk ||22 .

Differentiating (11.6), we have ∂ ∂ H (x, y, t) H (x, y, t) dy H (x, x, 2t) = 2 ∂t ∂t M H (x, y, t) H (x, y, t) dy =2 M



= −2

|∇ H (x, y, t)|2 dy.

(11.8)

M

The Sobolev inequality (10.5) asserts that 

M

|∇ H (x, y, t)|2 dy ≥ C SD

|H (x, y, t)|2m/(m−2) dy

(m−2)/m

,

M

(11.9)

114

Geometric Analysis

where CSD > 0 is the Sobolev constant. On the other hand, H¨older inequality implies that  (m−2)/(m+2) 2 2m/(m−2) H (x, y, t) dy ≤ H (x, y, t) dy M

M



4/(m+2)

×

H (x, y, t) dy M

 ≤

H

2m/(m−2)

(x, y, t) dy

(m−2)/(m+2)

.

M

Combining this inequality with (11.8) and (11.9), we have ∂ H (x, x, 2t) ≤ −2C SD H (m+2)/m (x, x, 2t), ∂t implying ∂ −2/m 2CSD (x, x, s) ≥ H . ∂s m Integrating over  ≤ s ≤ t yields H −2/m (x, x, s) − H −2/m (x, x, ) ≥

2CSD (s − ). m

Since Theorem 11.1 asserts that H (x, x, ) ∼ (4π )−m/2 , we conclude that lim H 2/m (x, x, ) = 0,

→0

hence

 H (x, x, s) ≤

2CSD s m

−m/2

.

Inequality (11.7) then completes the first part of the theorem. The second part follows from the fact that ∞  e−μi t = H (x, x, t) d x, i=1

M

hence combining this identity with the estimate from the first part gives   ∞  2C SD −m/2 −m/2 −μi t e ≤ t V (M). m i=1

11

Properties and estimates of the heat kernel

115

Setting t = μ−1 k , we have ke−1 ≤

k 

e−μi /μk

i=1

 ≤

2CSD m

−m/2

m/2

μk

V (M),

yielding the lower bound for μk . We also have e−μk t φk2 (x) ≤ H (x, x, t)   2CSD −m/2 −m/2 t , ≤ m where φk is the kth eigenfunction with ||φk ||2 = 1. Again setting t = μ−1 k yields the estimate  φk 2∞ ≤ e

2CSD m

−m/2

m/2

μk



.

Note that the estimate on μk is an improvement from that obtained in the proof of Theorem 10.1. We also have the corresponding theorem for the Neumann heat kernel. In this case, the Sobolev inequality required is of the form  (m−2)/m |∇ f |2 ≥ C S N | f |2m/(m−2) (11.10) M

M

!

for all f ∈ H 1,2 (M) satisfying M f = 0. Observe that the Sobolev inequality given in Corollary 9.9 implies the above one since the L 2 -norm of f can be estimated by 2 λ1 (M) f ≤ |∇ f |2 . M

M

Theorem 11.3 Let M be a Riemannian manifold with boundary. The Neumann heat kernel is bounded from above and below by 1 − V (M)



2C S N m

−m/2

t −m/2 ≤ K (x, y, t) 1 ≤ + V (M)



2C S N m

−m/2

t −m/2 ,

116

Geometric Analysis

for all x, y ∈ M and t > 0, where C S N > 0 is the Sobolev constant from (11.10). Moreover, the kth Neumann eigenvalue must satisfy the estimate  2/m k 2(m−4)/m eC S N , λk ≥ m V (M) and the kth Neumann eigenfunction must satisfy

−m/2 2(m−4)/m C S N m/2 2 λk ||ψk ||22 . ψk ∞ ≤ e m Proof

We will estimate the function K 1 (x, y, t) = K (x, y, t) − =

∞ 

1 V (M)

e−λi t ψi (x) ψi (y)

i=1

instead, since ψ0 = V − 2 (M). One checks readily that K 1 also satisfies the semi-group property. Moreover, K 1 (x, y, t) dy = 0 1

M

and



|K 1 (x, y, t)| dy ≤ M

K (x, y, t) dy + 1 M

≤ 2. The same argument as in the proof of Theorem 11.2 now works on K 1 and produces

−m/2 2(m−4)/m C S N . s K 1 (x, x, s) ≤ m Hence the above inequality together with the inequality 1/2

1/2

|K 1 (x, y, t)| ≤ K 1 (x, x, t) K 1 (y, y, t) yields the desired estimate on K 1 . The rest of the proof is identical to that of Theorem 11.2.  To summarize, we have proved that Sobolev inequalites of the form (10.6) and (11.10) imply upper bounds of the Dirichlet and the Neumann heat kernel,

11

Properties and estimates of the heat kernel

117

respectively. The converse is also true as we will demonstrate in this chapter. The following theorem was proved by Varopoulos in [V]. Let us first state the Marcinkiewicz interpolation theorem as Lemma 11.5 without proof. The interested reader should refer to Appendix B of [St] for further details. Definition 11.4 Let T be a subadditive operator on L p (M), i.e., |T ( f + g)(x)| ≤ |T ( f )(x)| + |T (g)(x)|, for all f, g ∈ L p (M) and for all x ∈ M. T is of weak ( p, q) type if there exists a constant A, such that for f ∈ L p (M),   A  f p q m{x | |T ( f )(x)| > β} ≤ β for all β > 0. If q = ∞, this means T ( f )∞ ≤ A  f  p for all f ∈ L p (M). Lemma 11.5 Suppose that 1 ≤ pi ≤ qi ≤ ∞, for i = 1, 2 is such that p1 < p2 and q1 = q2 . Assume that T is an operator of weak ( p1 , q1 ) type and weak ( p2 , q2 ) type. Then for any 0 < θ < 1 with 1 θ 1−θ + = p p1 p2

and

1 θ 1−θ + , = q q1 q2

T is a bounded operator from L p (M) to L q (M). Theorem 11.6 Let M be a Riemannian manifold with boundary. Suppose there is a constant A1 > 0, such that the Dirichlet heat kernel satisfies the estimate H (x, y, t) ≤ A1 t −m/2 for all x, y ∈ M and for all t ∈ (0, ∞). Then there exists a constant C1 > 0 depending only on m such that

−2/m

M

|∇ f |2 ≥ C 1 A1

for all f ∈ H 1,2 (M) with f |∂ M = 0.

 | f |2m/(m−2) M

(m−2)/m

118

Geometric Analysis

If there is a constant A2 > 0, such that the Neumann heat kernel satisfies the estimate K (x, y, t) ≤

1 + A2 t −m/2 V (M)

for all x, y ∈ M and for all t ∈ (0, ∞), then there exists a constant C 2 > 0 depending only on m, such that  (m−2)/m −2/m 2 2m/(m−2) |∇ f | ≥ C2 A2 |f| M

for all f ∈ H 1,2 (M) with

M

! M

f = 0.

Proof We will first prove the case for the Dirichlet heat kernel. Let {φi } be the set of orthonormal eigenfunctions with eigenvalues {μi }. For all f ∈ H 1,2 (M) with f |∂ M = 0, we write ∞ 

f =

ai φi

i=1

and hence f = −

∞ 

μi ai φi .

i=1

For any α ∈ R, we define the psuedo-differential operator (−)α by (−)α f =

∞ 

μiα ai φi .

i=1

We can write



|∇ f |2 = − M

f f M

=

∞ 

μi ai2

i=1

=

|(−)1/2 f |2 , M

hence it suffices to show that  (m−2)/m −2/m 1/2 2 2m/(m−2) |(−) f | ≥ C 1 A1 |f| . M

M

11

Properties and estimates of the heat kernel

119

Let g = (−)1/2 f , hence f = (−)−1/2 g. We need to show that the operator (−)−1/2 : L 2 (M) → L 2m/(m−2) (M) is bounded and satisfies the bound (m−2)/m  2/m −1/2 2m/(m−2) |(−) g| ≤ C A1 |g|2 . M

M

Expressing (−)1/2 g in terms of the heat kernel, if g = have ∞ 

(−)−1/2 g(x) =

−1/2

μi

∞

i=1 bi

φi then we

bi φi (x)

i=1

 ∞

=

M i=1



∞

=B

−1/2

μi

0

=B

t −1/2 e−μi t φi (x) φi (y) g(y) dy M

∞

t

−1/2



0

where B −1 = T1 and T2 by

!∞ 0

φi (x) φi (y) g(y) dy

H (x, y, t) g(y) dy, M

s −1/2 e−s ds. For 0 < T < ∞, let us define the functionals

T

T1 (g) = B

t

−1/2

H (x, y, t) g(y) dy dt

0

M

and

∞

T2 (g) = B T

t −1/2

H (x, y, t) g(y) dy dt, M

hence (−)−1/2 = T1 + T2 . For β > 0, obviously the measure of the sublevel set {(−)−1/2 g ≥ β} can be estimated by $ # β m{x | |(−)−1/2 g(x)| > β} ≤ m x | |T1 g(x)| > 2 # $ β + m x | |T2 g(x)| > . 2

120

Geometric Analysis

On the other hand, for 1/ p + 1/q = 1, using the upper bound of H , we have

∞

|T2 g(x)| ≤ B



t −1/2

T



∞

≤ B gq

H (x, y, t) |g(y)| dy dt M

T

1/q

1/ p H p (x, y, t) dt

dt

 t −1/2 sup H ( p−1)/ p (x, y, t)

T

≤ C A1

 M

∞

≤ Bgq

t −1/2

1/ p H (x, y, t) dt

dt

M

y∈M

T (q−m)/2q gq .

For 1 < q < m, by choosing T such that 1/q

C A1

T (q−m)/2q gq =

β , 2

(11.11)

we have $ # β m{x | |(−)−1/2 g(x)| > β} ≤ m x | |T1 g(x)| > 2  −q β q ≤ T1 (g)q . 2

(11.12)

To estimate T1 gq , we consider  |T1 g(x)| ≤ B q

q

T

t

−1/2

0

 ≤ Bq

q

H (x, y, t) |g(y)| dy dt M

T

0

t −1/2 dt

≤ C T q /2 p

T

q/ p

t −1/2

t −1/2

H (x, y, t) |g(y)| dy M

H (x, y, t) |g(y)|q dy dt, M

! M

H (x, y, t) dy ≤ 1. This implies that





T

|T1 g(x)|q d x ≤ C T q/2 p M

q



0



0

where we have used the fact that

T

0 q

= C T q/2 gq .

t −1/2 gq dt q

dt

11

Properties and estimates of the heat kernel

121

Combining this identity with (11.11) and (11.12), we have  −q β q T q/2 gq m{x | |(−)−1/2 g(x)| > β} ≤ C 2 q/(m−q)

≤ C A1

β −mq/(m−q) gq

mq/(m−q)

.

This establishes that the operator (−)−1/2 is of weak (q, mq/(m − q)) type, for 1 < q < m. We now choose 1 < q1 < 2 < q2 < m and 0 < θ < 1, such that, 1 θ 1−θ + , = 2 q1 q2 implying m−2 (1 − θ )(m − q1 ) θ (m − q2 ) + , = 2m mq1 mq2 and the theorem follows from Lemma 11.5. The same argument also works for the Neumann heat kernel ! by defining the operator (−)−1/2 on the subspace of S = {g ∈ L 2 (M) | M g = 0} by ∞  −1/2 (−)−1/2 g(x) = λ1 ψi (x) ψi (y) g(y) dy, M

i=1

where {ψi } is an orthonormal basis of eigenfunctions corresponding to the nonzero eigenvalues {λi }. In this case, we also have ∞ 1 1 t − 2 K 1 (x, y, t) g(y) dy dt. (−)− 2 g(x) = B 0

M

Using the estimate on K 1 , the theorem follows for the Neumann boundary condition. 

12 Gradient estimate and Harnack inequality for the heat equation

In this chapter, we will derive estimates for positive solutions of the heat equation. The gradient estimate given by Theorem 12.2, and the Harnack inequality that follows (Corollary 12.3), have fundamental importance in the study of parabolic equations. There are many further developments of similar types of estimates for other nonlinear partial differential equations. The estimates presented herein were proved by Li and Yau in [LY2]. Lemma 12.1 Let M m be a manifold whose Ricci curvature is bounded by Ri j ≥ −(m − 1)R, for some constant R. Suppose g(x, t) is a smooth function defined on M × [0, ∞) satisfying the differential equation   ∂ g(x, t) = −|∇g|2 (x, t). − ∂t Then for any given α ≥ 1, the function G(x, t) = t (|∇g|2 (x, t) − αgt (x, t)) satisfies the inequality   ∂ G ≥ −2∇g, ∇G − t −1 G + 2α −2 m −1 t −1 G 2 − ∂t + 4(α − 1)m −1 α −2 |∇g|2 G − 2−1 m(m − 1)2 α 2 (α − 1)−2 t R 2 . 122

12

Gradient estimate and Harnack inequality for the heat equation

123

Proof Let {e1 , . . . em } be a set of local orthonormal frame on M. Differentiating in the direction of ei , we have G i = t (2g j g ji − α gti ). Covariant differentiating again, we obtain   G = t 2g 2ji + 2g j g jii − α gtii  ≥t

 2 2 2 (g) + 2∇g, ∇g − 2(m − 1)R|∇g| − α(g)t , (12.1) m

where we have used the inequality 

g 2ji ≥

(



i, j

gii )2 m

i

and the commutation formula g j g jii = g j gii j + Ri j gi g j ≥ ∇g, ∇g − (m − 1)R |∇g|2 . However, since g = −|∇g|2 + gt = −α −1 t −1 G − α −1 (α − 1) |∇g|2 (12.1) becomes G ≥

2tα −2 −1 (t G + (α − 1) |∇g|2 )2 − 2α −1 ∇g, ∇G m − 2α −1 (α − 1)t∇g, ∇|∇g|2  − 2(m − 1)R t|∇g|2 + G t − t −1 G + t (α − 1) (|∇g|2 )t

=

2tα −2 −2 2 (t G + 2(α − 1)t −1 |∇g|2 G + (α − 1)2 |∇g|4 ) m − 2α −1 ∇g, ∇G − 2α −1 (α − 1)∇g, ∇G − 2(m − 1)R t|∇g|2 + G t − t −1 G.

124

Geometric Analysis

The lemma follows by applying the inequality (α − 1)2 α −2 |∇g|4 − m(m − 1)R |∇g|2 ≥ − 4−1 m 2 (m − 1)2 α 2 (α − 1)−2 R 2 .



Theorem 12.2 Let M m be a complete manifold with boundary. Assume that p ∈ M and ρ > 0 so that the geodesic ball B p (4ρ) does not intersect the boundary of M. Suppose the Ricci curvature of M on B p (4ρ) is bounded from below by Ri j ≥ −(m − 1)R, for some constant R ≥ 0. If f (x, t) is a positive solution of the equation   ∂ f (x, t) = 0 − ∂t on M × [0, T ], then for any 1 < α, the function g(x, t) = log f (x, t) must satisfy the estimate |∇g|2 − αgt ≤

m 2 −1 α t + C 1 α 2 (α − 1)−1 (ρ −2 + R) 2

on B p (2ρ) × (0, T ], where C 1 is a constant depending only on m. Proof Let φ(r (x)) be a cutoff function, depending only on the distance to p, defined on B p (4ρ) with the properties . 1 on B p (2ρ), φ(r (x)) = 0 on M \ B p (4ρ), −C ρ −1 ≤ φ − 2 φ  ≤ 0, 1

and φ  ≥ −C ρ −2 , where φ  denotes differentiation with respect to r. For G = t (|∇g|2 − αgt ), we consider the function φ G with support on B p (4ρ) × (0, T ]. Let (x0 , t0 ) ∈ B p (4ρ) × (0, T ] be the maximum point of φ G. We may assume that φ G is

12

Gradient estimate and Harnack inequality for the heat equation

125

positive at (x 0 , t0 ) otherwise the theorem is trivial. Note that at (x0 , t0 ) we have the properties that ∇(φ G) = 0,

(12.2)

∂ (φ G) ≥ 0, ∂t

(12.3)

(φ G) ≤ 0.

(12.4)

and

Also using the comparison theorem and the properties of φ we have  √  φ ≥ −C2 ρ −2 ρ R + 1 . Applying Lemma 12.1 and this estimate to the identity (φ G) = (φ) G + 2∇φ, ∇G + φ (G), we obtain

 √  (φ G) ≥ −C 2 ρ −2 ρ R + 1 G + 2φ −1 ∇φ, ∇(φ G) − 2φ −1 |∇φ|2 G  + φ G t − 2∇g, ∇G − t −1 G + 2α −2 m −1 t −1 G 2 +4(α − 1)m −1 α −2 |∇g|2 G



− 2−1 m(m − 1)2 α 2 (α − 1)−2 tφ R 2 . Evaluating at (x0 , t0 ) and using (12.2), (12.3), and (12.4) yields  √  0 ≥ −C 2 ρ −2 ρ R + 1 G − 2φ −1 |∇φ|2 G + 2G∇φ, ∇g − φt0−1 G + 2α −2 m −1 t0−1 φ G 2 + 4(α − 1)m −1 α −2 |∇g|2 φ G − 2−1 m(m − 1)2 α 2 (α − 1)−2 t0 φ R 2 . Multiplying through by φ t0 and using the estimate for |∇φ|, we conclude that  √  0 ≥ −C 3 ρ −2 ρ R + 1 t0 φ G − φ 2 G − C4 t0 φ 3/2 ρ −1 G |∇g| + 2α −2 m −1 φ 2 G 2 + 4(α − 1)m −1 α −2 t0 |∇g|2 φ 2 G − 2−1 m(m − 1)2 α 2 (α − 1)−2 t02 φ 2 R 2 .

(12.5)

126

Geometric Analysis

Observe that there exists a constant C5 > 0 such that 4m −1 (α − 1)α −2 φ |∇g|2 − C 4 ρ −1 φ 2 |∇g| ≥ −C 5 α 2 (α − 1)−1 ρ −2 . (12.6) 1

Hence combining (12.5) and (12.6), we conclude that  √    0 ≥ 2m −1 α −2 (φ G)2 − C6 ρ −2 t0 ρ R + 1 + α 2 (α − 1)−1 + 1 φ G − 2−1 m(m − 1)2 α 2 (α − 1)−2 t02 R 2 , for some constant C 6 > 0. This implies that at the maximum point (x0 , t0 ), φG ≤

mα 2 2 + C 7 t0

  √   ρ −2 α 2 ρ R + 1 + α 2 (α − 1)−1 + α 2 (α − 1)−1 R .

In particular, since t0 ≤ T, when restricted on B p (2ρ) × {T } we have |∇g|2 − α gt   √   mα 2 + C7 ρ −2 α 2 ρ R + 1 + α 2 (α − 1)−1 + α 2 (α − 1)−1 R . 2T The theorem follows by letting t = T .  ≤

Corollary 12.3 Under the same hypotheses as Theorem 12.2,  mα/2 t2 f (x, t1 ) ≤ f (y, t2 ) t1  2  α r (x, y) −1 −2 × exp + C 1 α (α − 1) (ρ + R) (t2 − t1 ) 4(t2 − t1 ) for any x, y ∈ B p (ρ) and 0 < t1 < t2 ≤ T, where r (x, y) is the geodesic distance between x and y. Proof Let γ : [0, 1] → M be a minimizing geodesic joining y to x. Since x, y ∈ B p (ρ), the triangle inequality asserts that (γ ) = r (x, y) ≤ r (x) + r (y) ≤ 2ρ, where (γ ) denotes the length of γ and r (x) is the distance from p to x. We now claim that γ ⊂ B p (2ρ). Indeed, if this were not the case and γ (s) ∈ / B p (2ρ) for some s ∈ [0, 1], then the minimizing property of γ would imply that (γ |[0,s] ) > ρ and (γ |[s,1] ) > ρ violating the fact that (γ ) = r (x, y) ≤ 2ρ. We define η : [0, 1] → B p (ρ) × [t1 , t2 ] by η(s) = (γ (s), (1 − s)t2 + st1 ).

12

Gradient estimate and Harnack inequality for the heat equation

127

Clearly, η(0) = (y, t2 ) and η(1) = (x, t1 ). Integrating along η, we obtain  1 d log f (x, t1 ) − log f (y, t2 ) = log f ds ds 0 1

  γ , ∇(log f ) − (t2 − t1 )(log f )t ds. = 0

(12.7) On the other hand, Theorem 12.2 implies that m −(log f )t ≤ αt −1 + A − α −1 |∇(log f )|2 , 2 where A = C1 α (α − 1)−1 (ρ −2 + R). Hence (12.7) becomes  1   f (x, t1 ) |γ  | |∇(log f )| − (t2 − t1 ) α −1 |∇(log f )|2 ds ≤ log f (y, t2 ) 0 1  m + (t2 − t1 ) A + α t −1 ds, 2 0 where t = (1 − s)t2 + st1 . Using the fact that −(t2 − t1 ) α −1 z 2 + |γ  | z ≤

α |γ  |2 4(t2 − t1 )

for any z, we conclude that     f (x, t1 ) α r 2 (x, y) mα t2 log ≤ + log + A(t2 − t1 ), f (y, t2 ) 4(t2 − t1 ) 2 t1 where we have used r (x, y) = (γ ) = |γ  |. The corollary follows by exponentiating both sides.  We are now ready to prove the existence of a minimal heat kernel on any complete (not necessarily compact) manifold. Theorem 12.4 Let M m be a complete, noncompact manifold. There exists a positive, symmetric heat kernel H (x, y, t) on M with the property that, for any f 0 ∈ L 2 (M), the function given by H (x, y, t) f 0 (y) dy f (x, t) = M

solves the heat equation   ∂ f (x, t) = 0 − ∂t

on

M × (0, ∞),

128

Geometric Analysis

with the initial condition lim f (x, t) = f 0 (x).

t→0

Moreover, H (x, y, t) satisfies H (x, y, t) dy ≤ 1, M

and it is the unique minimal heat kernel on M × M × (0, ∞); namely, if H¯ (x, y, t) is another positive heat kernel defined on M, then H (x, y, t) ≤ H¯ (x, y, t), for all x, y ∈ M and t ∈ (0, ∞). Proof Let {i } be a compact exhaustion of M satisfying i ⊂  j for i ≤ j and ∪i i = M. On each i , let Hi (x, y, t) be the Dirichlet heat kernel given by Theorem 10.1. The maximum principle implies that Hi (x, y, t) ≤ H j (x, y, t) for i ≤ j. For a fixed point p ∈ M and a fixed radius ρ > 0, there exists i 0 sufficiently large such that B p (4ρ) ⊂ i for i ≥ i 0 . Corollary 12.3 asserts that there exists a constant C depending only on ρ, t, and the lower bound of the Ricci curvature on B p (4ρ), such that Hi (z, x, t) ≤ C Hi (z, y, 2t) for all z ∈ i and x, y ∈ B p (ρ). Combined with the fact that dy ≤ 1, this yields

! i

Hi (z, y, 2t)

Hi (z, x, t) ≤ C V p−1 (ρ). Hence the monotonically increasing sequence {Hi (z, x, t)} converges uniformly on compact subsets of M × M × (0, ∞) to a kernel function H (z, x, t). Obviously, H (z, x, t) is positive and symmetric in the space variables, and since H (x, y, t) dy ≤ lim Hi (x, y, t) dy i→∞ B p (ρ)

B p (ρ)

≤ 1, we also obtain

H (x, y, t) dy ≤ 1. M

Moreover, the local gradient estimate of Theorem 12.2 asserts that |∇ f |2 − α f f t ≤ C f 2

on

B p (ρ)

12

Gradient estimate and Harnack inequality for the heat equation

129

for f (x, t) = Hi (z, x, t). In particular, if φ is a nonnegative cutoff function supported on B p (ρ), then φ 2 |∇ f |2 ≤ α φ 2 f ft + C φ2 f 2 M



M

≤α

M

φ2 f  f + C

φ2 f 2

M

M



≤ −α





φ |∇ f | − 2α 2

φ f ∇φ, ∇ f  + C

2

M

M

φ2 f 2. M

Applying the Schwarz inequality φ f ∇φ, ∇ f  ≤ φ 2 |∇ f |2 + |∇φ|2 f 2 , −2 M

M

we obtain the estimate





|∇ f | ≤ C

φ f +α

2

B p (ρ/2)

M

2

|∇φ|2 f 2

2

M

M

≤ C1 . by choosing φ = 1 on B p (ρ/2). Hence the sequence Hi (z, x, t) converges in H 1,2 -norm over any compact subset of M. In particular, H (z, x, t) solves the heat equation. The semi-group property of Hi (z, x, t) asserts that Hi (z, y, t) Hi (x, y, s) dy ≤ Hi (z, y, t) Hi (x, y, s) dy j

i

= Hi (z, x, t + s), for i ≥ j. Hence letting i → ∞, we conclude that H (z, y, t) H (x, y, s) dy ≤ H (z, x, t + s). j

In particular, by setting x = z and t = s, this implies that H (z, y, t) is in L 2 with respect to the y variable, hence also the z variable because of symmetry. Letting j → ∞, we also conclude that H (z, y, t) H (x, y, s) dy ≤ H (z, x, t + s). M

130

Geometric Analysis

On the other hand, since {Hi } is monotonically increasing, we have H (z, y, t) H (x, y, s) dy ≥ Hi (z, y, t) Hi (x, y, s) dy i

i

= Hi (z, x, t + s), hence

H (z, y, t) H (x, y, s) dy ≥ H (z, x, t + s). M

Therefore H also satisfies the semi-group property H (z, y, t) H (x, y, s) dy = H (z, x, t + s). M

Obviously, if f 0 is a continuous function with compact support, then H (x, y, t) f 0 (y) dy f (x, t) = M

will satisfy the heat equation with initial data f0 , since this is the case for the function f i (x, t) = Hi (x, y, t) f 0 (y) dy M

when i is sufficiently large. For f 0 just continuous and in L 2 (M), we see that this is also the case by an approximation argument since H ∈ L 2 (M).  In the above proof, the estimates on Hi (x, y, t) are interior estimates and the upper bound does not depend on the boundary condition. In particular, the same estimates are valid for Neumann heat kernels K i (x, y, t) on i . Though the sequence {K i (x, y, t)} is not necessarily monotonic, the kernels are uniformly bounded on compact subsets, and one can extract a convergenct subsequence replacing the original sequence K i (x, y, t) converging to a kernel K (x, y, t). However, Corollary 10.5 asserts that Hi (x, y, t) < K i (x, y, t) and by taking the limit, we conclude that H (x, y, t) ≤ K (x, y, t).

(12.8)

!An interesting question to ask is if H (x, y, t) = K !(x, y, t). Recall that M H (x, y, t) dy ≤ 1, and one can also show that M K (x, y, t) dy ≤ 1. Hence if H (x, y, t) dy = 1, (12.9) M

12

Gradient estimate and Harnack inequality for the heat equation

131

then this identity combined with (12.8) implies that H (x, y, t) = K (x, y, t). We will address the validity of (12.9) in a later chapter. The estimates in Theorem 12.2 and Corollary 12.3 take on much simpler forms when the function is defined globally on a complete manifold with nonnegative Ricci curvature. In fact, the estimates are sharp on Rm . Corollary 12.5 Let M m be a complete manifold with nonnegative Ricci curvature. If f (x, t) is a positive solution of the heat equation   ∂ f (x, t) = 0 − ∂t on M × [0, ∞), then ft |∇ f |2 m − ≤ 2 f f 2t on M × [0, ∞). Moreover,   m/2  2 t2 r (x, y) exp f (x, t1 ) ≤ f (y, t2 ) t1 4(t2 − t1 ) for all x, y ∈ M and 0 < t1 < t2 < ∞. Proof

Apply Theorem 12.2 by first taking ρ → ∞, then α → 1.



The gradient estimate is also valid for manifolds with boundary when the boundary is assumed to satisfy some convexity hypothesis. In the case of the Dirichlet boundary condition, one needs to assume that the boundary has nonnegative mean curvature with respect to the outward normal. For the Neumann boundary condition, one needs to assume that the boundary is convex in the sense that it has nonnegative second fundamental form with respect to the outward normal. We will state and prove the Neumann boundary condition case since this will be used in Chapter 14. Theorem 12.6 Let M be a complete manifold with convex boundary ∂ M. Suppose the Ricci curvature of M is bounded from below by Ri j ≥ −(m − 1)R for some constant R ≥ 0. If f (x, t) is a positive solution of the equation   ∂ f (x, t) = 0 − ∂t

132

Geometric Analysis

on M × [0, T ] with Neumann boundary condition f ν (x, t) = 0 on

∂ M × (0, T ],

then for any 1 < α, the function g(x, t) = log f (x, t) satisfies the estimate m |∇g|2 − αgt ≤ α 2 t −1 + C 1 α 2 (α − 1)−1 R 2 on M × (0, T ], where C 1 is a constant depending only on m. Moreover,  mα/2   t2 α 2 (γ ) exp + C1 α (α − 1)−1 R (t2 − t1 ) f (x, t1 ) ≤ f (y, t2 ) t1 4(t2 − t1 ) for any x, y ∈ M and 0 < t1 < t2 ≤ T, where (γ ) is the length of the shortest curve γ ⊂ M joining x to y. Proof The proof for the gradient estimate is again based on applying the maximum principle to the function G(x, t) = t (|∇g|2 (x, t) − α gt (x, t)). In this case, since M is compact, we will not need to use a cutoff function. In particular, if (x 0 , t0 ) is the maximum point for G on M × [0, T0 ] with T0 ≤ T and if t0 = 0, then G(x, t) ≤ 0 for all (x, t) ∈ M × [0, T0 ]. Hence we may assume that t0 > 0. If x0 ∈ ∂ M, then the Hopf boundary point lemma asserts that G ν (x0 , t0 ) > 0,

(12.10)

where ν is the outward normal to ∂ M. However, since t (2giν gi − αgtν ) = t (2 f −2 f iν f i − 2 f −3 |∇ f |2 f ν − α f −1 f tν + α f −2 f t f ν ), the boundary condition on f implies that G ν (x0 , t0 ) = 2t0 f −2 (x0 , t0 ) f iν (x0 , t0 ) f i (x0 , t0 ). On the other hand, the computation in the proof of Corollary 5.8 implies that 2 f iν f i = −2h βγ f β f γ ≤ 0, where h βγ is the second fundamental form of ∂ M. This gives a contradiction / ∂ M. to (12.10), hence x0 ∈

12

Gradient estimate and Harnack inequality for the heat equation

133

We can now apply the maximum principle to G at the point (x0 , t0 ). At this point, Lemma 12.1 implies that 0 ≥ −G + 2α −2 m −1 G 2 − 2−1 m(m − 1)2 α 2 (α − 1)−2 t02 R 2 , hence we conclude that G(x, t) ≤ G(x 0 , t0 ) ≤

m 2 α + C 1 α 2 (α − 1)−1 t0 R. 2

In particular, m 2 −1 α T0 + C1 α 2 (α − 1)−1 R, 2 and the first part of the theorem follows. The Harnack inequality follows from the same argument as in the proof of Corollary 12.3.  |∇g|2 (x, T0 ) − αgt (x, T0 ) ≤

13 Upper and lower bounds for the heat kernel

In this chapter, we will derive estimates for positive solutions of the heat equation. In particular, upper and lower bounds for the fundamental solution of the heat equation will be established for manifolds with Ricci curvature bounded from below. Much of the following argument was developed by Li and Yau in [LY2]. However, part of the proof has since been simplified and generalized. In particular, the integral estimate of the heat kernel given in Lemma 13.3 is due to Davies [D]. Lemma 13.1 Let M be a compact domain with boundary. Suppose f (x, t) is a nonnegative subsolution for the heat equation on M × (0, T ) with the Dirichlet boundary condition f (x, t) = 0

∂ M × (0, T ).

on

Assume that g(x, t) is a function defined on M × (0, T ) satisfying the differential inequality |∇g|2 + gt ≤ 0, and μ1 (M) > 0 denotes the first eigenvalue of the Dirichlet Laplacian on M. Then the function F(t) = exp(2μ1 (M) t)

exp(2g(x, t)) f 2 (x, t) d x M

is a nonincreasing function of t ∈ (0, T ). 134

13

Upper and lower bounds for the heat kernel

135

Using the variational characterization of μ1 (M), we have that

Proof





μ1 (M)

exp(2g) f 2 ≤ M

|∇(exp(g) f )|2

M

=

|∇ exp(g)|2 f 2 M



+2

exp(g) f ∇ exp(g), ∇ f  M



exp(2g) |∇ f |2

+ M



|∇ exp(g)|2 f 2

= M

+



∇ exp(2g), ∇( f 2 ) +

1 2

exp(2g) |∇ f |2 .

M

M

Integration by parts yields

∇ exp(2g), ∇( f ) = −

exp(2g) ( f 2 )

2

M

M





= −2

exp(2g) |∇ f |2 ,

exp(2g) f  f − 2 M

M

hence





exp(2g) f 2 ≤

μ1 (M) M

|∇ exp(g)|2 f 2 −

M

exp(2g) f  f M



exp(2g) f 2 |∇g|2 −

≤ M

exp(2g) f f t . (13.1) M

On the other hand, ∂ ∂t



 exp(2g) f

2

M





=2

exp(2g) f gt + 2 2

M

exp(2g) f f t M



≤ −2



exp(2g) f 2 |∇g|2 + 2 M

exp(2g) f f t , M

hence combining this inequality with (13.1), we have

∂ 2μ1 (M) exp(2g) f ≤ − ∂t M





2

exp(2g) f M

2

.

136

Geometric Analysis

This implies that ∂ ∂t





exp(2g) f 2

exp(2μ1 (M)t)

≤0

M



and the lemma follows.

!Corollary 13.2 Let M be a complete Riemannian manifold and f (x, t) = M H (x, y, t) u(y) dy be a solution of the heat equation defined on M × [0, ∞). Suppose μ1 (M) ≥ 0 denotes the greatest lower bound of the L 2 -spectrum of the Laplacian on M. Assume that g(x, t) is a function defined on M × (0, ∞) satisfying |∇g|2 + gt ≤ 0, then the function



F(x, t) = exp(2μ1 (M) t)

exp(2g(x, t)) f 2 (x, t) d x M

is nonincreasing in t ∈ (0, ∞). Proof

Let {i } be a compact exhaustion of M such that i ⊂ i+1

and !

∪i i = M.

Define f i (x, t) = i Hi (x, y, t) u(y) dy, where Hi (x, y, t) is the Dirichlet heat kernel defined on i . According to Lemma 13.1, the function exp(2g(x, t)) f i2 (x, t) d x Fi (x, t) = exp(2μ1 (i ) t) i

is a nonincreasing function in t. The corollary follows by letting i → ∞ and  observing that μ1 (i ) → μ1 (M) and f i → f. Lemma 13.3 (Davies) Let M be a complete manifold. Suppose B1 and B2 are bounded subsets of M. Then H (x, y, t) d y d x ≤ exp(−μ1 (M) t)V 1/2 (B1 ) V 1/2 (B2 ) B1

B2



 −d 2 (B1 , B2 ) × exp , 4t

13

Upper and lower bounds for the heat kernel

137

where V (Bi ) denotes the volume of the set Bi for i = 1, 2 and d(B1 , B2 ) denotes the distance between the sets B1 and B2 . If M is a manifold with boundary, and H (x, y, t) is the Dirichlet heat kernel on M, then the same estimate still holds provided that d(B1 , B2 ) is now interpreted as the distance between the two sets of curves in M. In particular, d(B1 , B2 ) is given by the infimum of the length of all curves contained in M joining B1 to B2 . Proof

For i = 1, 2, let f i (x, t) =

H (x, y, t) dy Bi

and d 2 (x, Bi ) , 4(t + )

gi (x, t) =

where d(x, Bi ) is the distance between x and the set Bi . A direct computation shows that    d∇d 2 2   |∇gi | =  2(t + )    2  d (x, Bi )    ≤ 4(t + )2  and −d 2 (x, Bi ) ∂gi = . ∂t 4(t + )2 Applying Corollary 13.2 to f i and gi , we conclude that exp(2gi (x, t)) f i2 (x, t) d x ≤ exp(−2μ1 (M) t) M

× M

exp(2gi (x, 0)) fi2 (x, 0) d x.

However, since fi (x, 0) =

H (x, y, 0) dy Bi

= χ (Bi )

138

Geometric Analysis

is the characteristic function of Bi , we conclude that exp(2gi (x, 0)) f i2 (x, 0) = V (Bi ) M

and

M

exp(2gi (x, t)) f i2 (x, t) d x ≤ V (Bi ) exp(−2μ1 (M) t)

(13.2)

for i = 1, 2. On the other hand, the triangle inequality asserts that d 2 (B1 , B2 ) ≤ (d(x, B1 ) + d(x, B2 ))2 ≤ 2d 2 (x, B1 ) + 2d 2 (x, B2 ), hence implying that  exp

d 2 (B1 , B2 ) 8(t + )

 ≤ exp(g1 + g2 ).

Combining the above estimate with (13.2), we have   2 d (B1 , B2 ) f1 f2 ≤ exp exp(g1 ) exp(g2 ) f 1 f 2 8(t + ) M M 1/2 

 ≤ M

exp(2g1 ) f 12

1/2 M

exp(2g2 ) f 22

≤ V 1/2 (B1 ) V 1/2 (B2 ) exp(−2μ1 (M) t). (13.3) Using the semi-group property, the left-hand side can be written as  2  d (B1 , B2 ) exp f1 f2 8(t + ) M 

d 2 (B1 , B2 ) = exp 8(t + )  = exp

d 2 (B1 , B2 ) 8(t + )



H (x, z, t) H (x, y, t) dz dy d x

M



B1

B2

H (y, z, 2t) dz dy. B1

B2

Lemma 13.3 follows by combing this with (13.3) and letting  → 0.



We are now ready to give an upper bound on the fundamental solution of the heat equation.

13

Upper and lower bounds for the heat kernel

139

Theorem 13.4 (Li–Yau) Let M m be a complete manifold and H (x, y, t) denote the minimal symmetric heat kernel defined on M × M × (0, ∞) with the properties that   ∂ H (x, y, t) = 0 y − ∂t and lim H (x, y, t) = δx (y).

t→0

For any p ∈ M,  > 0, and ρ > 0, if the Ricci curvature of M on B p (4ρ) is bounded from below by Ri j ≥ −(m − 1)R, then there exist constants C1 , C 2 > 0 depending only on m and , such that   −1/2 √ t H (x, y, t) ≤ C1 exp(−μ1 (M) t) Vx ×

−1/2 Vy

  / √  r 2 (x, y) −2 t exp − + C2 (ρ + R) t 4(1 + 2)t

for any x, y ∈ B p (ρ) and t ≤ ρ 2 /4. In particular, if the Ricci curvature of M is bounded by Ri j ≥ −(m − 1)R, then −1/2

H (x, y, t) ≤ C1 exp(−μ1 (M) t) Vx ×

−1/2 Vy

√  t

  √  √ r 2 (x, y) + C2 R t t exp − 4(1 + 2)t

for all x, y ∈ M and t ∈ (0, ∞). Proof For a fixed y ∈ B p (ρ) and δ > 0, applying Corollary 12.3 to the positive solution f (x, t) = H (x, y, t) by taking t1 = t and t2 = (1 + δ)t, we have   2 α r (x, x  ) mα/2 + Aδt H (x  , y, (1 + δ)t), exp H (x, y, t) ≤ (1 + δ) 4δt

140

Geometric Analysis

where A = C1 α(α − 1)−1 (ρ −2 + R). Integrating over x  ∈ Bx gives

√  t , this

 √  α + Aδt Vx−1 H (x, y, t) ≤ (1 + δ)mα/2 exp t 4δ   × √ H (x , y, (1 + δ)t) d x . Bx ( t )

(13.4)

Applying Corollary 12.3 and the same argument to the positive solution f (y, s) =

√ Bx ( t )

H (x  , y, s) d x 

by taking t1 = (1 + δ)t and t2 = (1 + 2δ)t, we obtain √ Bx ( t )

H (x  , y, (1 + δ)t) d x 



≤

 √  α  1 + 2δ mα/2 exp t + Aδt Vy−1 1+δ 4δ     × √ √ H (x , y , (1 + 2δ)t) d x dy . B y ( t ) Bx ( t )

Hence combining the upper bound with (13.4), we have α  + 2Aδt H (x, y, t) ≤ (1 + 2δ)mα/2 exp 2δ √  √  t Vy−1 t × Vx−1     × √ √ H (x , y , (1 + 2δ)t) d x dy . B y ( t ) Bx ( t )

(13.5)

On the other hand, Lemma 13.3 implies that

√ By ( t )

√ Bx ( t )

H (x  , y  , (1 + 2δ)t) d x  dy 

    1/2 √ 1/2 √ ≤ exp(−μ1 (1 + 2δ)t) Vx t Vy t

 

√

√ t , By t −d 2 Bx . × exp 4(1 + 2δ)t

(13.6)

13

Upper and lower bounds for the heat kernel

141

Observing that √  √  # t , By t = d Bx 

0 √ r (x, y) − 2 t

√ if r (x, y) ≤ 2 t, √ if r (x, y) > 2 t,

hence we obtain the estimate

√ 

√  t , By t −d 2 Bx =0 4(1 + 2δ)t ≤1−

r 2 (x, y) , 4(1 + 4δ)t

√ when r (x, y) ≤ 2 t, and

√ 

√  √ 2 − r (x, y) − 2 t t , By t −d 2 Bx ≤ 4(1 + 2δ)t 4(1 + 2δ)t ≤

1 −r 2 (x, y) + 4(1 + 4δ)t 2δ

√ when r (x, y) > 2 t. In any case, there exists a constant C3 > 0, such that when combining the above estimate with (13.5) and (13.6) we obtain 

 (α + 1) + (2δ At − μ1 (1 + 2δ)t 2δ     −r 2 (x, y) −1/2 √ −1/2 √ ( t) Vy t exp . (13.7) × Vx 4(1 + 4δ)t)

H (x, y, t) ≤ C3 (1 + 2δ)mα/2 exp

√ When At > 1, we choose 2δ = / AT , and (13.7) can be estimated by   √ (α + 1) + At exp(−μ1 t)  AT ⎛ ⎞ √  √  2 −r (x, y) −1/2 −1/2 t Vy t exp ⎝  × Vx √  ⎠ 4 1 + 2/ AT t    √ (α + 1) mα/2 + ≤ C3 (1 + ) exp At exp(−μ1 t)        −r 2 (x, y) −1/2 √ −1/2 √ . t Vy t exp × Vx 4(1 + 2)t) 

H (x, y, t) ≤ C3

1+ √



mα/2



exp

142

Geometric Analysis

√ When At ≤ 1, we choose 2δ =  and conclude from (13.7) and At ≤ At that   √ (α + 1) mα/2 exp +  At H (x, y, t) ≤ C3 (1 + )      − 1 √  −r 2 (x, y) − 12 √ 2 × exp(−μ1 t) Vx t Vy t exp . 4(1 + 2)t) In either case, the theorem follows by choosing α = 2.



Due to the fact that both Corollary 12.3 and Lemma 13.3 are valid on manifolds with boundary after suitable localization, a corresponding upper bound for the Dirichlet heat kernel can also be derived following the above proof. Corollary 13.5 Let M m be a manifold with boundary and H (x, y, t) be its Dirichlet heat kernel. Let us assume that B p (4ρ) ∩ ∂ M = ∅ and the Ricci curvature of M on B p (4ρ) is bounded from below by Ri j ≥ −(m − 1)R for some constant R ≥ 0. For any  > 0, there exist constants C 1 > 0 depending only on m and , and C2 depending only on m, such that     −1/2 √ −1/2 √ H (x, y, t) ≤ C1 exp(−μ1 (M) t) Vx t Vy t   / r 2 (x, y) + C2 (ρ −2 + R) t × exp − 4(1 + 2)t for any x, y ∈ B p (ρ) and t ≤ ρ 2 /4. We will now derive a lower bound for the heat kernel. The following comparison type theorems were first proved by Cheeger and Yau [CY] and adapted to the present form in [LY2]. Lemma 13.6 Let M m be a complete manifold with boundary. Suppose B p (ρ) is the geodesic ball centered at some fixed point p ∈ M with radius 0 < ρ ≤ d( p, ∂ M), at most the distance from p to ∂ M. Assume that the Ricci curvature of M is bounded from below by Ri j ≥ (m − 1)K

on

B p (ρ),

for some constant K . Suppose M¯ is a simply connected space form with constant sectional curvature K and B p¯ (ρ) is a geodesic ball centered at some fixed point p¯ ∈ M¯ of radius ρ. Let H (x, y, t) and H¯ (x, ¯ y¯ , t) be the Dirichlet

13

Upper and lower bounds for the heat kernel

143

heat kernels on M and B p¯ (ρ), respectively. Then for any nonnegative function f 0 (r ) defined on [0, ρ] satisfying the properties f 0 (0) = 0, f 0 (ρ) = 0, and f 0 (r ) ≤ 0, the inequality H¯ (¯z , y¯ , t) f 0 (¯r (¯z ))d z¯ H (z, y, t) f 0 (r (z)) dz ≥ B p¯ (ρ)

M

is valid for all y ∈ B p (ρ) with y¯ ∈ B p¯ (ρ) such that r (y) = r¯ ( y¯ ), where ¯ r (y) and r¯ ( y¯ ) denote the distance to p ∈ M and the distance to p¯ ∈ M, respectively. In particular, the inequalities H¯ (¯z , y¯ , t) d z¯ H (z, y, t) dz ≥ B p (ρ1 )

B p¯ (ρ1 )

for all ρ1 ≤ ρ, and H ( p, y, t) ≥ H¯ ( p, ¯ y¯ , t) are valid for r (y) = r¯ ( y¯ ) and for t ∈ (0, ∞). Proof We will prove the inequality for ρ < d( p, ∂ M) and the general case follows by taking the limit as ρ → d( p, ∂ M). For ρ < d( p, ∂ M), let f0 (x) = f 0 (r (x)) be a nonnegative function of the distance r to the center point p satisfying the stated properties. Let f (x, t) = H (z, x, t) f 0 (z) dz M

be the solution of the Dirichlet heat equation with f 0 as the initial datum on M. We also consider the solution of the heat equation on B p¯ (ρ) given by ¯ f (x, ¯ t) = H¯ (¯z , x, ¯ t) f 0 (¯r (¯z )) d z¯ , B p¯ (ρ)

where r¯ (¯z ) is the distance function on M¯ to the point p. ¯ By the uniqueness is rotationally symmetric, f¯ must of the heat equation and the fact that f0 ¯ also be rotationally symmetric. Hence, we can write f ( y¯ , t) = f¯(¯r ( y¯ ), t). In particular, ∂ f¯ f¯ (0, t) = (0, t) = 0, ∂ r¯ and also by the Dirichlet boundary condition ∂ f¯ (ρ, t) < 0. f¯ (ρ, t) = ∂ r¯

(13.8)

144

Geometric Analysis

We claim that f¯ (¯r , t) ≤ 0

(13.9)

for all 0 ≤ r¯ ≤ ρ and for all t ≥ 0. Indeed, since f¯ is a function of r¯ and t alone, the heat equation on M¯ can be written as   ∂ ¯ − f¯(¯r , t) 0=  ∂t ¯ r + f¯ − f¯t , = f¯ ¯ with

√  ⎧ √ ⎪ Kr ⎨ (m − 1) K cot ¯ r = (m − 1)r −1 ¯ ⎪

√  √ ⎩ (m − 1) −K coth −K r

(13.10)

for

K > 0,

for for

K = 0, K < 0.

Differentiating this with respect to r¯ again yields ¯ r + u (¯ ¯ r ) + u  − u t 0 = u  ¯ with u = f¯ and

√  ⎧ 2 ⎪ Kr ⎨ −(m − 1)K cosec ¯ r ) = −(m − 1)r −2 (¯ ⎪ 

√ ⎩ (m − 1)K cosech2 −K r

for

K > 0,

for for

K = 0, K < 0.

¯ r ) ≤ 0 for any choices of K . Hence, the boundary conditions Note that (¯ u(0, t) = 0 and u(ρ, t) < 0 and the initial condition u(¯r , 0) ≤ 0, together with the maximum principle imply that u(¯r , t) ≤ 0. We now consider the transplanted function defined by f¯(y, t) = f¯(r (y), t). The Laplacian comparison theorem, (13.9), and (13.10) assert that   ∂ − f¯(r (y), t) = f¯ r + f¯ − f¯t ∂t ¯ r + f¯ − f¯t ≥ f¯ ¯ =0 in the sense of distribution on M, with the initial condition f¯(y, 0) = f 0 (y). The difference F = f − f¯ is therefore a supersolution of the heat equation of B p (ρ) × (0, ∞) which vanishes on B p (ρ) × {0}. Moreover, the Dirichlet

13

Upper and lower bounds for the heat kernel

145

boundary condition yields F(y, t) = f (y, t) ≥ 0 on

∂ B p (ρ) × (0, ∞).

The parabolic maximum principle implies that F ≥ 0, hence f (y, t) ≥ f¯(r (y), t). The second and the third inequalities follow by approximating the characteristic function of B p (ρ1 ) and the point mass function at p by a sequence of smooth functions satisfying the properties required for f 0 .  Lemma 13.7 Let M m be a complete manifold with boundary. Suppose there exists a point p ∈ M such that M is geodesically star-shaped with respect to p, and ρ0 is the maximum distance from p to any point in ∂ M. Assume that the Ricci curvature of M is bounded from below by Ri j ≥ (m − 1)K for some constant K . Suppose M¯ is a simply connected space form with constant sectional curvature K and B p¯ (ρ0 ) is a geodesic ball centered at some fixed point p¯ ∈ M¯ of radius ρ0 . Let K (x, y, t) and K¯ (x, ¯ y¯ , t) be the Neumann heat kernels on M and B p¯ (ρ0 ), respectively. Suppose ρ1 ≤ d( p, ∂ M), then for any nonnegative function f 0 (r ) defined on [0, ρ1 ] satisfying properties that f 0 (0) = 0, f 0 (ρ1 ) = 0, and f 0 (r ) ≤ 0, the inequality K¯ (¯z , y¯ , t) f 0 (¯r (¯z ))d z¯ K (z, y, t) f 0 (r (z)) dz ≥ B p¯ (ρ0 )

M

is valid for all y ∈ M with y¯ ∈ B p¯ (ρ0 ) such that r (y) = r¯ ( y¯ ), where r (y) and ¯ respectively. In r¯ ( y¯ ) denote the distance to p ∈ M and the distance to p¯ ∈ M, particular, the inequalities K¯ (¯z , y¯ , t) d z¯ K (z, y, t) dz ≥ B p (ρ1 )

B p¯ (ρ1 )

for all ρ1 ≤ d( p, ∂ M), and K ( p, y, t) ≥ K¯ ( p, ¯ y¯ , t) are valid for all r (y) = r¯ ( y¯ ) and for all t ∈ (0, ∞). Proof The proof for the Neumann boundary condition follows using the same argument as the previous lemma. By solving the heat equation with Neumann

146

Geometric Analysis

boundary condition on B p¯ (ρ0 ), we consider the rotationally symmetric solution ¯ K (¯z , y¯ , t) f 0 (¯z ) d z¯ . f ( y¯ , t) = B p¯ (ρ0 )

The Neumann boundary condition asserts that f¯ (ρ0 , t) = 0 instead of (13.8). As in the previous proof, the maximum principle implies that u(¯r , t) = f¯ (¯r , t) ≤ 0. Again we consider the difference F(y, t) = f (y, t) − f¯(r (y), t) with

on

M × (0, ∞),

f (y, t) =

K (z, y, t) f 0 (z) dz. M

We now claim that the normal derivative ∂ F/∂ν with respect to the outward normal must be nonnegative on ∂ M. Clearly, by the boundary condition of F, it suffices to check that ∂ f¯ ≤ 0. ∂ν Indeed, using the fact that f¯ is rotationally symmetric, for any y ∈ ∂ M we have

0 1 ∂ ∂ f¯  ¯ (y, t) = f (r (y), t) , ν (y). ∂ν ∂r

The assertion follows from the fact that f¯ ≤ 0, and the fact that ∂/∂r , ν ≥ 0 on ∂ M because M is geodesically star-shaped with respect to p. Hence ∂F (y, t) ≥ 0 on ∂ M × (0, ∞), ∂ν implying that the minimum of F cannot occur on ∂ M × (0, ∞), because of the Hopf boundary lemma. Hence, the minimum of F must be achieved on M × {0}, which has value 0. This implies that f ≥ f¯ and the rest of the lemma follows.  Theorem 13.8 Let M m be a complete manifold with boundary. Suppose p ∈ M and ρ > 0 are such that B p (4ρ) ∩ M = ∅. Assume that the Ricci curvature is bounded from below by Ri j ≥ −(m − 1)R

on

B p (4ρ),

13

Upper and lower bounds for the heat kernel

147

for some constant R ≥ 0. Suppose H (x, y, t) is Dirichlet heat kernel on ¯ y¯ , t) is the Dirichlet heat kernel on Bx¯ (3ρ) in the simply B p (4ρ) and H¯ (x, connected, constant −R curvature space form. Then for  > 0, there exists constant C4 > 0 depending on m, such that √      −1/2 √ −1/2 √ H (x, y, t) ≥ (1 − )m V¯x¯ t Vx t Vy t   ¯ y¯ , (1 − )t) × exp −2 −1 −  C4 (1 + Rt) H¯ (x, and

√  √  H (x, y, t) ≥ (1 − )m V¯x¯ t Vx−1 t   ¯ y¯ , (1 − )t) × exp −2 −1 −  C4 (1 + Rt) H¯ (x,

for x, y ∈ B p (ρ) and 0 < t ≤ ρ 2 . Proof Let x, y ∈ B p (ρ). The comparison theorem of Lemma 13.6 asserts ¯ then that if H¯ (¯z , y¯ , t) is the Dirichlet heat kernel on Bx¯ (3ρ) ⊂ M, H¯ (¯z , y¯ , (1 − δ)t) d z¯ ≤ H (z, y, (1 − δ)t) dz (13.11) Bx¯ (ρ1 )

Bx (ρ1 )

for ρ1 ≤ 3ρ. However, Corollary 12.3 implies that

αρ12 −αm/2 exp + Aδt H (x, y, t) H (z, y, (1 − δ)t) ≤ (1 − δ) 4δt for z ∈ Bx (ρ1 ), with A = C1 α(α − 1)−1 (ρ −2 + R). Integrating over Bx (ρ1 ), we conclude that H (z, y, (1 − δ)t) dz ≤ (1 − δ)−αm/2 Bx (ρ1 )



αρ12 × exp + Aδt 4δt

H (x, y, t) Vx (ρ1 ).

Applying the same argument to H¯ (x, ¯ y¯ , t), we have H¯ (x, ¯ y¯ , (1 − 2δ)t) Vx¯ (ρ1 ) ≤



×

1−δ 1 − 2δ

αm/2

Bx¯ (ρ1 )



αρ12 exp + Aδt 4δt

H¯ (¯z , y¯ , (1 − δ)t) d z¯ .

(13.12)

148

Geometric Analysis

√ Combining this estimate with (13.11) and (13.12), and setting ρ1 = t, we obtain √  √   α  H (x, y, t) ≥ V¯x¯ t Vx−1 t (1 − 2δ)αm/2 exp − − 2Aδt 2δ × H¯ (x, ¯ y¯ , (1 − 2δ)t). Setting α = 2,  = 2δ, and using t ≤ ρ 2 , we conclude that   √  √ 2 t Vx−1 ( t) exp − −  C4 (1 + Rt) H (x, y, t) ≥ (1 − )m V¯x¯  × H¯ (x, ¯ y¯ , (1 − )t) for some constant C4 > 0 depending only on m. If y ∈ B p (ρ) also, then we can symmetrize the estimate and obtain √      −1/2 √ −1/2 √ t Vx t Vy t H (x, y, t) ≥ (1 − )m V¯x¯   2 ¯ y¯ , (1 − )t). × exp − −  C4 (1 + Rt) H¯ (x,   Corollary 13.9 Let M m be a complete, noncompact manifold without boundary. Suppose M has nonnegative Ricci curvature. For any  > 0, there exists a constant C5 > 0 depending on m and , such that the minimal heat kernel H (x, y, t) satisfies the lower bounds       r 2 (x, y) −1/2 √ −1/2 √ t Vy t exp − H (x, y, t) ≥ C5 Vx 4(1 − )t and H (x, y, t) ≥ C5 Vx−1

  √  r 2 (x, y) t exp − 4(1 − )t

for all x, y ∈ M and t ∈ (0, ∞). Proof According to Theorem 13.8, by setting ρ = ∞, it suffices to estimate the heat kernel H¯ (x, ¯ y¯ , t) on M¯ from below. Note that if R = 0, then   √  ¯ y¯ ) r¯ 2 (x, ¯ ¯ t = C exp − H (x, ¯ y¯ , (1 − )t) Vx¯ 4(1 − )t and our estimate becomes H (x, y, t) ≥

−1/2 C4 Vx

since r¯ (x, ¯ y¯ ) = r (x, y).

  √    r 2 (x, y) −1/2 √ t Vy t exp − 4(1 − )t 

14 Sobolev inequality, Poincar´e inequality and parabolic mean value inequality

In this chapter, we will use the heat kernel to estimate the Sobolev constant and the first nonzero Neumann eigenvalue for geodesic balls. These estimates are useful in the studies of various partial differential equations. In particular, they will provide control over the constants involved in the De Giorgi–Nash– Moser theory in terms of the lower bound of the Ricci curvature and the diameter of the underlying geodesic ball. The Sobolev inequality was proved by Saloff-Coste [SC] adapting Varopoulos’ argument (Theorem 11.6) [V] to this situation. We will also use the heat kernel estimate to prove a parabolic mean value inequality established by Li and Tam in [LT5]. A corollary to this parabolic version will be a generalization to the mean value inequality given in Chapter 7. We will first establish a lemma on the decay rate of the heat kernel. Lemma 14.1 Let M be a manifold with boundary. If μ1 (M) is the first Dirichlet eigenvalue of M, then the Dirichlet heat kernel must satisfy H (x, x, t) ≤ exp(−μ1 (M) (t − t0 )) H (x, x, t0 ) for all x ∈ M and t ≥ t0 > 0. If λ1 (M) is the first nonzero Neumann eigenvalue of M, then the Neumann heat kernel must satisfy K (x, x, t) − V −1 (M) ≤ exp(−λ1 (M) (t − t0 )) (K (x, x, t0 ) − V −1 (M)) for all x ∈ M and t ≥ t0 > 0.

149

150 Proof

Geometric Analysis The semi-group property of the heat kernel implies that ∂ ∂ H (x, x, 2t) = H 2 (x, y, t) dy ∂t ∂t M H (x, y, t) H (x, y, t) dy =2 M



= −2

|∇ H |2 (x, y, t) dy M



≤ −2μ1 (M)

H 2 (x, y, t) dy M

= −2μ1 (M)H (x, x, 2t). Rewriting the inequality in the form ∂ (log H (x, x, t)) ≤ −μ1 (M) ∂t and integrating from t0 to t yields that desired estimate. The estimate on the Neumann heat kernel follows in exactly the same manner by considering K 1 (x, x, t) = K (x, x, t) − V −1 (M) that has the property

K 1 (x, y, t) dy = 0, M

and using the variational principle for λ1 (M).



Theorem 14.2 Let M m be a complete manifold and B p (ρ) be a geodesic ball centered at p with radius ρ. Assume that M has Ricci curvature bounded from below by Ri j ≥ −(m − 1)R for some constant R ≥ 0. Let H (x, y, t) be the Dirichlet heat kernel defined on B p (ρ). Then there exist constants C15 , C16 > 0 depending only on m such that for all x ∈ B p (ρ) and t > 0 we have  √  H (x, x, t) ≤ C15 t −m/2 V p−1 (ρ) ρ m exp C16 ρ R , where V p (ρ) is the volume of B p (ρ).

14

Inequalities of Sobolev, Poincar´e and parabolic mean value

151

Proof Let HM (x, y, t) be the heat kernel on the complete manifold M. Recall that Corollary 13.5 implies that  √  √  t exp C2 Rt HM (x, x, t) ≤ C1 Vx−1 for some constant C 1 , C2 > 0. By the monotonicity property of the heat kernel, we conclude that for x ∈ B p (ρ) and for t ≤ ρ 2 √   √  √  √ H (x, x, t) ≤ C1 V¯ −1 t V¯ t Vx−1 t exp(C2 ρ R),

√  √ t is the volume of the ball of radius t in the simply connected where V¯ space form of constant curvature −R. By the volume comparison theorem, since t ≤ ρ 2 , we have √  √  V¯ t Vx−1 t ≤ V¯ (ρ) Vx−1 (ρ) , hence H (x, x, t) ≤ C1 V¯ −1

√   √  t (V¯ (ρ)Vx−1 (ρ)) exp C2 ρ R

 √  ≤ C1 t −m/2 (V¯ (ρ)Vx−1 (ρ)) exp C2 ρ R .

(14.1)

On the other hand, for t ≥ ρ 2 we can use the estimate given by Lemma 14.1 and conclude that H (x, x, t) ≤ H (x, x, ρ 2 ) exp(−μ1 (t − ρ 2 ))   √ ≤ C 1 Vx−1 (ρ) exp C2 ρ R − μ1 (t − ρ 2 ) .

(14.2)

Let us now consider the function f (t) = t m/2 exp(−μ1 (t − ρ 2 )). It has a local maximum at the point t = m/2μ1 with maximum value  f

m 2μ1



 =

m 2μ1

m/2

  m exp − + μ1 ρ 2 . 2

Moreover, f (t) is a decreasing function for t > m/2μ1 . Hence if m/2μ1 ≤ ρ 2 , then f (t) ≤ f (ρ 2 ) = ρ m

(14.3)

152

Geometric Analysis

for t ≥ ρ 2 . On the other hand, if m/2μ1 ≥ ρ 2 , then   m f (t) ≤ f 2μ1  =  ≤

m 2μ1 m 2μ1

m/2

  m exp − + μ1 ρ 2 2

m/2 .

(14.4)

However, by the estimate of Li–Schoen (Theorem 5.9), μ1 ≥ C3 ρ −2  √  exp −C4 ρ R . Applying this to (14.4) and taking it together with (14.3) yields  √  f (t) ≤ C5 ρ m exp C6 ρ R for all t ≥ ρ 2 . Therefore, substituting this into (14.2), we have  √  H (x, x, t) ≤ C7 t −m/2 Vx−1 (ρ) ρ m exp C8 ρ R for t ≥ ρ 2 .  √  On the other hand, using V¯ (ρ) ≤ C 9 ρ m exp C10 ρ R for some constants C 9 , C10 > 0 depending only on m, the estimate (14.1) becomes √ m (14.5) H (x, x, t) ≤ C11 t − 2 Vx−1 (ρ) ρ m exp(C 12 ρ R) for all t ≤ ρ 2 , hence it is valid for all t > 0. The Bishop volume comparison theorem and the fact that x ∈ B p (ρ) imply that Vx (ρ) ≥ Vx (2ρ)

V¯ (ρ) ¯ V (2ρ)

 √  ≥ C13 V p (ρ) exp −C14 ρ R . The theorem follows by combining this with (14.5).



Combining Theorem 14.2 and Theorem 11.6, we obtain the following Sobolev inequality on a geodesic ball. Theorem 14.3 Let M m be a complete manifold, of dimension m ≥ 3, with Ricci curvature bounded from below by Ri j ≥ −(m − 1)R

14

Inequalities of Sobolev, Poincar´e and parabolic mean value

153

for some constant R ≥ 0. There exist constants C17 , C18 > 0 depending only on m such that for any function f ∈ Hc1,2 (B p (ρ)) with compact support in some geodesic ball B p (ρ), f must satisfy the Sobolev inequality B p (ρ)

|∇ f |2

≥ C17 ρ

−2

√  2/m exp −C18 ρ R (V p (ρ))

(m−2)/m



B p (ρ)

|f|

.

2m/(m−2)

Remark 1: The Sobolev inequality can be written in the form —

|∇ f | ≥ C17 ρ 2

B p (ρ)

−2

√  exp −C 18 ρ R —

(m−2)/m



B p (ρ)

|f|

,

2m/(m−2)

! ! where —B p (ρ) g denotes V p−1 (ρ) B p (ρ) g. This has the advantage that the  √  constant C17 ρ −2 exp −C18 ρ R is now independent on the volume of the ball. Remark 2: When M has nonnegative Ricci curvature, the Sobolev inequality now takes the form

(m−2)/m B p (ρ)

|∇ f |2 ≥ C 17 ρ −2 (V p (ρ))2/m

B p (ρ)

| f |2m/(m−2)

.

In particular, when M has maximal volume growth, i.e., V p (ρ) ≥ C ρ m , the Sobolev constant is uniformly bounded independent of ρ, as in the case of Rm . Conversely, we show in Chapter 16 that if a complete manifold has a uniformly bounded Sobolev constant of the form

(m−2)/m



|∇ f | ≥ C 2

B p (ρ)

|f|

2m/(m−2)

B p (ρ)

,

then the manifold must have volume growth bounded by V p (ρ) ≥ C ρ m . Theorem 14.4 Let M m be a compact manifold without boundary of dimension m ≥ 3. Suppose the Ricci curvature of M is bounded below by Ri j ≥ −(m − 1)R

154

Geometric Analysis

for some constant R ≥ 0, and let d(M) be diameter of M. Then there exist constants C1 , C2 > 0 depending only on m, such that |∇ f | + d 2

−2



 √  f 2 ≥ C 1 exp −C 2 d(M) R V 2/m (M)

(M)

M

M

 | f |2m/(m−2)

×

(m−2)/m

M

for all f ∈ H 1,2 (M). Moreover, there are also constants C3 , C4 > 0 depending only on m, such that the inequality (m−2)/m  √  2/m 2m/(m−2) |∇ f | ≥ C 1 exp −C2 d(M) R V (M) |f|



2



M

M

is also valid for f ∈ H 1,2 (M) satisfying the extra condition

! M

f = 0.

k be a maximal set of points in M with the properties that Proof Let {xi }i=1 r (xi , x j ) ≥ d/4 for all i = j, where d = d(M). Clearly this implies that

Bx i

    d d ∩ Bx j =∅ 8 8

(14.6)

for i = j, hence k 

Vxi

i=1

  d ≤ V (M). 8

(14.7)

 k On the other hand, if x ∈ M \∪i=1 Bxi d4 , then r (x, xi ) ≥ d/4, contradicting the maximal property of the set {xi }, hence we have M=

k 2

  d 4

(14.8)

  d Vxi . 4

(14.9)

Bx i

i=1

and V (M) ≤

k  i=1

However, the volume comparison theorem asserts that

14

Inequalities of Sobolev, Poincar´e and parabolic mean value

155

        d d ¯ d ¯ −1 d V V ≤ Vxi , Vxi 4 8 4 8 where V¯ (r ) denotes the volume of the geodesic ball of radius r in the simply connected space form with constant −R curvature. Combining the volume bound with (14.7), we conclude that k 

kV (M) ≤

Vxi (d)

i=1

≤ V¯ (d) V¯ −1

    k d d Vxi 8 8 i=1

  −1 d ¯ ¯ V (M), ≤ V (d) V 8 implying the bound     d ¯ −1 d V 4 8  √  ≤ C exp Cd R .

k ≤ V¯

To prove the Sobolev inequality, we may consider only nonnegative functions f ∈ H 1,2 (M). Let φ be a nonnegative cutoff function with the properties that .

 1 on Bxi d4 , φ=

 0 on M \ Bxi d2 , and |∇φ|2 ≤ Cd −2 . Applying Theorem 14.3 to the function φ f yields    √  2/m d 2 −2 exp −C18 d R Vxi   |∇(φ f )| ≥ C17 d 2 Bxi d2

(m−2)/m

×

  |φ Bx i

d 2

f|

2m/(m−2)

.

(14.10)

156

Geometric Analysis

Using |∇ f |2 + 2Cd −2 2

f2 ≥ 2

M



 φ

M

d 2

Bx i

2

|∇ f |2 + 2

2   |∇φ| Bx i

d 2

f2

≥

  |∇(φ d 2

Bx i

and



f )|2

Bxi (d/2)

|φ f |2m/(m−2) ≥

(14.10) becomes 2 −2 |∇ f | + Cd M

2m/(m−2)  |f| d 4

Bx i

f ≥ C 19 d 2

−2

M

  √  2/m d exp −C18 d R Vxi 2 

(m−2)/m

2m/(m−2)  |f|

×

.

d 4

Bx i

Note that the volume comparison theorem allows us to estimate    √  d Vxi ≥ C exp −Cd R Vxi (d) 2  √  = C exp −Cd R V (M). Summing over i = 1, . . . , k and using the inequality k  i=1

(m−2)/m  |f|

≥

2m/(m−2)

Bx i

d 4

(m−2)/m

k 

 |f| B xi

i=1

2m/(m−2)

d 4

 | f |2m/(m−2)

=

(m−2)/m

M

because of (14.8), we conclude that  √  k |∇ f |2 + kCd −2 f 2 ≥ C 19 d −2 exp −C20 d R V 2/m (M) M

M

 |f|

×

2m/(m−2)

(m−2)/m

.

M

The first part of the theorem follows by applying the upper bound for k.

14

Inequalities of Sobolev, Poincar´e and parabolic mean value

157

The second part follows from the Poincar´e inequality |∇ f |2 ≥ λ1 (M) f2 M

H 1,2 (M) satisfying

for f ∈ by Theorem 5.7.

M

! M

f = 0, and the lower bound of λ1 (M) provided 

We will next present an estimate of the first nonzero Neumann eigenvalue for star-shaped domains. As a consequence, it gives a lower bound of the Neumann eigenvalue for geodesic balls. This was first proved by Buser [Bu] using a geometric argument via the isoperimetric inequality. The analytic argument which we present below was due to Chen and Li [CnL]. Theorem 14.5 Let M be an m-dimensional manifold with boundary ∂ M. Assume that M is geodesically star-shaped with respect to a point p ∈ M. Suppose that the Ricci curvature of M is bounded from below by Ri j ≥ −(m − 1)R for some constant R ≥ 0. Let ρ be the radius of the largest geodesic ball centered at p contained in M, and ρ0 be the radius of the smallest geodesic ball centered at p containing M. Then there exists a constant C1 > 0 depending only on m, such that the first nonzero Neumann eigenvalue λ1 (M) has a lower bound given by   √  ρm λ1 (M) ≥ m+2 exp −C1 1 + ρ0 R . ρ0 Proof By the variational principle, it suffices to show that there exists a constant C 1 > 0 such that   √  ρm 2 |∇ f | ≥ m+2 exp −C1 1 + ρ0 R inf ( f − a)2 a∈R M ρ0 M for any smooth function f 0 defined on M. Let K (x, y, t) be the Neumann heat kernel defined on M. The function f (x, t) = K (x, y, t) f 0 (y) dy M

solves the heat equation 

∂ − ∂t

 f (x, t) = 0

158

Geometric Analysis

on M, with the Neumann boundary condition ∂f =0 ∂ν on ∂ M, and the initial condition f (x, 0) = f 0 (x). Let us now consider the function g(x, t) = K (x, y, t) ( f 0 (y) − f (x, t))2 dy. M

Clearly,





g(x, t) d x = M



M

M

M



= −2



f 02 (y) dy −

= =−

K (x, y, t) f 02 (y) d y d x −

∂ 0 ∂s t t

0

=2

M

f 2 (x, t) d x M

f 2 (x, s) d x M

f (x, s)  f (x, s) d x M

t 0

f 2 (x, t) d x

|∇ f |2 (x, s) d x. M

However, if we consider ∂ |∇ f |2 (x, t) d x = 2 ∇ f, ∇ f t (x, t) d x ∂t M M = −2  f, f t (x, t) d x

M

= −2 M

we conclude that



f t2 (x, t) d x,

|∇ f |2 (x, t) d x ≤ M

|∇ f 0 |2 (x) d x. M

Hence, we have the estimate g(x, t) d x ≤ 2t |∇ f 0 |2 (x) d x. M

M

(14.11)

14



Inequalities of Sobolev, Poincar´e and parabolic mean value

159

On the other hand, since g is also nonnegative, we have g(x, t) d x ≥ g(x, t) d x B p (ρ/2)

M



=

B p (ρ/2)

K (x, y, t) f 0 (y) − f (x, t))2 d y d x M



inf K (x, y, t)

≥

( f 0 (y) − f (x, t))2 d y d x

B p (ρ/2) y∈M

M



≥ inf

a∈R M

( f 0 (y) − a)2 dy

inf K (x, y, t) d x. (14.12)

B p (ρ/2) y∈M

Using (14.11) and (14.12), we conclude that −1 inf K (x, y, t) d x λ1 (M) ≥ (2t) B p (ρ/2) y∈M

for all t > 0. Note that the Harnack inequality (Corollary 12.3) of Li and Yau asserts that, for any x, z ∈ B p (ρ/2), we have  2     ρ t t C4 exp −C 5 + Rt + 2 K (x, y, t) ≥ K z, y, 2 t ρ for some constants C4 , C 5 > 0 depending only on m. Hence inf K (x, y, t) d x B p (ρ/2) y∈M

≥ Vp

ρ  2

inf

x∈B p (ρ/2),y∈M

K (x, y, t)

 2     ρ t t inf K z, y, ≥ C4 exp −C5 + Rt + 2 dz, y∈M B p (ρ/2) t 2 ρ and λ1 (M) ≥

   2   t ρ C4 t exp −C5 + Rt + 2 dz. inf K z, y, y∈M B p (ρ/2) t t 2 ρ (14.13)

To estimate the right-hand side, we apply Lemma 13.7 and obtain     t t ¯ K z, y, dz ≥ φ r¯ ( y¯ ), , 2 2 B p (ρ/2)

(14.14)

160

Geometric Analysis

where φ¯ is defined by     t t ¯ ¯ K z¯ , y¯ , = d z¯ φ r¯ ( y¯ ), 2 2 B p¯ (ρ/2) with K¯ being the Neumann heat kernel on B p¯ (ρ0 ) with r¯ ( y¯ ) = r (y). The proof of Lemma 13.7 asserts that φ¯  ≤ 0, hence     t t ¯ ¯ ≥ φ ρ0 , . (14.15) φ r¯ ( y¯ ), 2 2 On the other hand, since B p¯ (ρ0 ) is convex, we can apply Theorem 12.6 to the function φ¯ and obtain

    ρ02 t t ¯ ¯ ≥ C 6 φ 0, exp −C7 + Rt (14.16) φ ρ0 , 2 4 t for some constants C6 , C 7 > 0 depending only on m. Using φ¯  ≤ 0 again, we conclude that p¯ is a maximum point of φ¯ for any t ∈ (0, ∞). Therefore, the heat equation implies that ∂ ¯ ¯ φ(0, ¯ t) φ= ∂t ≤ 0, hence

  t ¯ t) ¯ ≥ lim φ(0, φ 0, t→∞ 4 = V¯ p−1 ¯ (ρ0 ) d z¯ , B p¯ (ρ/2)

where V¯ p¯ (ρ0 ) is the volume of B p¯ (ρ0 ). Combining the above inequality with (14.13), (14.14), (14.15), and (14.16), we have

ρ02 V¯ p¯ (ρ/2) ρ2 t −1 λ1 (M) ≥ C21 (t) exp −C22 + 2+ + Rt , t ρ t V¯ p¯ (ρ0 ) for some constants C21 , C 22 > 0. Setting t=

ρ02 1 + ρ0

√

and using the estimates t −1 ≥ ρ0−2

R

14

Inequalities of Sobolev, Poincar´e and parabolic mean value

161

and √ ρ02 R √ ≤ 1 + ρ0 R, 1 + ρ0 R we conclude that   √  V¯ p¯ (ρ/2) λ1 (M) ≥ ρ0−2 exp −C23 1 + ρ0 R V¯ p¯ (ρ0 ) for some constant C23 > 0 depending only on m. The theorem now follows by observing that there exist constants C24 > 0 and C25 > 0, depending only on m, such that   √  r¯ m exp C24 1 + r¯ R ≤ V¯ p¯ (¯r )   √  ≤ r¯ m exp C25 1 + r¯ R . This completes the proof of Theorem 14.5.



Corollary 14.6 Let B p (ρ) be a geodesic ball centered at a point p ∈ M with radius ρ > 0 such that B p (ρ) ∩ ∂ M = ∅. Suppose that the Ricci curvature of B p (ρ) is bounded from below by Ri j ≥ −(m − 1)R for some constant R ≥ 0. Then there exists a constant C2 > 0 depending only on m, such that the first nonzero Neumann eigenvalue λ1 (B p (ρ)) has a lower bound given by   √  λ1 (B p (ρ)) ≥ ρ −2 exp −C2 1 + ρ R . The following theorem gives a parabolic mean value inequality for subsolutions of the heat equation. It was first proved by Li and Tam in [LT5]. Theorem 14.7 Let M m be a complete noncompact Riemannian manifold with boundary. Let p ∈ M and ρ > 0 be such that B p (4ρ) ∩ ∂ M = ∅. Suppose g(x, t) is a nonnegative function defined on B p (4ρ) × [0, T ] for some 0 < T ≤ ρ 2 /4 satisfying the differential inequality g −

∂g ≥ 0. ∂t

If the Ricci curvature of B p (4ρ) is bounded by Ri j ≥ −(m − 1)R

162

Geometric Analysis

for some constant R ≥ 0, then for any q > 0, there exist positive constants C 1 and C2 depending only on m and q, such that for any 0 < τ < T , 0 < δ < 12 , and 0 < η < 12 we have sup

B p ((1−δ)ρ)×[τ,T ]

  √  V¯ (2ρ)  √ ρ R + 1 exp C2 R T V p (ρ)

gq ≤ C1

 ×

1 1 +√ δρ ητ

m+2



T (1−η)τ

ds

B p (ρ)

g q (y, s) dy,

where V¯ (r ) is the volume of the geodesic ball of radius r in the m-dimensional, simply connected space form with constant sectional curvature −R. Proof For any 0 < δ, η < 14 , let φ˜ and ψ˜ be two smooth functions on [0, ∞) ˜ ψ˜ ≤ 1, with so that 0 ≤ φ, . 1 for 0 ≤ s ≤ 1 − δ, ˜ φ(s) = 0 for s ≥ 1, and

. ˜ ψ(s) =

0

for

0 ≤ s ≤ 1 − 2η,

1

for

s ≥ 1 − η.

We can choose φ˜ and ψ˜ such that −C δ −1 ≤ φ˜  φ˜ − 2 ≤ 0, 1

φ˜  ≥ −C δ −2 , and 0 ≤ ψ˜  ≤ C η−1 , for some constant C > 0 which is independent of δ and η. Let r (x) denote the distance function from p, and we define φ(x) = ˜ (x) R −1 ) and ψ(t) = ψ(t ˜ τ −1 ). Since g is a nonnegative subsolution of φ(r the heat equation, for q > 1 we have   ∂ (φ(x) ψ(t) g q (x, t)) − ∂t ≥ (φ) ψ g q + 2qψ g q−1 ∇φ, ∇g + q(q − 1)φ ψ g q−2 |∇g|2 − φ ψ  g q ≥ (φ) ψ g q −

q |∇φ|2 ψ gq − φ ψ  gq . q −1 φ

14

Inequalities of Sobolev, Poincar´e and parabolic mean value

On the other hand, the Laplacian comparison theorem asserts that    √  q |∇φ|2 ≥ −C3 ρ R + 1 δ −2 ρ −2 , φ − q −1 φ

163

(14.17)

for some constant C3 depending only on m and q. If H (x, y, t) is the Dirichlet heat kernel of M, by using (14.17) and the definition of φ and ψ, there exists a constant C 4 depending only on m and q so that gq (x, t) = g q (x, t) φ(x) ψ(t) 

 ∂ =− ds H (x, y, t − s)  y − (g q (y, s) φ(y) ψ(s)) dy ∂s 0 M   t ∂ =− ds H (x, y, t − s)  y − ∂s (1−2η)τ B p (ρ)

t



× (g q (y, s) φ(y) ψ(s)) dy .  √  −2 −2 ≤ C4 ρ R + 1 δ ρ



T (1−2η)τ

ds

B p (ρ)\B p ((1−δ)ρ)

× H (x, y, t − s) g q (y, s) dy 1 + ητ



(1−2η)τ

3



(1−η)τ

ds

q

B p (ρ)

H (x, y, t − s) g (y, s) dy (14.18)

for any x ∈ B p ((1 − 2δ)ρ) and τ < t < T . Using the upper bound of the heat kernel given by Corollary 13.5, if x, y ∈ B p (R) and t − s ≤ R 2 /4, then we have  − 1 √  − 1 √ t − s Vy 2 t −s H (x, y, t − s) ≤ C5 Vx 2   / r 2 (x, y) × exp −C6 + C7 (ρ −2 + R)(t − s) , t −s where C 5 , C6 , and C7 are constants depending only on m. On the other hand, the volume comparison theorem implies that for x ∈ B p (ρ) and 0 < s < t ≤ T ≤ ρ 2 /4 V p (ρ) Vx (2ρ) V¯ (2ρ)  ≤  ≤ √

√

√  ≤ C8 (t − s)−m/2 V¯ (2ρ) Vx t −s Vx t −s t −s V¯

164

Geometric Analysis

for some constant C8 depending only on m. Hence V¯ (2ρ) (t − s)−m/2 V p (ρ)   / r 2 (x, y) −2 × exp −C6 + C7 (ρ + R)(t − s) t −s   / V¯ (2ρ) ≤ C10 (δρ)−m exp C7 (ρ −2 + R)(t − s) (14.19) V p (ρ)

H (x, y, t − s) ≤ C 9

for x ∈ B p ((1 − 2δ)ρ), y ∈ B p (ρ)\ B p ((1 − δ)ρ) and 0 < s ≤ t ≤ T ≤ ρ 2 /4, where C9 and C10 are constants depending only on m. Also for t > τ , and s ≤ (1 − η)τ , we can find a constant C11 depending only on m so that m

H (x, y, t − s) ≤ C11 (ητ )− 2

  / V¯ (2ρ) −2 (ρ + R)(t − s) . (14.20) exp C7 V p (ρ)

Combining (14.18), (14.19), and (14.20), we conclude that sup

B((1−2δ)ρ)×[τ,T ]

g ≤ C 12 q

  /  V¯ (2ρ)  √ −2 ρ R + 1 exp C13 (ρ + R)T V p (ρ)

  × (δρ)−(m+2) + (ητ )−(m+2)/2 ×



T (1−2η)τ

ds

B p (ρ)

g q (y, s) dy

(14.21)

for 0 < τ < T ≤ ρ 2 /4, 0 < δ, η ≤ 1/4, and q > 1, where C12 and C13 depend only on m and q. This completes the proof of the theorem for the case when q > 1 if we replace 2δ by δ and 2η by η. In order to prove the case when q ≤ 1, we can proceed as in [LS]. Taking q = 2 in (14.21), we have sup

B p ((1−δ)ρ)×[τ,T ]

g ≤ C12 2

  /  V¯ (2ρ)  √ −2 ρ R + 1 exp C13 (ρ + R)T V p (ρ)

  × (δρ)−(m+2) + (ητ )−m+2/2 ×



T (1−η)τ

ds

B p (ρ)

g 2 (y, s) dy

(14.22)

14

Inequalities of Sobolev, Poincar´e and parabolic mean value

165

for 0 < δ, η < 12 , and T ≤ ρ 2 /4. For k ≥ 0 and l ≥ 1, let us define M(k, l) =

sup

B p ((1−δk )ρ)×[τl ,T ]

g2,

   where δk = 2−k δ, and τl = 1 − η li=1 2−i τ . Inequality (14.22) implies that M(k, l) ≤ C12

  /  V¯ (2ρ)  √ ρ R + 1 exp C13 (ρ −2 + R)T V p (ρ)

  × 2(k+1)(m+2) (δρ)−(m+2) + 2(l+1)(m+2)/2 (ητ )−(m+2)/2 × M λ (k + 1, l + 1)





T

(1−η)τ

ds

B p (ρ)

g q (y, s) dy,

˜ where λ = 1 − q/2. Let M(k) = M(k, 2k + 1), then ˜ M(k) ≤ A(2m+2 )k+1 M˜ λ (k + 1), where  /   V¯ (2ρ)  √ −2 A = C 12 ρ R + 1 exp C13 (ρ + R)T V p (ρ) 

× (δρ)

−(m+2)

−(m+2)/2

+ (ητ )





T (1−η)τ

ds

B p (ρ)

g q (y, s) dy.

Iterating this inequality, we get ˜ M(0) ≤ C 14 A1/(1−λ) , where C14 is a constant depending only on m and q. From this, it is easy to conclude that the theorem is also true for 0 < q ≤ 1.  Corollary 14.8 Let M m be a complete manifold with boundary and p be a fixed point in M. Suppose ρ > 0 is such that B p (4ρ) ∩ ∂ M = ∅ and assume that the Ricci curvature of M on B p (4ρ) satisfies the bound Ri j ≥ −(m − 1)R

166

Geometric Analysis

for some constant R ≥ 0. Let 0 < δ < 12 , q > 0, and λ ≥ 0 be fixed constants. √ Then there exists a constant C > 0 depending only on δ, q, λ ρ 2 , m, and ρ R, such that for any nonnegative function f defined on B p (2ρ) satisfying the differential inequality  f ≥ −λ f we have f q ≤ C V p−1 (ρ)

sup

B p ((1−δ)ρ)

Proof

B p (ρ)

f q (y) dy.

Let us consider the function g(x, t) = e−λt f (x)

which satisfies the differential inequality   ∂ g(x, t) ≥ 0 − ∂t on B p (2ρ) × [0, ∞). Applying Theorem 14.7 to g(x, t) and setting T = ρ 2 /4, τ = ρ 2 /8, and η = 14 , we obtain sup

e

− pλ t

B p ((1−δ)ρ)×[ρ 2 /8,ρ 2 /4]

f (x) ≤ C 1 V¯ (2ρ) ρ −(m+2)



q

ρ 2 /4 3ρ 2 /32

e−qλ s ds

 √   √  × ρ R + 1 exp C2 ρ R V p−1 (ρ) ×

B p (ρ)

f q (y) dy,

where C1 > 0 depends on m, q, and δ. This implies that   √ sup f q (x) ≤ C 4 exp C2 ρ R + C 3 λ ρ 2 V¯ (2ρ) ρ −m V p−1 (ρ) B p ((1−δ)ρ)

×

B p (ρ)

f q (y) dy.

Obviously the corollary follows from this estimate.

(14.23) 

Corollary 14.9 Let M m be a complete manifold with boundary and p be a fixed point in M. Suppose ρ > 0 be such that B p (4ρ) ∩ ∂ M = ∅ and assume that the Ricci curvature of M is nonnegative. Let 0 < δ < 12 and q > 0 be fixed constants. Then there exists a constant C > 0 depending only on δ, q, and m,

14

Inequalities of Sobolev, Poincar´e and parabolic mean value

167

such that for any nonnegative function f defined on B p (2ρ) satisfying the differential inequality f ≥ 0 we have sup

B p ((1−δ)ρ)

Proof

f q ≤ C V p−1 (ρ)

B p (ρ)

f q (y) dy.

Applying the assumptions to (14.23) by setting αm−1 m V¯ (2ρ) = ρ m

and λ = 0, we obtain sup

B p ((1−δ)ρ)

f q (x) ≤ C1 ρ −2



ρ 2 /4 3ρ 2 /32

ds V p−1 (ρ)

B p (ρ)

f q (y) dy. 

This proves the corollary.

Another form of the mean value inequality similar to that of Theorem 14.7 is to allow information on the subsolution at t = 0 to be used. The proof is similar and slightly easier than the proof of Theorem 14.7 and is by choosing ψ = 1. We will only state the theorem, leaving the proof as an exercise. Theorem 14.10 Let M m be a complete noncompact Riemannian manifold with boundary. Let p ∈ M and ρ > 0 be such that B p (4ρ) ∩ ∂ M = ∅. Suppose g(x, t) is a nonnegative function defined on B p (4ρ) × [0, T ] for some 0 < T ≤ ρ 2 /4 satisfying the differential inequality g −

∂g ≥ 0. ∂t

If the Ricci curvature of B p (4ρ) is bounded by Ri j ≥ −(m − 1)R for some constant R ≥ 0, then for any q > 0 and  > 0, there exist positive constants C15 and C16 depending on m and q, with C15 also depending on , such that for any 0 < δ < 12 we have

168

Geometric Analysis

sup

B p ((1−δ)ρ)×[0,T ]

g q ≤ C 15

  √  V¯ (2ρ)  √ ρ R + 1 exp C2 R T (δρ)−(m+2) V p (ρ)





T

×

ds 0

B p (ρ)

g q (y, s) dy + (1 + ) sup g q (˙,0), B p (ρ)

where V¯ (r ) is the volume of the geodesic ball of radius r in the m-dimensional, simply connected space form with constant sectional curvature −R.

15 Uniqueness and the maximum principle for the heat equation

As alluded to in our discussion preceding Definition 10.3, if H (x, y, t) is the minimal heat kernel on a complete manifold, constructed by Theorem 12.4, then the question is if H (x, y, t) dy = 1 (15.1) M

is related to the uniqueness of the heat kernel. The validity of (15.1) implies that H (x, y, t) = K (x, y, t) where K (x, y, t) is a limiting heat kernel obtained from a sequence of Neumann heat kernels. Note that the inequality H (x, y, t) ≤ 1 (15.2) M

implies that the (minimal) heat semi-group is contractive on L ∞ , because      H (x, y, t) f 0 (y) dy  | f (x, t)| =  M



≤  f 0 ∞

H (x, y, t) dy M

≤  f 0 ∞ , implying  f (·, t)∞ ≤  f 0 ∞ . Observe that the constant function 1 and ! M H (x, y, t) dy are solutions to the heat equation with the same initial data, hence if H (x, y, t) dy < 1 M

169

170

Geometric Analysis

for some x ∈ M, then there are two distinct L ∞ solutions to the heat equation with initial datum 1. In general, let f (x, t) be any L ∞ solution to the heat equation with f (x, 0) = f 0 (x). If | f (x, t)| < A on M × [0, T ], then the functions A + f and A − f are both positive solutions to the heat equation with initial data given by A + f 0 and A − f 0 , respectively. The minimality of the heat kernel H implies that H (x, y, t) (A + f 0 (y)) dy ≤ A + f (x, t) M

and

H (x, y, t) (A − f 0 (y)) dy ≤ A − f (x, t). M

Therefore, we conclude that  H (x, y, t) f 0 (y) dy − A 1 − M

 H (x, y, t) dy M

 H (x, y, t) f 0 (y) dy + A 1 −



≤ M

≤ f (x, t)

 H (x, y, t) dy . M

! In particular, the validity of (15.1) implies that f (x, t) = M H (x, y, t) f 0 (y) dy, asserting that the uniqueness of all bounded solutions is equivalent to (15.1). Inequality (15.2) also implies that the semi-group is contractive on L 1 , since | f (x, t)| dy ≤ H (x, y, t) | f 0 |(y) d y d x M



M

≤

M

| f 0 |(y) dy. M

When (15.1) is valid, then for nonnegative initial datum f 0 , we have f (x, t) dy = H (x, y, t) f 0 (y) d y d x M



M

=

M

f 0 (y) dy, M

asserting that the heat semi-group preserves L 1 -norm on nonnegative functions. Also, note that the minimal heat semi-group is contractive on L p for 1 < p < ∞. To see this, let f 0 ∈ L p and let H (x, y, t) f 0 (y) dy f (x, t) = M

15

Uniqueness and the maximum principle for the heat equation 171

be the solution of the heat equation with initial datum f0 . We will first assume that f 0 ≥ 0. In this case, since H (x, y, t) is the limit of a sequence of Dirichlet heat kernels H ! i (x, y, t) on a compact exhaustion {i }, f (x, t) is the limit of f i (x, t) = M Hi (x, y, t) f 0 dy, where we extend Hi to be 0 on M \i . However, since ∂ p p−1 fi = p fi  fi ∂t M M p−2 = − p( p − 1) fi |∇ f i |2 M

≤ 0, we have  f i (·, t) p ≤  f i (·, 0) p ≤  f0  p for all t. Taking the limit as i → ∞, we conclude that  f (·, t) p ≤  f 0  p , as claimed. In general, we observe that     | f (x, t)| =  H (x, y, t) f 0 (y) dy 

M

≤

H (x, y, t) | f 0 (y)| dy. M

Hence the contractivity of the minimal heat equation on L p is established. Proposition 15.1 Let M m be a complete manifold, and let f (x, t) be a nonnegative subsolution of the heat equation. If f (·, t) ∈ L p (M) for all [0, T ], for 1 < p < ∞, then  f (·, t) p ≤  f (·, 0) p for all 0 ≤ t ≤ T. Moreover, the heat equation (not necessarily given by the minimal heat kernel) is contractive in L p and it is uniquely determined by its initial condition and hence has to be given by f (x, t) = H (x, y, t) f 0 (y) dy. M

172

Geometric Analysis

Proof Using the fact that f (x, t) is a nonnegative subsolution, if φ is a nonnegative cutoff function, we have ∂ φ 2 (x) f p (x, t) d x = p φ 2 (x) f p−1 (x, t) f t (x, t) d x ∂t M M ≤p φ 2 (x) f p−1 (x, t)  f (x, t) d x M



= − p( p − 1)

φ 2 (x) f p−2 (x, t) |∇ f |2 (x, t) d x M

− 2p

φ(x) f p−1 (x, t) ∇φ(x), ∇ f (x, t) d x. M

(15.3) Schwarz inequality asserts that p−1 φ f ∇φ, ∇ f  d x ≤ p( p − 1) φ 2 f p−2 |∇ f |2 d x −2 p M

M

+ ( p − 1)−1 p

|∇φ|2 f p . M

Combining this inequality with (15.3), we conclude that ∂ φ 2 (x) f p (x, t) d x ≤ ( p − 1)−1 p |∇φ|2 f p . ∂t M M Choosing the cutoff function φ to satisfy # 1 on Bq (ρ), φ= 0 on M \ Bq (2ρ), and |∇φ|2 ≤ 2 ρ −2 , (15.4) yields



f (x, t) d x −

p

p

Bq (ρ)



Bq (2ρ)

f 0 (x) d x

p

φ (x) f (x, t) d x −

≤

2

M

≤ ( p − 1)−1 p

p

M

t 0

≤ 2 p( p − 1)−1 ρ −2

φ 2 (x) f 0 (x) d x

|∇φ|2 f p (x) d x dt M

t 0

f p (x) d x dt. M

(15.4)

15

Uniqueness and the maximum principle for the heat equation 173

Using the assumption that f (·, t) ∈ L p (M) and letting ρ → ∞, we obtain f p (x, t) d x ≤ f p (x, 0) d x. M

M

Observing that taking the absolute value of a solution yields a nonnegative subsolution, we conclude that the heat equation (not necessarily minimal) is contractive on nonnegative L p functions. In particular, by taking the difference of two L p solutions, this implies that L p solutions are uniquely determined by their initial conditions.  We are now ready to address the issue of the uniqueness of L ∞ solutions. The theorem presented below was proved by Karp and Li [KL] in an unpublished paper. It was also independently proved by Grigor’yan [G1] with a slightly weaker assumption. Theorem 15.2 (Karp–Li) Let M m be a complete manifold. Suppose f (x, t) is a subsolution of the heat equation satisfying   ∂ − f (x, t) ≥ 0 on M × (0, ∞) ∂t with f (x, 0) ≤ 0

on

M.

If there exist p ∈ M and a constant A > 0 such that the L 2 -norm of f over the geodesic ball centered at p of radius ρ satisfies f 2 (x, t) ≤ exp(A ρ 2 ) for all t ∈ (0, T ), B p (ρ)

then f (x, t) ≤ 0. Proof

Let us define the positive part of the function f (x, t) by f + (x, t) = max{ f (x, t), 0},

which is a subsolution satisfying   ∂ − f + (x, t) ≥ 0 on ∂t with f + (x, 0) = 0.

M × (0, ∞)

174

Geometric Analysis

It suffices to show that f + = 0 on M × [0, ∞]. For simplicity of notation, we will replace f + by f . Let us consider the function g(x, t) =

−r 2 (x) 4(2T − t)

(15.5)

for x ∈ M and 0 ≤ t < T, where r (x) is the distance function to the fixed point p ∈ M. One can verify that |∇g|2 + gt = 0.

(15.6)

For  > 0 and ρ > 0, let φ be a nonnegative cutoff function with the properties that # 1 on B p (ρ), φ= (15.7) 0 on M \ B p (ρ + 1), 0 ≤ φ ≤ 1 and |∇φ|2 ≤ 2. Using the fact that f is a nonnegative subsolution and its initial condition, we have τ τ 2 0≤ φ exp(g) f  f − φ 2 exp(g) f f t 0

=−

0 τ

−



φ 2 exp(g) |∇ f |2 − 2

M

 φ 2 exp(g) f 2 t=τ +

1 2

However, Schwarz inequality implies that τ − φ 2 exp(g) f ∇g, ∇ f  ≤ M

0

φ exp(g) f ∇φ, ∇ f 

0

M

M



1 2

M τ



M

φ 2 exp(g) f ∇g, ∇ f 

0

−

0

M τ





τ

φ 2 exp(g) f 2 gt .

0

M

1 2

+



τ 0

φ 2 exp(g) f 2 |∇g|2 M τ



1 2

0

φ 2 exp(g) |∇ f |2 M

and

τ

− 0



φ exp(g) f ∇φ, ∇ f  ≤ M

1 4

τ 0

+

0

φ 2 exp(g) |∇ f |2 M

τ



|∇φ|2 exp(g) f 2 , M

(15.8)

15

Uniqueness and the maximum principle for the heat equation 175

hence (15.8) becomes τ |∇φ|2 exp(g) f 2 + 0≤2 0

−



M

φ exp(g) 2

1 2

M



f 2 t=τ

1 2

+

1 2

τ

0 τ 0

φ 2 exp(g) f 2 |∇g|2 M



φ 2 exp(g) f 2 gt . M

Combining this inequality with (15.6) and (15.7), we obtain   2 exp(g) f t=τ ≤ φ 2 exp(g) f 2 t=τ B p (ρ)

M



τ

≤4 0

Note that (15.5) implies that   ρ2 exp(g) ≤ exp − 8T

on

B p (ρ+1)\B p (ρ)

exp(g) f 2 .

(15.9)

(B p (ρ + 1)\ B p (ρ)) × [0, T ],

and (15.9) yields   τ ρ2 2 exp(g) f (x, τ ) d x ≤ 4 exp − f 2 (x, t) d x dt 8T 0 B p (ρ) B p (ρ+1)\B p (ρ) for all 0 ≤ τ ≤ T. Applying the growth assumption on f , we conclude that   ρ2 2 2 exp(g) f (x, τ ) d x ≤ 4τ exp − + A(ρ + 1) . 8T B p (ρ) Choosing T < 1/8A and letting ρ → ∞, this implies that exp(g) f 2 (x, τ ) d x ≤ 0 for all 0 < τ ≤ T, M

hence f = 0 on M × [0, T ]. Using f (x, T ) as the initial condition, we argue inductively that f is identically 0.  Corollary 15.3 Let M m be a complete manifold. Suppose there exists a constant A > 0 such that the volume growth of M satisfies the estimate V p (ρ) ≤ exp(Aρ 2 ) for all ρ > 0. Then any L ∞ solution of the heat equation is uniquely determined by its initial data. In particular, H (x, y, t) = 1. M

176

Geometric Analysis

Proof By considering the difference of two L ∞ solutions, it suffices to prove that if  f (·, t)∞ ≤ A and is a solution of the heat equation on M with f (x, 0) = 0, then f must be identically 0. This follows by applying Theorem 15.2 to the function g = | f |.  We have yet to address the uniqueness of L 1 solutions. It turns out that with some mild curvature assumptions one can prove that the L 1 solution to the heat equation is also unique. We will delay our discussion until Chapter 31. Note that Theorem 15.2 can be viewed as a form of the maximum principle on complete manifolds. Various generalizations based on the argument given in the proof of Theorem 15.2 have been developed. Interested readers should refer to [LoT] and [NT] for further discussion.

16 Large time behavior of the heat kernel

In this chapter, we will discuss the behavior of the heat equation when t → ∞. When M is a compact manifold with (or without) boundary, due to the eigenfunction expansion, the Dirichlet and Neumann heat kernels are asymptotically given by the first eigenfunction and the first eigenvalue. In particular, H (x, y, t) ∼ e−μ1 t ψ1 (x) ψ1 (y) and K (x, y, t) ∼ V −1 (M) as t → ∞. However, if M is complete and noncompact, then the behavior of H as t → ∞ is not so clear. If the bottom of the L 2 spectrum μ1 (M) is positive, it is still rather easy to see that t −1 log H (x, y, t) ∼ −μ1 (M). The following discussion gives a sharp asymptotic behavior of H for manifolds with nonnegative Ricci curvature with maximal volume growth. As a corollary (Corollary 16.3), one concludes that such manifolds must have finite fundamental group. This was proved by the author in [L5]. Lemma 16.1 Let M m be a complete manifold with nonnegative Ricci curvature. If there exists a constant θ > 0 such that lim inf ρ→∞

V p (ρ) =θ ρm

177

178

Geometric Analysis

for some point p ∈ M, then lim

ρ→∞

A p (ρ) = θ, m ρ m−1

where A p (ρ) denotes the area of the geodesic sphere ∂ B p (ρ). ¯ Proof The volume comparison theorem asserts that if A(ρ) is the area of the sphere of radius ρ in Rm , then A p (ρ) ¯ A(ρ) is a nonincreasing function of ρ. In particular, the limit lim

ρ→∞

A p (ρ) m ρ m−1

exists. Note that since V p (ρ) V¯ (ρ) is also a nonincreasing function, we have the limit lim

ρ→∞

V p (ρ) = θ. ρm

L’Hˆopital’s rule implies that lim

ρ→∞

V p (ρ) A p (ρ) = lim , m ρ→∞ ρ m ρ m−1 

which proves the lemma.

Theorem 16.2 (Li) Let M m be a complete manifold with nonnegative Ricci curvature. If there exists a constant θ > 0 such that lim inf ρ→∞

V p (ρ) =θ ρm

for some point p ∈ M, then √ lim V p ( t) H (x, y, t) = m −1 αm−1 (4π )−m/2 , t→∞

where αm−1 denotes the area of the unit sphere Sm−1 ⊂ Rm .

16 Proof

Large time behavior of the heat kernel

179

The estimate of Corollary 12.5 asserts that  m/2 t2 H ( p, p, t1 ) ≤ H ( p, p, t2 ) t1

for t1 ≤ t2 . This implies that t m/2 H ( p, p, t) is a nondecreasing function of t. However, Theorem 13.4 asserts that √ H ( p, p, t) ≤ C V p−1 ( t). From this and the assumption on the volume growth, we conclude that t m/2 H ( p, p, t) is bounded from above. Hence the limit lim t m/2 H ( p, p, t) = α

t→∞

exists and t m/2 H ( p, p, t)  α. For x, y ∈ M, Corollary 12.5 asserts that

 r 2 (x) 4δt  2  r (x) + r 2 (y) ≤ ((1 + 2δ)t)m/2 H ( p, p, (1 + 2δ)t) exp 4δt  2  r (x) + r 2 (y) ≤ α exp 4δt 

t m/2 H (x, y, t) ≤ ((1 + δ)t)m/2 H ( p, y, (1 + δ)t) exp

for δ > 0, where r (·) denotes the distance function to the point p. This implies that lim sup t m/2 H (x, y, t) ≤ α. t→∞

The same argument also shows that t

m 2

 2  r (x) + r 2 (y) H (x, y, t) ≥ ((1 − 2δ)t)m/2 H ( p, p, (1 − 2δ)t) exp − 4δt

implying that lim inf t m/2 H (x, y, t) ≥ α, t→∞

180

Geometric Analysis

hence lim t m/2 H (x, y, t) = α.

(16.1)

t→∞

To compute the value of α, we use the heat kernel comparison in Lemma 13.6 and obtain H¯ ( p, ¯ y¯ , t) d y¯ , H ( p, y, t) dy ≥ B¯ p¯ (ρ)

B p (ρ)

where H¯ is the heat kernel of Rm with r (y) = r¯ ( y¯ ). In particular H ( p, y, t)dy ≥ lim t m/2 lim t m/2 H¯ ( p, ¯ y¯ , t)d y¯ t→∞

t→∞

B p (ρ)

= lim t t→∞

B¯ p¯ (ρ)

−m/2

(4π t)

m/2

B¯ p¯ (ρ)

 2  r¯ ( y¯ ) exp − d y¯ 4t

= (4π )−m/2 V¯ (ρ), implying αV p (ρ) ≥ (4π )−m/2 V¯ (ρ). Letting ρ → ∞, we conclude that α ≥ (4π )−m/2 m −1 αm−1 θ −1 .

(16.2)

To obtain the upper bound for α, we apply Corollary 12.5 again and get  2  r (y) 2 ≤ (1 + δ)m H 2 ( p, y, (1 + δ)t). H ( p, p, t) exp − 2δt Integrating over ∂ B p (ρ) yields 

ρ2 A p (ρ) exp − 2δt



H 2 ( p, p, t) ≤ (1 + δ)m

∂ B p (ρ)

H 2 ( p, y, (1 + δ)t) dy.

Integrating over 0 ≤ ρ ≤ ∞ and using the semi-group property, we obtain  2 ∞ ρ A p (ρ) exp − H 2 ( p, y, (1 + δ)t)dy H 2 ( p, p, t)dρ ≤ (1 + δ)m 2δt 0 M = (1 + δ)m H ( p, p, 2(1 + δ)t). (16.3)

16

Large time behavior of the heat kernel

181

However, Lemma 16.1 asserts that A p (ρ)  θ, m ρ m−1 hence A p (ρ) ≥ θ m ρ m−1 . ¯ Observe that since A(ρ) = αm−1 ρ m−1 , when combined with (16.3), we conclude that   ∞ ρ2 −1 ¯ A(ρ) exp − H 2 ( p, p, t) dρ θ m αm−1 2δt 0 ≤ (1 + δ)m H ( p, p, 2(1 + δ)t). Multiplying both sides by (2(1 + δ)t)m/2 and using (16.1), we get   ∞ ρ2 −1 ¯ A(ρ) exp − (2(1 + δ)t)m/2 θ m αm−1 H 2 ( p, p, t) dρ ≤ (1 + δ)m α. 2δt 0 (16.4) Noting that (8π δt)−m/2 = H¯ ( p, ¯ p, ¯ 2δt) H¯ 2 ( p, ¯ y¯ , δt) d y¯ = Rm



∞

= 0

  ρ2 −m ¯ dρ, A p¯ (ρ)(4π δt) exp − 2δt

we can rewrite (16.4) as −1 (2(1 + δ))m/2 (4π δt)m H 2 ( p, p, t) (8π δ)−m/2 ≤ (1 + δ)m α. θ m αm−1

Letting t → ∞ and using (16.1), we obtain −1 θ m αm−1 (1 + δ)−m/2 (4π δ)m/2 α 2 ≤ α,

hence α ≤ θ −1 m −1 αm−1 (1 + δ)m/2 (4π δ)−m/2 . Letting δ → ∞ and combining with (16.2), we arrive at α = θ −1 m −1 αm−1 (4π )−m/2 , and the theorem follows by substituting the definition of θ.



182

Geometric Analysis

Corollary 16.3 Let M be a complete manifold with nonnegative Ricci curvature. Suppose there exists a constant θ > 0 such that lim inf ρ→∞

V p (ρ) =θ ρm

for some point p ∈ M. Then |π1 (M)| < ∞. Moreover, |π1 (M)| = lim

ρ→∞

V p˜ (ρ) , V p (ρ)

where p˜ ∈ M˜ is a preimage point in the universal cover M˜ of M and V p˜ (ρ) is ˜ the volume of a geodesic ball of radius ρ centered at p˜ in M. Proof Let π : M˜ → M be the projection map from the universal cover M˜ to M. We equip M˜ with the pull-back metric from M so that π is a local isometry ˜ If and the elements of the fundamental group π1 (M) act as isometries on M. ˜ ˜ we let H and H be minimal heat kernels on M and M, respectively, we then claim that  H (π(x), ˜ π( y˜ ), t) = H˜ (x, ˜ g( y˜ ), t) g∈π1 (M)

˜ In fact, we first claim that for any x, ˜ y˜ ∈ M.  H˜ (x, ˜ g( y˜ ), t) H1 (x, y, t) =

(16.5)

g∈π1 (M)

for x, y ∈ M with π(x) ˜ = x and π( y˜ ) = y is well defined. To see this, for any x, y ∈ M, let B y (ρ) be a sufficiently small geodesic ball around y such that their preimages {Bg( y˜ ) (ρ) | g ∈ π1 (M)} are all disjoint. In particular,  H˜ (x, ˜ z˜ , t) d z˜ ≤ H˜ (x, ˜ z˜ , t) d z˜ M˜

g∈π1 (M) Bg( y˜ ) (ρ)

≤ 1.

(16.6)

On the other hand, Corollary 12.5 asserts that ˜ y˜ , t) ≤ C Vg( y˜ ) (ρ) H˜ (x,



 2 ρ ˜ H (x, ˜ z˜ , 2t) d z˜ exp . 4t Bg( y˜ ) (ρ)

Combining this inequality with (16.6), we conclude that (16.5) is summable and hence well defined.

16

Large time behavior of the heat kernel

183

Obviously  y −

∂ ∂t

 H1 (x, y, t) = 0

on

M × (0, ∞)

with H1 (x, y, 0) = δx (y). The Duhamel principle now implies that for y ∈ Bx (ρ), H (x, y, t) − H1 (x, y, t) t ∂ φ 2 (z) H1 (x, z, t − s) H (z, y, s) dz ds = 0 ∂s M t φ 2 (z) H1 (x, z, t − s) H (z, y, s) dz ds =− 0

+ =2

t

φ 2 (z) H1 (x, z, t − s) H (z, y, s) dz ds

0

M

0

M

t

−2

M

φ(z) H (z, y, s) ∇ H1 (x, z, t − s), ∇φ(z) dz ds

t 0

φ(z) H1 (x, z, t − s) ∇ H (z, y, s), ∇φ(z) dz ds

(16.7)

M

for any nonnegative cutoff function φ satisfying . 1 on Bx (ρ), φ= 0 on M \ Bx (2ρ), and |∇φ|2 ≤ 2ρ −2 . However, the Schwarz inequality asserts that   t    φ(z) H (z, y, s) ∇ H1 (x, z, t − s), ∇φ(z) dz ds   0 M √ −1 t ≤ 2ρ H (z, y, s) |∇ H1 (x, z, t − s)| dz ds 0

≤

√

2ρ −1

0

 ×

Bx (2ρ)\Bx (ρ)

t 

Bx (2ρ)\Bx (ρ)

1/2 H 2 (z, y, s) dz 1/2

|∇ H1 (x, z, t − s)| dz 2

Bx (2ρ)\Bx (ρ)

ds.

(16.8)

184

Geometric Analysis

Similarly, we also have   t    φ(z) H1 (x, z, t − s) ∇ H (z, y, s), ∇φ(z) dz ds   0

≤

M

√

2ρ

−1

t Bx (2ρ)\Bx (ρ)

0

≤

√

2ρ −1

1/2

t 

Bx (2ρ)\Bx (ρ)

0

 ×

H1 (x, z, t − s) |∇ H (z, y, s)| dz ds H12 (x, z, t − s) dz 1/2

|∇ H (z, y, s)| dz 2

Bx (2ρ)\Bx (ρ)

ds.

(16.9)

On the other hand, the semi-group property asserts that H 2 (z, y, s) dz = H (y, y, 2s), M

hence H ∈ L 2 (M). Also, for the cutoff function φ defined above, we have φ 2 (z) |∇ H |2 (z, y, s)dz = − φ 2 H (z, y, s) H (z, y, s) dz M

M

−2

φ(z)H (z, y, s)∇φ(z), ∇ H (z, y, s)dz M

  1 ∂ φ 2 (z) H 2 (z, y, s) dz 2 ∂s M 1 + φ 2 (z) |∇ H (z, y, s)|2 dz 2 M +2 |∇φ|2 (z) H 2 (z, y, s) dz,

≤−

M

implying that   ∂ φ 2 (z) |∇ H (z, y, s)|2 dz ≤ − φ 2 (z) H 2 (z, y, s) dz ∂s M M +4 |∇φ|2 (z) H 2 (z, y, s) dz M

∂ ≤− ∂s





+ 2ρ −2

φ (z) H (z, y, s) dz 2

2

M



H 2 (z, y, s) dz. M

16

Large time behavior of the heat kernel

185

Letting ρ → ∞ and using the fact that H ∈ L 2 , we conclude that





∂ |∇ H | (z, y, s) dz ≤ − ∂s M

H (z, y, s) dz

2

=

2

M

∂ H (y, y, 2s), ∂s

hence |∇ H | ∈ L 2 (M). In the case of H1 , we also observe that if M˜ 1 is a fixed fundamental domain of M in M˜ and use the fact that H˜ (g(x), ˜ g( y˜ ), t) = H˜ (x, ˜ y˜ , t) for any g ∈ π1 (M), then M

H12 (x, z, t − s) dz = =





˜ g, h∈π1 (M) M1





˜ g, h∈π1 (M) M1

H˜ (x, ˜ g(˜z ), t − s) H˜ (x, ˜ h(˜z ) t − s) d z˜ H˜ (x, ˜ g(˜z ), t − s)

× H˜ (g ◦ h −1 (x), ˜ g(˜z ) t − s) d z˜  H˜ (x, ˜ z˜ , t − s) = ˜ g, h∈π1 (M) g( M1 )

× H˜ (g ◦ h −1 (x), ˜ z˜ , t − s) d z˜ . However, since 



H˜ (g ◦ h −1 (x), ˜ z˜ , t − s) =

h∈π1 (M)

H˜ (k(x), ˜ z˜ , t − s),

k∈π1 (M)

we conclude that M

H12 (x, z, t

− s) dz = = =





˜ g, k∈π1 (M) g( M1 )



˜ k∈π1 (M) M



H˜ (x, ˜ z˜ , t − s) H˜ (k(x), ˜ z˜ , t − s) d z˜

H˜ (x, ˜ z˜ , t − s) H˜ (k(x), ˜ z˜ , t − s) d z˜

H˜ (x, ˜ k(x), ˜ 2(t − s))

k∈π1 (M)

= H1 (x, ˜ x, ˜ 2(t − s)),

186

Geometric Analysis

hence H1 ∈ L 2 (M). The same argument we used before also implies that |∇ H1 | ∈ L 2 (M). Therefore the right-hand sides of (16.8) and (16.9) tend to 0 as ρ → ∞, and (16.7) yields H (x, y, t) = H1 (x, y, t). Applying Theorem 16.2, we have √ m −1 αm−1 (4π )−m/2 = lim V p ( t) H ( p, p, t) t→∞



= lim

t→∞

g∈π1 (M)



= lim

t→∞

√ V p ( t) H˜ ( p, ˜ g( p), ˜ t)

g∈π1 (M)

√ √ V p ( t) ˜ g( p), ˜ t). (16.10) √ V p˜ ( t) H˜ ( p, V p˜ ( t)

On the other hand, we observe that M˜ also has nonnegative Ricci curvature √ √ and V p˜ ( t) ≥ V p ( t), hence we can apply Theorem 16.2 to H˜ and obtain √  V p ( t) √ ˜ g( p), ˜ t) = |π1 (M)| m −1 αm−1 (4π )−m/2 √ V p˜ ( t) H˜ ( p, lim t→∞ V ( t) p ˜ g∈π (M) 1

× lim

t→∞

√ V p ( t) √ . V p˜ ( t)

Combining this identity with (16.10), we conclude that √ V p ( t) |π1 (M)| lim √ = 1, t→∞ V p˜ ( t) as claimed. In particular, since lim inf ρ→0

V p (ρ) =θ ρm

and V p˜ (ρ) ≤ m −1 αm−1 ρ m , we deduce that |π1 (M)| ≤ m −1 αm−1 θ −1 is finite.



16

Large time behavior of the heat kernel

187

The following lemma proved in [L5] asserts a mean value type property at infinity of a complete manifold with nonnegative Ricci curvature. This will be useful in Chapter 27 when we study linear growth harmonic functions. Lemma 16.4 (Li) Let M m be a complete manifold with nonnegative Ricci curvature. Suppose f is a bounded subharmonic function defined on M, then f (x) d x = sup f. lim V p−1 (ρ) ρ→∞

Proof

B p (ρ)

M

Let us define g0 (x) = sup M f − f (x), then g0 satisfies g0 ≤ 0

and g0 ≥ 0. It suffices to show that lim V −1 (ρ) ρ→∞ p

B p (ρ)

g0 (x) d x = 0.

Let us consider the solution of the heat equation given by g(x, t) = H (x, y, t) g0 (y) dy. M

Observe that if Hi (x, y, t) is a Dirichlet heat kernel on a compact subdomain i ⊂ M, then ∂ ∂ Hi (x, y, t) g0 (y) dy Hi (x, y, t) g0 (y) dy = ∂t i i ∂t =  y Hi (x, y, t) g0 (y) dy =

i

i

Hi (x, y, t)  y g0 (y) dy



+

∂i

∂ Hi (x, y, t) g0 (y) dy, ∂ν

≤ 0, ! implying the function gi (x, t) = i Hi (x, y, t) g0 (y) dy is nonincreasing in t. Since H (x, y, t) is the limit of a sequence of Dirichlet heat kernels, we conclude that g(x, t) is also nonincreasing in t. However, the maximum principle,

188

Geometric Analysis

Theorem 15.2, asserts that g(x, t) ≥ 0, hence g(x, t) converges to a nonnegative, bounded function g∞ (x) as t → ∞. Moreover, g∞ is harmonic because ∂g(x, t) , ∂t which tends to 0 as t → ∞ by the fact that ∂g/∂t ≤ 0 and ∞ ∂g (x, t) dt = g∞ (x) − g0 (x). ∂t 0 g(x, t) =

On the other hand, since M has nonnegative Ricci curvature, a bounded harmonic function must be constant. The facts that g∞ ≤ g0 and inf M g0 = 0 imply that g∞ = 0, hence g(x, t) → 0. Note that for p ∈ M, we have H ( p, y, t) g0 (y) dy g( p, t) = ≥

M √ B p ( t)

H ( p, y, t) g0 (y) dy.

(16.11)

Applying the lower bound (Corollary 13.9), we obtain √ H ( p, y, t) ≥ C V p−1 ( t) √ for all y ∈ B p ( t). Hence combining this inequality with (16.11) yields √ g( p, t) ≥ C V p−1 ( t) √ g0 (y) dy, B p ( t)

and the lemma follows.



17 Green’s function

Let M m be a compact manifold of dimension m with boundary ∂ M. Let  be the Laplacian defined on functions with a Dirichlet boundary condition on M. Then standard elliptic theory asserts that there exists a Green’s function G(x, y) defined on M × M \ D, where D = {(x, x) | x ∈ M}, so that G(x, y)  f (y) dy = − f (x) (17.1) M

for all functions f satisfying the Dirichlet boundary condition f |∂ M = 0. Moreover, G(x, y) = 0 for y ∈ ∂ M and x ∈ M \∂ M. Since both G and f satisfy the Dirichlet boundary condition, after integration by parts, (17.1) becomes  y G(x, y) f (y) dy = − f (x), (17.2) M

which is equivalent to saying that  y G(x, y) = −δx (y),

(17.3)

where δx (y) is the delta function at x. If we let f (x) = G(z, x), then (17.2) yields G(z, x) = −  y G(x, y) G(z, y) dy. M

However, applying (17.1) to the right-hand side, we obtain G(z, x) = G(x, z).

(17.4) 189

190

Geometric Analysis

This shows that G(x, y) must be symmetric in the variables x and y. We also observe that by letting f (x) = x G(z, x) in (17.2), we have x G(z, x) = −  y G(x, y)  y G(z, y) dy. M

Since the right-hand side is symmetric in x and z, we conclude that x G(z, x) = z G(x, z). On the other hand, z G(x, z) = z G(z, x) by the symmetry of G, and we conclude that x G(z, x) = z G(z, x). Therefore (17.2) can be written as x G(x, y) f (y) dy = x G(x, y) f (y) dy M

M

= − f (x).

(17.5)

Hence if we define the operator −1 by −1 f (x) = − G(x, y) f (y) dy, M

then (17.1) and (17.5) give  −1 = I and −1  = I, respectively. When M m is a complete, noncompact manifold without boundary, one would like to obtain a Green’s function G(x, y) that also satisfies the properties (17.1), (17.5) and (17.4) for any compactly supported function f ∈ Cc∞ (M). For the case M = Rm , it is well known that the function . ((m − 2)αm−1 )−1 r (x, y)2−m for m ≥ 3, G(x, y) = −(α1 )−1 log(r (x, y)) for m = 2,

17

Green’s function

191

where αm−1 is the area of the unit (m − 1)-sphere in Rm , satisfies these properties. In fact, these formulas, when interpreted appropriately, reflect the ideal situations for manifolds also. Let us now assume that M m has a point p around which the metric is rotationally symmetric. This is equivalent to saying that if we take polar coordinates (r, θ2 , . . . , θm ) around p, then 2 = dr 2 + gαβ dθα dθβ ds M

with gαβ (r ) being functions of r alone. The Laplacian with respect to this coordinate system will take the form =

√ ∂2 1 ∂ g ∂ + ∂ B p (r ) , + √ g ∂r ∂r ∂r 2

where g = det(gαβ ) and ∂ B p (r ) denotes the Laplacian defined on the sphere, ∂ B p (r ), of radius r centered at p. If we let A p (r ) be the area of ∂ B p (r ), then "

Since

√

g(r ) dθ = A p (r ).

g is independent of the θα s, √ Ap (r ) 1 ∂ g = , √ g ∂r A p (r )

and we have =

Ap (r ) ∂ ∂2 + + ∂ B p (r ) . A p (r ) ∂r ∂r 2

Let us now consider the function

r (y)

G(y) = − 1

dt , A p (t)

where r (y) is the distance from p to y. Direct computation gives G =

Ap (r ) ∂G ∂ 2G + A p (r ) ∂r ∂r 2

=0

192

Geometric Analysis

for y = p. Moreover, if f ∈ Cc∞ (M) is a compactly supported smooth function, then G(y)  f (y) dy = G(y) f (y) dy M\B p ()

M\B p ()

∂G ∂f G(y) dθ + f dθ − ∂r ∂ B p () ∂ B p () ∂r  1 dt ∂f f dθ. dθ − = A p () ∂ B p () 1 A p (t) ∂ B p () ∂r

(17.6) Note that the continuity of f implies that 1 f dθ → f ( p) A p () ∂ B p () as  → 0. Also the continuity of  f and the divergence theorem imply that 1 ∂f 1 dθ =  f dy V p () ∂ B p () ∂r V p () B p () →  f ( p) as  → 0, where V p (r ) is the volume of B p (r ). This implies that ∂f dθ ∼  f ( p) V p (). ∂ B p () ∂r On the other hand, since A p (t) ∼ αm−1 t m−1 , where αm−1 is the area of the unit (m − 1)-sphere in Rm , we have ⎧ 1 ⎪ ⎪  2−m for m ≥ 3,  ⎨ − (m − 2) α dt m−1 ∼ ⎪ 1 1 A p (t) ⎪ ⎩ log() for m = 2. α1 Therefore, by letting  → 0, the right-hand side of (17.6) becomes

 dt 1 ∂f dθ − f dθ = − f ( p), lim →0 A p () ∂ B p () 1 A p (t) ∂ B p () ∂r

17 hence

Green’s function

193

G(y)  f (y) dy = − f ( p). M

In particular, this implies that one can take r (y) G( p, y) = − 1

dt A p (t)

(17.7)

to be the Green’s function. Note that if ∞ dt < ∞, A p (t) 1 then by adding this constant to G, we can use the formula ∞ dt . G( p, y) = A p (t) r (y)

(17.8)

The Green’s functions in Rm are given in the form of (17.7) and (17.8) for dimensions m = 2 and m ≥ 3, respectively. We are now ready to construct a Green’s function on a complete manifold. We should point out that an existence proof was first given by Malgrange [M]. However, for the purpose of application, a constructive argument is a key step in getting the appropriate estimates. The first constructive proof was published in [LT2]. ∞ be a compact exhaustion of M, such that Let {i }i=1 i   j

for i < j,

∞ i ∪i=1

= M,

and each i is a sufficiently smooth compact subdomain of M. For each i, let G i (x, y) be the symmetric, positive, Green’s function with a Dirichlet boundary condition of i . Moreover, . ((m − 2)αm−1 )−1 r 2−m ( p, y) for m ≥ 3, G i ( p, y) ∼ for m = 2, −(α1 )−1 log r ( p, y) as y → p. Lemma 17.1 Let  ⊂ M be a connected open subset of a complete manifold M. Suppose { f i } is a sequence of positive harmonic functions defined on . If there exists a point p ∈  such that the sequence f i ( p) is bounded, then after passing through a subsequence fi converges uniformly on compact subsets of  to a harmonic function f .

194

Geometric Analysis

Proof For any point x ∈ , let 2ρ(x) be the distance from x to ∂, then Theorem 6.1 asserts that |∇ log f i | ≤ C(m, K , ρ(x))

on

Bx (ρ(x))

for some constant C(m, K , ρ(x)) depending only on m, K , and ρ(x). Let D be any compact, connected subset of  containing p. By compactness, D can be covered by a finite number of geodesic balls of the form B xi (ρ(xi )). Hence the functions |∇ log f i | are uniformly bounded on D. Integrating this along a curve γ in D joining any point x ∈ D to p yields d log f i log f i (x) − log f i ( p) = ds ds γ ≤ |∇ log f i |ds γ

≤ C (γ ). This implies that ˜ p, x)), f i (x) ≤ fi ( p) exp(C d( ˜ p, x) is the distance from p to x within D. The compactness of D and where d( the fact that the sequence { f i ( p)} is uniformly bounded for all i implies that the sequence of functions { f i } is uniformly bounded on D. Hence by passing through a subsequence, fi → f uniformly on compact subsets of . Since |∇ f i | = |∇ log f i | fi are bounded on D, f i → f must converge in the C 0,1 sense and f is Lipschitz. Moreover, f i being harmonic implies that 0= φ  fi 



=−



∇φ ∇ f i 

for any compactly supported function φ defined on . Passing to the limit, we conclude that 0 = − ∇φ, ∇ f  

and f is weakly harmonic. Regularity theory then implies that f is smooth and harmonic. 

17

Green’s function

195

Lemma 17.2 Let p ∈ M be a fixed point and D be any compact subset of M \{ p}. For sufficiently large i, so that p and D are properly contained in i , the sequence of Green’s functions G i ( p, y) as functions of y must have uniformly bounded oscillations on D. Proof It suffices to show that the oscillations of the G i s are uniformly bounded on sets of the form B p (ρ2 )\ B p (ρ1 ), where 0 < ρ1 < ρ2 < ∞. Let us denote the oscillation of G i ( p, y) by ωi (ρ1 , ρ2 ) = osc y∈B p (ρ2 )\B p (ρ1 ) G i ( p, y) =

sup

y∈B p (ρ2 )\B p (ρ1 )

G i ( p, y) −

inf

y∈B p (ρ2 )\B p (ρ1 )

G i ( p, y).

We will prove the uniform oscillation bound by contradiction. Let us assume that there is a subsequence of the {ωi } such that ωi → ∞. We define the set of functions gi (y) by gi (y) = ωi−1 G i ( p, y) − ωi−1

inf

z∈B p (ρ2 )

G i ( p, z).

Each gi is a harmonic function defined on i \{ p} with the properties that osc y∈B p (ρ2 )\B p (ρ1 ) gi (y) = 1, gi (y) ∼ ωi−1 G i ( p, y)

as

y → p,

(17.9) (17.10)

and inf

y∈B p (ρ2 )

gi (y) = 0.

Note that since the function G i ( p, y) is harmonic when restricted to B p (ρ2 )\ { p} and G i ( p, y) → ∞ as y → p, the maximum principle asserts that its minimum must occur on ∂ B p (ρ2 ), hence inf

y∈∂ B p (ρ2 )

gi (y) = 0.

It is also convenient to define ii (r ) =

inf

y∈∂ B p (r )

G i ( p, y)

(17.11)

196

Geometric Analysis

and si (r ) =

sup

y∈∂ B p (r )

G i ( p, y).

Applying the maximum principle to the domain i \ B p (r ), we conclude that si (r ) is a montonically decreasing function since G i ( p, y) = 0 on ∂i . ¯ p, y), then we If we denote the Dirichlet Green’s function on B p (ρ2 ) by G( claim that gi must satisfy the estimates ¯ p, y) ≤ gi (y) ωi−1 G( ¯ p, y) + 1. ≤ ωi−1 G(

(17.12)

To see this, let α > 0 be any positive constant, then the function ¯ p, y) (1 + α) gi (y) − ωi−1 G( tends to ∞ as y → p because of the asymptotic behavior of G¯ and (17.10). It is also nonnegative on ∂ B p (ρ2 ). Hence the maximum principle implies that it must be nonnegative on B p (ρ2 ). Letting α → 0, we obtain the lower bound for gi . The upper bound follows similarly by considering the function ¯ p, y) − 1 (1 − α) gi (y) − ωi−1 G( and using (17.9) and (17.11). The assumption that ωi → ∞, (17.12), and Lemma 17.1 imply that there is a subsequence of gi (y) that converges uniformly on compact subsets of B p (ρ2 )\{ p}. We also denote this subsequence by gi (y). Moreover, the limiting function g is harmonic and bounded between 0 and 1. However, the removable singularity theorem for harmonic functions asserts that g can be extended to a harmonic function defined on B p (ρ2 ). On the other hand, the property that si (r ) is monotonically decreasing implies that sup

y∈∂ B p (r )

gi (y)

is decreasing in r . Passing to the limit, we conclude that sup

y∈∂ B p (r )

g(y)

is nonincreasing in r. In particular, g( p) ≥ sup y∈∂ B p (r ) g(y) for all r > 0, and p must be a local maximum for g. However, this violates the maximum principle unless g is identically constant. On the other hand, the fact that si (r ) is monotonically decreasing implies that the supremum of gi on M \ B p (ρ1 ) is achieved on ∂ B p (ρ1 ). Using (17.9) and (17.11), we conclude that 1 − gi (y) is nonnegative on M \ B p (ρ1 ). Using

17

Green’s function

197

Lemma 17.1, we conclude that 1 − gi must have a uniformly convergent subsequence since gi → g on B p (ρ2 )\{ p}. Hence gi → g, which is a constant function on M. However, after passing to the limit (17.9) implies that osc y∈B p (ρ2 )\B p (ρ1 ) g(y) = 1, which is a contradiction. Hence the oscillation of G( p, y) must be bounded on  B p (ρ2 )\ B p (ρ1 ). Theorem 17.3 Let M m be a complete manifold without boundary. There exists a Green’s function G(x, y) which is smooth on (M × M)\ D satisfying properties (17.1), (17.4), and (17.5). Moreover, G(x, y) can be taken to be positive if and only if there exists a positive superharmonic function f on M \ B p (ρ) with the property that lim inf f (x) < x→∞

inf

x∈∂ B p (ρ)

f (x).

Proof Let us first observe that if G(x, y) is a positive symmetric Green’s function, then the maximum principle asserts that the infimum of G( p, y) must be achieved at infinity of M. The function f (y) given by the restriction of G( p, y) on M \ B p (ρ) is a positive harmonic function with the desired property. We will first construct G( p, y) assuming that there is a nonconstant, positive, superharmonic function f defined on M \ B p (ρ) whose infimum is ∞ be a compact exhaustion of M with achieved at infinity of M. Let {i }i=1 the property that B p (ρ) ⊂ i for all i. Let G i ( p, y) be the Dirichlet Green’s function defined on i . Note that G i ( p, y) are monotonically increasing. In fact, the maximum principle and the asymptotic behavior of G i ( p, y) as y → p assert that (1 + α)G j ( p, y) ≥ G i ( p, y) for α > 0 and i < j, and the monotonicity follows by taking α → 0. As before, let si (ρ) =

sup

y∈∂ B p (ρ)

G i ( p, y),

then si (ρ) is a monotonically increasing sequence in i. We now claim that si (ρ) is bounded from above. To see this, let us consider the sequence of functions si−1 (ρ) G i ( p, y).

198

Geometric Analysis

By addition and multiplication by constants, we may assume that inf

x∈∂ B p (ρ)

f (x) = 1

and lim inf f (x) = 0.

(17.13)

x→∞

The maximum principle asserts that si−1 (ρ) G i ( p, y) ≤ f (y)

M \ B p (ρ).

on

By passing to a subsequence, the sequence of functions {si−1 (ρ) G i ( p, y)} converges uniformly on compact subsets of M \ B p (ρ) to a harmonic function g satisfying the properties that g(y) ≤ f (y)

on

M \ B p (ρ)

and sup

x∈∂ B p (ρ)

g(x) = 1.

(17.14)

Lemma 17.1 asserts that the subsequence, in fact, converges on a compact subset of M \{ p} and g is a harmonic function on M \{ p}. However, the maximum principle implies that ¯ p, y) ≤ s −1 (ρ) G i ( p, y) ≤ s −1 (ρ) G( ¯ p, y) + 1, si−1 (ρ) G( i i where G¯ is the Dirichlet Green’s function on B p (ρ). If si (ρ) → ∞, then similarly to the argument of Lemma 17.2 we conclude that g must be identically 0 due to (17.13), hence contradicting (17.14). Therefore, si (ρ) must be bounded. Let s(ρ) = limi→∞ si (ρ). The maximum principle again implies that G i ( p, y) ≤ s(ρ) f (y)

on

M \ B p (ρ).

Arguing as before, we conclude that the monotonic sequence G i ( p, y) converges uniformly on compact subsets of M \{ p} to a harmonic function G( p, y). In particular, given any  > 0, for sufficiently large i, we have 0 ≤ G( p, y) − G i ( p, y) ≤ 

on

∂ B p (1).

The maximum principle then asserts that G i ( p, y) ≤ G( p, y) ≤ G i ( p, y) + 

on

B p (1)\{ p}.

17

Green’s function

199

In particular, if f ∈ Cc∞ (M), then      G( p, y)  f (y) dy − G i ( p, y)  f (y)  M

M



≤

|G( p, y) − G i ( p, y)| | f |(y) dy M



≤

B p (1)

| f |(y) dy +

M\B p (1)

|G( p, y) − G i ( p, y)| | f |(y) dy.

Note that the second term on the right-hand side tends to 0 as i → ∞, and using G i ( p, y)  f (y) dy = − f ( p) M

we conclude that       G( p, y)  f (y) dy + f ( p) ≤   M

B p (1)

| f |(y) dy.

Letting  → 0, this yields property (17.1) for G( p, y). Observe that the above argument shows that G i (q, y) → G(q, y) as long as q ∈ B p (ρ/2). For arbitrary q ∈ M, we just need to take ρ sufficiently large such that q ∈ B p (ρ/2). Hence G i (q, y) → G(q, y) for any q ∈ M. Since each G i satisfies (17.4) and (17.5), G will also inherit the same properties. Note that the function G obtained in this manner will have the minimal property, namely, ˜ G(x, y) ≥ G(x, y), ˜ Indeed, since the maximum principle for any positive Green’s function G. implies that ˜ G(x, y) ≥ G i (x, y)

on

i ,

the minimal property of G follows by taking the limit as i → ∞. We now assume that f does not exist on M \ B p (ρ). This is equivalent to saying that si (ρ) → ∞

200

Geometric Analysis

as i → ∞ for all fixed ρ. Let ai =

inf

y∈∂ B p (1)

G i ( p, y).

The sequence of functions {h i ( p, y) = G i ( p, y) − ai } then has the property that inf

y∈∂ B p (1)

h i ( p, y) = 0.

Also, Lemma 17.2 asserts that there

is  a constant ω such that the oscillations of G i ( p, y) over the set B p (2)\ B p 12 is bounded by ωi (1, 2) ≤ ω. In particular, sup

y∈∂ B p (1)

h i ( p, y) ≤ ω

for all i. Hence the sequence of positive harmonic functions {h i + ω} when restricted to B p (2)\{ p} has a convergent subsequence, also denoted by {h i + ω}, such that {h i ( p, y) + ω} converges uniformly on compact subsets of B p (2)\{ p} to a harmonic function G( p, y) + ω. Similarly, the sequence  of positive harmonic functions {ω − h i } when restricted to M \ B p 12 has

a subsequence converging to ω − G 1 ( p, y) on compact subsets of M \ B p 12 . Since G 1 ( p, y) = G( p, y) on B p (2)\ B p 12 , we can denote G 1 by G also, and G( p, y) is the uniform limit of h i ( p, y) on compact subsets of M \{ p}. Moreover, since h i ( p, y)  f (y) dy = G i ( p, y)  f (y) dy − ai  f (y) dy M

M

M

= − f ( p) for f ∈ Cc∞ (M), the previous argument implies that G( p, y) also satisfies ¯ p, y) is the Dirichlet Green’s property (17.1). We also observe that if G( function on B p (1), then the maximum principle implies that ¯ p, y) ≤ h i ( p, y) ≤ G( ¯ p, y) + ω. G( Hence by passing through the limit, we conclude that ¯ p, y) ≤ G( p, y) ≤ G( ¯ p, y) + ω. G(

(17.15)

17

Green’s function

201

For another point q = p, the same argument implies that there is a subsequence i j of i and a sequence of constants bi j such that G i j (q, y) − bi j converges uniformly on compact subset of M \{q}. Setting y = p, we observe that since (G i j (q, p) − bi j ) = (G i j ( p, q) − ai j ) + (ai j − bi j ), by taking the limit as j → ∞, we conclude that lim G i j (q, p) = G( p, q) + lim (ai j − bi j ).

j→∞

j→∞

Hence the limit lim (ai j − bi j ) = c

j→∞

must exist and the sequence of functions G i j (q, y) − ai j = (G i j (q, y) − bi j ) − (ai j − bi j ) also converges. Let lim G i j (q, y) − ai j = J (q, y).

j→∞

Again, setting y = p, we conclude that G( p, q) = J (q, p). We now claim that the original sequence G i (q, y) − ai must converge to J (q, y) also. To see this, it suffices to show that if G ik (q, y) − aik is any other convergent subsequence of i, then it must converge to J (q, y). Let lim (G ik (q, y) − aik ) = K (q, y)

k→∞

denote its limit. The same argument as above shows that K (q, p) = G( p, q), hence K (q, p) = J (q, p).

(17.16)

202

Geometric Analysis

On the other hand, if we let ρ be sufficiently large so that Bq (1) ⊂ B p (ρ), then G( p, y) − K (q, y) = lim (G ik ( p, y) − aik ) − lim (G ik (q, y) − aik ) k→∞

k→∞

= lim (G ik ( p, y) − G ik (q, y)). k→∞

Since G ik satisfies the Dirichlet boundary condition on ik , for sufficiently large k such that B p (ρ) ⊂ ik , the maximum principle asserts that the harmonic function G ik ( p, y) − G ik (q, y) when restricted on ik\ B p (ρ) must have its maximum achieved on ∂ B p (ρ). Passing to the limit, we conclude that sup

(G( p, y) − K (q, y)) =

y∈M\B p (ρ)

sup (G( p, y) − K (q, y))

y∈∂ B p (ρ)

and G( p, y) − K (q, y) is bounded on M \ B p (ρ). A similar argument implies that G( p, y) − J (q, y) is also bounded on M \ B p (ρ) and therefore K (q, y) − J (q, y) is bounded on M \ B p (ρ). This together with estimate (17.15), when applied to both K and J at the point q, implies that K (q, y) − J (q, y) is bounded on M \{q}. Again the removable singularity theorem asserts that K (q, y) − J (q, y) is a bounded harmonic function in y. However, this will imply that M admits a harmonic function on M \ B p (ρ) with its infimum achieved at infinity unless K (q, y) − J (q, y) is identically constant. The identity (17.16) implies that K (q, y) = J (q, y). In particular, we write this as G(q, y). Properties (17.4) and (17.5) obviously follow since G is the limit of functions satisfying these properties. 

18 Measured Neumann Poincar´e inequality and measured Sobolev inequality

In this chapter, we will adapt an argument by Hajtasz and Koskela [HK] that proved the validity of the Neumann Poincar´e inequality which when combined with a volume doubling property implies the validity of a Sobolev type inequality. In particular, we will use this argument to establish a measured Sobolev inequality to be used in the next chapter. Definition 18.1 We say that a subdomain  of a manifold satisfies the volume doubling property (VD) if there exists a constant CVD () > 0 such that V p (2ρ) ≤ C VD () V p (ρ) for all p ∈  and for all 0 < ρ ≤ d(y, ∂)/2, where d(y, ∂) is the distance from y to ∂. In particular, we call the smallest such constant, CVD () = sup

V p (2ρ) , V p (ρ)

the volume doubling constant on . In addition to the volume doubling property, we also assume that M admits a uniform lower bound for the scaled first nonzero Neumann eigenvalue for geodesic balls. Again, one can localize the argument by introducing the following local λ1 bound. Definition 18.2 We say that C P () > 0 is a scaled local λ1 bound for the first nonzero Neumann eigenvalues over the smooth subdomain  if inf(ρ 2 λ1 (B y (ρ))) = C P (),

203

204

Geometric Analysis

where the infimum is taken over all y ∈  and all ρ ≤ d(y, ∂)/2, where d(y, ∂) is the distance from y to ∂. We will now state the main theorem of this chapter. Theorem 18.3 Let M be a complete manifold (possibly with boundary) and B p (ρ) be a geodesic ball in M satisfying B p (ρ) ∩ ∂ M = ∅. For any fixed 0 < δ < 1, let φ(r ) be a function defined on B p (ρ) given by ⎧ 1 ⎪ ⎪ ⎪ ⎨ ρ −r φ(r ) = (1 − δ)ρ ⎪ ⎪ ⎪ ⎩ 0

if

r ≤ δρ,

if

δρ ≤ r ≤ ρ,

if

ρ ≤ r.

Then there exists a universal constant C 1 > 0 depending on CP (B p (ρ)) and CVD (B p (ρ)) such that the measured Neumann Poincar´e inequality of the form C1 ρ −2

B p (ρ)

| f (x) − f¯|2 φ 2 (x) d V ≤

is valid for all f ∈ C ∞ (B p (ρ)), where f¯ =

B p (ρ)

! B p (ρ)

|∇ f |2 (x) φ 2 (x) d V

f (x) φ 2 (x) d V.

To prove the theorem, we will first establish a local version of the measured Neumann Poincar´e inequality. Lemma 18.4 Let B p (ρ) ⊂ M be a geodesic ball satisfying B p (ρ) ∩ ∂ M = ∅. Assume that B p (ρ) has a scaled local λ1 bound given by CP (B p (ρ)) > 0. Then the measured Neumann Poincar´e inequality CP (B p (ρ)) 9τ 2

B y (τ )

| f (x) − f˜|2 φ 2 (x) d V ≤

B y (τ )

|∇ f |2 (x) φ 2 (x) d V (18.1)

y ∈ B p (ρ) and for all 0 ≤ τ ≤ d(y, ∂ B p (ρ))/2, where f˜ = !is valid for all 2 B y (τ ) f (x) φ (x) d V.

18 Proof B y (τ )

Measured Neumann Poincar´e and Sobolev inequalities

205

The local λ1 bound asserts that | f (x) − f˜|2 φ 2 d V = inf



k∈R B y (τ )

| f (x) − k|2 φ 2 d V

≤ sup φ inf 2

k∈R B y (τ )

B y (τ )

| f (x) − k|2 d V

−1

≤ C P (B p (ρ)) τ

sup φ

2

B y (τ )

−1 (B p (ρ)) τ 2 ≤ CP

2 B y (τ )

|∇ f |2 (x) d V

sup By (τ ) φ 2 inf B y (τ ) φ 2

×

B y (τ )

|∇ f |2 (x) φ 2 (x) d V.

(18.2)

We observe that

sup φ = 2

B y (τ )

⎧ ⎪ ⎨



1

ρ − r ( p, y) + τ ⎪ ⎩ (1 − δ)ρ

2

if

r ( p, y) − τ ≤ δρ,

if

r ( p, y) − τ ≥ δρ,

and ⎧ 2 ⎪ ⎨ ρ − r ( p, y) − τ (1 − δ)ρ inf φ 2 = ⎪ B y (τ ) ⎩ 1

if

r ( p, y) + τ ≥ δρ,

if

r ( p, y) + τ ≤ δρ,

(18.3)

implying that sup B y (τ ) φ 2 inf B y (τ ) φ 2 ⎧ 1 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ (1 − δ)ρ ⎨ = ρ − r ( p, y) − τ ⎪ ⎪  ⎪ ⎪ ρ − r ( p, y) + τ 2 ⎪ ⎪ ⎩ ρ − r ( p, y) − τ

if

r ( p, y) + τ ≤ δρ,

if

r ( p, y) − τ ≤ δρ ≤ r ( p, y) + τ,

if

δρ ≤ r ( p, y) − τ.

206

Geometric Analysis

In the case when r ( p, y) − τ ≤ δρ ≤ r ( p, y) + τ , we have ρ − r ( p, y) + τ (1 − δ)ρ ≤ ρ − r ( p, y) − τ ρ − r ( p, y) − τ ≤3 since (ρ − r ( p, y) + τ )/(ρ − r ( p, y) − τ ) is an increasing function of 0 ≤ τ ≤ 12 (ρ − r ( p, y)). The same estimate also applies to the case when δρ ≤ r ( p, y) − τ. Inequality (18.1) follows by combining these estimates with (18.2).  We will also need the following volume doubling property for the measure φ 2 d V, which follows as a consequence of the volume doubling property (VD) for the background metric. Lemma 18.5 Let B p (ρ) ⊂ M be a geodesic ball satisfying B p (ρ) ∩ ∂ M = ∅. Suppose C VD (B p (ρ)) is the volume doubling constant for B p (ρ). Then the inequality

B y (2τ )

φ 2 (x) d V ≤ 16C VD (B p (ρ))

B y (τ )

φ 2 (x) d V

is valid for all y ∈ B p (ρ) and τ ≤ d(y, ∂ B p (ρ))/2. Proof If y ∈ B p (ρ) and τ ≤ d(y, ∂ B p (ρ))/2, then the volume doubling property for the background metric asserts that B y (2τ )

φ 2 (x) d V ≤ Vy (2τ ) sup φ 2 B y (2τ )

≤ C VD (B p (ρ)) Vy (τ ) sup φ 2 B y (2τ )

≤ C VD (B p (ρ))

sup By (2τ ) φ 2 inf By (τ ) φ 2

B y (τ )

φ 2 (x) d V.

Since

sup φ = 2

B y (2τ )

⎧ ⎪ ⎨



1

ρ − r ( p, y) + 2τ ⎪ ⎩ (1 − δ)ρ

2

if

r ( p, y) − 2τ ≤ δρ,

if

r ( p, y) − 2τ ≥ δρ,

(18.4)

18

Measured Neumann Poincar´e and Sobolev inequalities

207

when combining this with (18.3), we have sup B y (2τ ) φ 2 inf B y (τ ) φ 2 ⎧ 1 ⎪ ⎪ ⎪  ⎪ 2 ⎪ ⎪ (1 − δ)ρ ⎨ = ρ − r ( p, y) − τ ⎪ ⎪   ⎪ ⎪ ρ − r ( p, y) + 2τ 2 ⎪ ⎪ ⎩ ρ − r ( p, y) − τ

if

r ( p, y) + τ ≤ δρ,

if

r ( p, y) − 2τ ≤ δρ ≤ r ( p, y) + τ,

if

δρ ≤ r ( p, y) − 2τ

Estimating the same way as in the proof of Lemma 18.4 by using the assumption that τ ≤ d(y, ∂ B p (ρ))/2, we conclude that sup B y (2τ ) φ 2 inf B y (τ ) φ 2

≤ 16 

and the lemma follows from (18.4).

Theorem 18.3 now follows by applying Lemma 18.4, Lemma 18.5, and a measured Hajtasz–Koskela [HK] theorem. Theorem 18.6 Let B p (ρ) be a geodesic ball in a complete manifold satisfying B p (ρ) ∩ ∂ M = ∅. Let φ be a nonnegative function defined on B p (ρ). Suppose CVD,φ (B p (ρ)) > 0 is a constant such that the volume doubling property with respect to the measure φ 2 d V of the form φ 2 d V ≤ CVD,φ (B p (ρ)) φ2 d V B y (2τ )

B y (τ )

is valid for all y ∈ B p (ρ) and for all τ ≤ d(y, ∂ B p (ρ))/2, where d(y, ∂ B p (ρ)) = ρ − r ( p, y). We also assume that there exists a constant CP ,φ (B p (ρ)) > 0 such that the measured Neumann Poincar´e inequality | f (x) − k|2 φ 2 d V ≤ |∇ f |2 φ 2 d V C P ,φ (B p (ρ)) τ −2 inf k∈R B y (τ )

B y (τ )

is valid for all y ∈ B p (ρ) and for all τ ≤ d(y, ∂ B p (ρ))/2. Then the measured Neumann Poincar´e inequality is valid on B p (ρ). In particular, there exists a constant C φ > 0 depending only on CP ,φ (B p (ρ)) and CVD,φ (B p (ρ)) such that 2 2 ¯ C φ ρ −2 | f − f | φ dV ≤ |∇ f |2 φ 2 d V B p (ρ)

B p (ρ)

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Geometric Analysis

! for all f ∈ C ∞ (B p (ρ)), where f¯ = B p (ρ) f φ 2 d V. Moreover, there exists α > 2 depending on CVD,φ (B p (ρ)) such that the measured Neumann Sobolev inequality of the form

(α−2)/α 2α −2 2/α 2 ¯ α−2 Cφ ρ V (B p (ρ)) | f − f | φ dV ≤ |∇ f |2 φ 2 d V φ

B p (ρ)

is valid, where Vφ (B p (ρ)) = respect to the measure

B p (ρ)

!

φ 2 d V.

B p (ρ) φ

denotes the volume of B p (ρ) with

2 dV

Proof In this proof, we will abbreviate our notation by using C VD,φ and CP ,φ for CVD,φ (B p (ρ)) and CP ,φ (B p (ρ)), respectively. For any y ∈ B p (ρ), let γ (t) be a normal geodesic with γ (0) = p and γ () = y. Let us define the

  i j sequence of points y0 = γ (0) = p, y1 = γ (ρ/2), and yi = γ j=1 ρ/2 . Let i 0 be the largest i such that i  ρ < . 2j j=1

Let us define

 ρ  ⎧ ⎪ ⎨ B yi i+1 2 Bi =  ρ  ⎪ ⎩ By 2i+1

For any f ∈ C ∞ (B p (ρ)), we define f Bi =

Vφ−1 (Bi )

where

for

i < i0 ,

for

i ≥ i0.

f φ 2 d V, Bi

Vφ (Bi ) =

φ2 d V Bi

denotes the volume of Bi with respect to the measure φ 2 d V. We observe that f Bi → f (y) as i → ∞, and | f B0 − f (y)| ≤

∞ 

| f Bi − f Bi+1 |

i=0

≤

∞  i=0

(| f Bi − f Di | + | f Di − f Bi+1 |),

(18.5)

18

Measured Neumann Poincar´e and Sobolev inequalities

209

where

 ρ  ⊂ Bi ∩ Bi+1 2i+3

i  j i+3 . However, since D ⊂ B , we have with z i = γ i i j=1 ρ/2 + 3ρ/2 | f Bi − f Di | ≤ Vφ−1 (Di ) | f − f Bi | φ 2 d V Di = Bz i

≤ Vφ−1 (Di ) ≤



Di

Bi

| f − f Bi | φ 2 d V

CVD,φ Vφ−1 (Bi )



3

Bi

| f − f Bi | φ 2 d V

and similarly | f Bi+1 − f Di | ≤

CVD,φ Vφ−1 (Bi+1 )



3

Bi+1

| f − f Bi+1 | φ 2 d V.

Therefore combining this inequality with (18.5), we conclude that 3 | f B0 − f (y)| ≤ 2CVD ,φ

∞ 

Vφ−1 (Bi )



i=0

≤ 2C VD,φ 3

∞  

Vφ−1 (Bi )

i=0

≤

−1/2 2C VD,φ CP ,φ 3

Bi

| f − f Bi | φ 2 d V 1/2

| f − f Bi | φ d V 2

Bi

2

 1/2 ∞  ρ −1 2 2 Vφ (Bi ) |∇ f | φ d V . 2i+1 Bi i=0

(18.6) Note that for  > 0, since ∞ 

2−(i+1) < ∞,

i=0

(18.6) implies that ∞  i=0

ρi (Vφ−1 (Bi )

|∇ f |2 φ 2 d V )1/2 Bi

−3 − ≥ C 1 CVD | f B0 − f (y)| ,φ C P ,φ ρ 1/2

∞  i=0

ρi ,

210

Geometric Analysis

where ρi = 2−(i+1) ρ and C1 > 0 is a universal constant. In particular, we deduce that there exists j ≥ 0 such that −6 −2 |∇ f |2 φ 2 d V ≥ C12 CVD | f B0 − f (y)|2 ρ 2−2 . Vφ−1 (B j ) j ,φ C P ,φ ρ Bj

(18.7) Note that since r (y, yi ) ≤ ρ − r ( p, yi ) ≤ρ−

i  ρ 2j j=1

=

ρ , 2i

we have Bi ⊂ B y (3ρi )

for all i.

Therefore taking this together with (18.7), we obtain |∇ f |2 φ 2 d V B y (3ρ j )∩B p (ρ)

−6 −2 ≥ C 12 CVD | f B0 − f (y)|2 ρ 2−2 j ,φ C P ,φ Vφ (B j ) ρ −8 −2 ≥ C 12 CVD | f B0 − f (y)|2 ρ 2−2 Vφ (B y (ρ j )). j ,φ C P ,φ ρ

In particular, we conclude that for all y ∈ B p (ρ), there exists ρ y > 0 such that |∇ f |2 φ 2 d V ≥ C φ ρ −2 | f B0 − f (y)|2 ρ y2−2 Vφ (B y (ρ y )), B y (ρ y )∩B p (ρ)

(18.8) −10 where Cφ = C12 C VD ,φ C P ,φ . We now claim that the volume doubling property with respect to the measure φ 2 d V implies that there exists a constant β > 0 depending only on CVD,φ such that  β ρ Vφ (B p (ρ)) ≤ CVD,φ Vφ (B y (ρ y )) (18.9) ρy

18

Measured Neumann Poincar´e and Sobolev inequalities

211

for all y ∈ B p (ρ). Indeed, let γ be the geodesic joining y to p and we choose a sequence of points w0 , w1 , . . . , wk with w0 = y and wi ∈ γ such that r (wi , wi+1 ) = 2i ρ y , where k is the first integer such that p ∈ Bwk (2k ρ y ). k 2i ρ y ≤ r ( p, y), hence In particular, ρ y + i=2  k ≤ log2

 ρ −1 . ρy

Note that the volume doubling property asserts that 3 i i C VD ,φ Vφ (Bwi (2 ρ y )) ≥ Vφ (Bwi (5(2 ρ y )))

≥ Vφ (Bwi+1 (2i+1 ρ y )) and CVD,φ Vφ (Bwk (2k ρ y )) ≥ Vφ (B p (2k ρ y )), hence we have 3k+1 2 k CVD ,φ Vφ (B y (ρ y ))φ d V ≥ Vφ (B p (2 ρ y )).

On the other hand, if we set k  to be the smallest integer such that 

2k (2k ρ y ) ≥ ρ, then 



k + k ≤ log2

ρ ρy

 +1

and 

k k CVD ,φ Vφ (B p (2 ρ y )) ≥ Vφ (B p (ρ)).

Combining this volume estimate with (18.10), we obtain 3 log (ρ/ρ y )+1

CVD,φ2

Vφ (Bwk (2k ρ y )) ≥ Vφ (B p (ρ)),

and the claim (18.9) is established with β = 3 log2 CVD,φ .

(18.10)

212

Geometric Analysis

Let us now define the sublevel set by At = {y ∈ B p (ρ) | | f B0 − f (y)| ≥ t}. Applying (18.9) to (18.8), we conclude that for all y ∈ At , there exists ρ y > 0 such that 2(−1)β −1

ρ 2 Vφ

 ≥ Cφ t 2

(B p (ρ)) ρ ρy

B y (ρ y )∩B p (ρ)

2−2

2(−1)β −1

Vφ

1+2(−1)β −1

≥ C φ t 2 Vφ

|∇ f |2 φ 2 d V

(B p (ρ)) Vφ (B y (ρ y ))

(B y (ρ y )).

Without loss of generality, we may assume that β ≥ 2. The Vitali covering lemma asserts that there is a countable (possibly finite), mutually disjoint, subcollection {B1 (ρ1 ), B2 (ρ2 ), . . . , Bi (ρi ), . . .} of the collection {B y (ρ y ) | y ∈ At } with the properties that ρ1 ≥ ρ2 ≥ · · · ≥ ρi ≥ · · · . Moreover, for any y ∈ At , there exists an i such that Bi (ρi ) ∩ B y (ρ y ) = ∅ and B y (ρ y ) ⊂ Bi (3ρi ). In particular, we have 1−2(1−)β −1

Vφ

2 (At ) ≤ C VD ,φ



1−2(1−)β −1

Vφ

(Bi (ρi ))

i −2(1−)β −1

≤ C φ−1 t −2 ρ 2 Vφ  × i

≤

Bi (ρi )∩B p (ρ)

C φ−1 t −2 ρ 2

(B p (ρ))

|∇ f |2 φ 2 d V

−2(1−)β −1 Vφ (B p (ρ))

B p (ρ)

|∇ f |2 φ 2 d V,

where C φ > 0 is a constant depending only on C VD,φ and C P ,φ . Setting B=

Cφ−1 ρ 2

−2(1−)β −1 Vφ (B p (ρ))

1/(1−2(1−)β −1 )

|∇ f | φ d V 2

B p (ρ)

2

,

18

Measured Neumann Poincar´e and Sobolev inequalities

213

then if q is a constant satisfying q < 2/(1 − 2(1 − )β −1 ) the coarea formula implies that

B p (ρ)

| f − f B0 |q φ 2 d V = q

∞ 0



t q−1 Vφ (At ) dt

T

=q

t

q−1

0

Vφ (At ) dt + q

≤ T q Vφ (B p (ρ)) + q B = T q Vφ (B p (ρ)) +

∞ T

∞

t q−1 Vφ (At ) dt

t q−1−2/(1−2(1−)β

−1 )

dt

T

qB q−

× T q−2/(1−2(1−)β

−1 )

2 1−2(1−)β −1

.

Optimizing by choosing T =ρ

−1/2 (B p (ρ)) Vφ



1/2 2

B p (ρ)

2

|∇ f | φ d V

,

we obtain

1/q q

B p (ρ)

≤ C φ−1 ρ Vφ

1/q−1/2

2

| f − f B0 | φ d V

(B p (ρ))

1/2

×

B p (ρ)

|∇ f |2 φ 2 d V

.

(18.11)

Since 2/(1 + 2( − 1)β −1 ) > 2, we can choose q = 2 and obtain

inf

k∈R B p (ρ)

| f − k| φ d V ≤ 2

2

≤

B p (ρ)

| f − f B0 |2 φ 2 d V

Cφ−1 ρ 2

B p (ρ)

|∇ f |2 φ 2 d V,

(18.12)

establishing the measured Neumann Poincar´e inequality. In fact, since we can take 2 < q < 2/(1 + 2( − 1)β −1 ), a measured Neumann Sobolev type inequality also follows. To see this, we apply (18.11) and obtain

214

Geometric Analysis

1/q



| f − f¯| φ d V q

B p (ρ)

2

≤

1/q | f − f B0 | φ d V q

B p (ρ)

+

1/q | f B0 − f¯| φ d V q

B p (ρ)

≤ C φ−1 ρ Vφ

1/q−1/2

×

2

B p (ρ)

2

(B p (ρ))

1/2

|∇ f |2 φ 2 d V

1/q + Vφ (B p (ρ)) | f B0 − f¯|.

However, 1/q Vφ (B p (ρ)) | f B0

1/q − f¯| ≤ Vφ (B p (ρ))Vφ−1 (B0 )



| f − f¯| φ 2 d V B0



≤

1/q −1/2 Vφ (B p (ρ))Vφ (B0 )

B p (ρ)

1/2 2 2 ¯ | f − f | φ dV



≤

(18.13)

1/q−1/2 C VD,φ Vφ (B p (ρ))

1/2 B p (ρ)

| f − f¯|2 φ 2 d V

,

hence combining this inequality with (18.12) and (18.13), we conclude that

(α−2)/α B p (ρ)

| f − f¯|2α/(α−2) φ 2 d V

−2/α

≤ Cφ−1 ρ 2 Vφ ×

with α = 2q/(q − 2).

(B p (ρ))

B p (ρ)

|∇ f |2 φ 2 d V 

Obviously, Theorem 18.3 follows from Lemma 18.4, Lemma 18.5, and Theorem 18.6. Let us also point out that if we set φ ≡ 1, then Theorem 18.6 asserts that a local λ1 bound and volume doubling implies a Neumann Sobolev inequality. Corollary 18.7 Let B p (ρ) be a geodesic ball in a complete manifold satisfying B p (ρ) ∩ ∂ M = ∅. Suppose CVD (B p (ρ)) > 0 is the volume doubling constant on B p (ρ) and CP (B p (ρ)) > 0 is the scaled local λ1 bound as given by Definition 18.1 and Definition 18.2, respectively. Then there exist α > 2 and a

18

Measured Neumann Poincar´e and Sobolev inequalities

215

constant C1 > 0 depending only on CP (B p (ρ)) and CVD (B p (ρ)) such that, the Neumann Sobolev inequality of the form

(α−2)/α 2α/(α−2) 2 −2/α ¯ |f − f| dV ≤ C1 ρ V (B p (ρ)) |∇ f |2 d V B p (ρ)

is valid, for all f ∈ C ∞ (B p (ρ)), where f¯ =

B p (ρ)

! B p (ρ)

f d V.

We would like to point out that both constants C V D () and CP () are monotonic in the sense that C VD (1 ) ≤ C VD (2 ) and CP (1 ) ≥ C P (2 ) if 1 ⊂ 2 . In particular, if M has nonnegative Ricci curvature, then the volume comparison theorem asserts that C VD () = 2m for all  ⊂ M. Moreover, Corollary 14.6 asserts that CP (B p (ρ)) ≥ C0 for all p ∈ M and ρ > 0, where C0 > 0 is a constant depending only on m.

19 Parabolic Harnack inequality and regularity theory

In this chapter, we will present Moser’s version of the Nash–Mosers Harnack inequality for parabolic equations. The elliptic version (also proved by De Giorgi) can be considered as a special case when the solution is time independent. The iteration procedure of Moser was particularly useful in the theory of geometric analysis. We will attempt to cover this in as much generality as possible while keeping explicit account of the dependency of various geometric and analytic constants. In applying this type of argument in the study of geometric partial differential equations, often the explicit geometric dependency is crucial. As a result of these estimates, one derives a mean value inequality for nonnegative subsolutions and a Harnack inequality for positive solutions of a fairly general class of parabolic operators. In particular, it gives a C α estimate for solutions of any second order parabolic (elliptic) operators of divergence form with only measurable coefficients. This regularity result was the original motivation for the development of this theory. We shall point out that the mean value inequality and the Harnack inequality derived from this argument are applicable to a more general class of equations, while the ones given in earlier chapters yield stronger results but require more smoothness from the operator. Both approaches are important in the theory of geometric analysis, but they are suited to different types of situation. The following account is a slightly modified version of Moser’s argument that has been adapted to a more geometrical setting. Let us define the average value of a function f on a geodesic ball B p (ρ) by —

B p (ρ)

216

f d V = V p (ρ)

−1

B p (ρ)

f d V.

19

Parabolic Harnack inequality and regularity theory

217

When the point p is fixed, the average L q -norm of f over B p (ρ) is denoted by

1/q

– f q,ρ = —

q

B p (ρ)

f dV

and the regular L q -norm is denoted by  f q,ρ =

1/q .

q

B p (ρ)

f dV

When the function is defined on B p (ρ) × [T0 , T1 ], we denote its L q -norm over the set B p (ρ) × [T0 , T1 ] by  f q,ρ,[T0 ,T1 ] =

1/q



T1

q

B p (ρ)

T0

and the average L q -norm by −1



– f q,ρ,[T0 ,T1 ] = (T1 − T0 )

T1

f d V dt

1/q

—

T0

q

B p (ρ)

f dV

.

For the sake of further generalization to Riemannian manifolds, we will introduce a normalized Sobolev inequality on a geodesic ball centered at a point p ∈ M of radius ρ. The inequality asserts that for a fixed constant μ ≤ m/(m − 2), there exists a constant CSD > 0, depending on B p (ρ), such that —

C SD |∇φ| ≥ 2 ρ B p (ρ) 2



1/μ

—

B p (ρ)

φ

2μ

,

(19.1)

c (B p (ρ)) that has boundary condition φ = 0 on ∂ B p (ρ). Typifor all φ ∈ H1,2 cally, the constant μ is given by μ = m/(m − 2) when m ≥ 3, and 2 < μ < ∞ when m = 2. However, in the following context, we are not restricting the value of μ.

Lemma 19.1 Let M be a complete manifold of dimension m. Let us assume that the geodesic ball B p (ρ) centered at p with radius ρ satisfies B p (ρ) ∩ ∂ M = ∅. Suppose that u(x, t) is a nonnegative function defined on B p (ρ) × [T0 , T1 ] such that   ∂ − u ≥ −fu ∂t

218

Geometric Analysis

in the weak sense. Assume that the function f is nonnegative on B p (ρ) × [T0 , T1 ] and its L q -norm on B p (ρ) is finite for some μ/(μ − 1) < q ≤ ∞ on [T0 , T1 ], with A = sup – f q,ρ [T0 ,T1 ]

= sup —

1/q

B p (ρ)

[T0 ,T1 ]

f (x, t) d V q

for μ/(μ − 1) < q < ∞ and A =  f ∞,ρ,[T0 ,T1 ] =

sup

B p (ρ)×[T0 ,T1 ]

f

for q = ∞. If α and ν are defined by α = q(μ − 1)/(μ(q − 1) − q) > 0 and ν = (2μ − 1)/μ > 1, then for any k > 1, there exists constant C2 > 0, depending only on k, μ, and q, such that −1 α ) u∞,θρ,[T,T1 ] ≤ C 2 (k A ρ 2 CSD

+ ((1 − θ )

−2



ρ

CSD ρ2

−2

(ν−1)/ν

−1

+ (T − T0 )



ρ2 ) CSD

1/ν ν/k(ν−1)

1

× (T1 − T0 ) k –uk,ρ,[T0 ,T1 ] for any θ ∈ (0, 1) and T ∈ (T0 , T1 ). For 0 < k ≤ 1, there exists constant C3 > 0, depending only on k, μ, and q, such that  u∞,θρ,[T,T1 ] ≤ C 3

A

ρ2 CSD

×ρ −1/k

× Vp

α 

−2(ν−1)/ν

ρ  2

CSD ρ2

(ν−1)/ν

−1/ν

+ CSD ρ

−1/ν

+ C SD (1 − θ )−2

ν/k(ν−1)

2/ν

−1

(T − T0 )

uk,ρ,[T0 ,T1 ] .

Proof Note that the normalized Sobolev inequality in the form described is scale invariant. Hence, without loss of generality, we may assume that V p (ρ) = 1. For any arbitrary constant a ≥ 1, the assumption on u implies that

φ 2 f u 2a −

φ 2 u 2a−1 u t ≥ −

φ 2 u 2a−1 u,

19

Parabolic Harnack inequality and regularity theory

219

for any nonnegative, compactly supported, Lipschitz function φ on B p (ρ). The right-hand side after integration by parts yields − φ 2 u 2a−1 u = 2 φ u 2a−1 ∇φ, ∇u + (2a − 1) φ 2 u 2a−2 |∇u|2 . (19.2) However, using the identity a 2 2 2a |∇(φu )| = |∇φ| u + 2a φ u 2a−1 ∇φ, ∇u +a

φ 2 u 2a−2 |∇u|2

2

(19.3)

and the assumption that a ≥ 1, we have 1 a φ 2 f u 2a + |∇φ|2 u 2a ≥ |∇(φu a )|2 + φ 2 (u 2a )t 2  1/μ CSD 1 (φ u a )2μ ≥ 2 + φ 2 (u 2a )t . ρ 2 (19.4) When q = ∞, (19.4) implies that aA

φ u 2

2a

+

|∇φ| u 2

2a

C SD ≥ 2 ρ



1/μ (φ u )

+

a 2μ

1 2

φ 2 (u 2a )t . (19.5)

When μ/(μ − 1) < q < ∞, by H¨older’s inequality, we have 

a

φ fu 2

2a

(φ u )

≤ aA

2 2a q/(q−1)

 ≤ aA

φ u

2 2a

(q−1)/q

(μ(q−1)−q)/q(μ−1) 

2 2a μ

(φ u )

1/q(μ−1) . (19.6)

However, applying the inequality 

x ≤δ

(−1)/

 x + δ

1/(1−)



1 −1 

220

Geometric Analysis

by setting  = (μ(q − 1) − q)/q(μ − 1) and 

 

(φ 2 u 2a )μ

φ 2 u 2a

x = (a A)q(μ−1)/(μ(q−1)−q)

−1/μ

,

we have  φ 2 u 2a

aA ≤δ

(−1)/

(μ(q−1)−q)/q(μ−1) 

(φ 2 u 2a )μ

 (a A)

q(μ−1)/(μ(q−1)−q)

 + δ 1/(1−)

(q−μ(q−1))/qμ(μ−1)  

2 2a

φ u

2 2a μ

(φ u )

 1 −1 . 

−1/μ

(19.7)

Multiplying through by 

2 2a μ

1/μ

(φ u ) and choosing δ so that  δ

1/(1−)

 CSD 1 −1 = ,  2ρ 2

(19.6) and (19.7) yield 

a

2

φ fu

2a

   CSD −μ/(μ(q−1)−q) q(μ−1)/(μ(q−1)−q) 2 2a φ u ≤ C1 (a A) ρ2  1/μ C SD 2 2a μ + (φ u ) 2ρ 2

for some constant C1 > 0 depending only on μ and q. Hence applying the above inequality to the first term of (19.4), we have  C1

CSD ρ2

−μ/(μ(q−1)−q)

CSD ≥ 2ρ 2



 (a A)

q(μ−1)/(μ(q−1)−q)

1/μ (φ u )

a 2μ

+

1 2

 φ u

2 2a

+

|∇φ|2 u 2a

φ 2 (u 2a )t .

(19.8)

19

Parabolic Harnack inequality and regularity theory

221

In any event, (19.5) and (19.8) imply the inequality  2C1

CSD ρ2

CSD ≥ 2 ρ

1−α



(a A)α

φ 2 u 2a + 2



1/μ (φ u )

+

a 2μ

|∇φ|2 u 2a

φ 2 (u 2a )t .

(19.9)

with α = q(μ − 1)/(μ(q − 1) − q) > 0 for all μ/(μ − 1) < q ≤ ∞. If ψ(t) is a Lipschitz function given by ⎧ 0 ⎪ ⎪ ⎪ ⎨t −s ψ(t) = ⎪ v ⎪ ⎪ ⎩ 1

for

T0 ≤ t ≤ s,

for

s ≤ t ≤ s + v,

for

s + v ≤ t ≤ T1 ,

then (19.9) implies that  2C 1

CSD ρ2

1−α

B p (ρ)

t

+2

B p (ρ)

+

t T0

C SD = 2 ρ +



t

B p (ρ)

ψ 2 (t) φ 2 (x) u 2a (x, t) d x dt

ψ 2 (t) |∇φ|2 (x) u 2a (x, t) d x dt ψ(t) ψt (t) φ 2 (x) u 2a (x, t) d x dt

ψ (t)

B p (ρ)

2μ

φ(x) u (x, t) a

ψ (t) φ (x) u (x, t) d x 2

B p (ρ)

t T0

1/μ dx

dt



∂ ∂t

B p (ρ)



2

T0







T0

CSD ≥ 2 ρ

t



T0





T0 t

+2

(a A)α

ψ (t)

2



2

B p (ρ)

2a

2μ

φ(x) u (x, t) a

ψ 2 (t  ) φ 2 (x) u 2a (x, t  ) d x

dt

1/μ dx

dt

(19.10)

222

Geometric Analysis

for all t  ∈ (T0 , T1 ]. First observe that since the first term on the right-hand side is nonnegative, we conclude that 

CSD ρ2 +2

2C1



1−α

T1

T0 T1

+2





T1 T0

B p (ρ)

B p (ρ)

T0

≥

(a A)α

s + v ≤ t  ≤ T1

B p (ρ)

ψ 2 (t) φ 2 (x) u 2a (x, t) d x dt

ψ 2 (t) |∇φ|2 (x) u 2a (x, t) d x dt ψ(t) ψt (t) φ 2 (x) u 2a (x, t) d x dt



sup



B p (ρ)

φ 2 (x) u 2a (x, t  ) d x.

(19.11)

By setting t  = T1 in (19.10), we also have 

CSD ρ2 +2

2C 1



1−α

T1

T0 T1

+2







T1 T0

B p (ρ)

B p (ρ)

T0

CSD ≥ 2 ρ

(a A)α

T1

s +v

B p (ρ)

ψ 2 (t) φ 2 (x) u 2a (x, t) d x dt

ψ 2 (t) |∇φ|2 (x) u 2a (x, t) d x dt ψ(t) ψt (t) φ 2 (x) u 2a (x, t) d x dt



1/μ

2μ



φ(x) u a (x, t)

B p (ρ)

dx

dt.

(19.12)

On the other hand, Schwarz inequality implies that  (φ u )

a 2

(μ−1)/μ 

1/μ (φ u )

≥

a 2μ

(φ u a )2(2μ−1)/μ ,

hence

(μ−1)/μ

sup

s + v ≤ t ≤ T1

≥

T1 s +v

B p (ρ)

(φu a )2 d x

B p (ρ)

(φ u a )2(2μ−1)/μ .

T1 s+v



B p (ρ)

φ(x)u a (x, t)

2μ

1/μ dx

dt

19

Parabolic Harnack inequality and regularity theory

223

Combining this inequality with (19.11) and (19.12), we obtain 

CSD ρ2 +2

2C1



1−α

T1

T0 T1

+2

≥

CSD ρ2





B p (ρ)

B p (ρ)



T1 T0



T0



(a A)α

T1

s+v

B p (ρ)

ψ 2 (t) φ 2 (x) u 2a (x, t) d x dt

ψ 2 (t) |∇φ|2 (x) u 2a (x, t) d x dt ψ(t) ψt (t) φ 2 (x) u 2a (x, t) d x dt

μ/(2μ−1)

(φ(x) u (x, t)) a

B p (ρ)

2(2μ−1)/μ

.

d x dt

(19.13)

Let us now choose φ(r (x)) to be a function of the distance to p, given by ⎧ 0 ⎪ ⎪ ⎪ ⎨ τ +σ −r φ(r ) = σ ⎪ ⎪ ⎪ ⎩ 1

on

B p (ρ)\ B p (τ + σ ),

on

B p (τ + σ )\ B p (τ ),

on

B p (τ ),

and (19.13) then yields 

 −1 α−1 −1 μ/(2μ−1) ) (a A)α + 2σ −2 + 2v −1 (ρ 2 CSD ) 2C 1 (ρ 2 CSD × s

T1



B p (τ + σ )

u

2a

≥

T1 s +v

μ/(2μ−1)

B p (τ )

u

2a(2μ−1)/μ

(19.14)

with α = q(μ − 1)/(μ(q − 1) − q) for all m/2 < q ≤ ∞. Let us now choose the sequences of ai , τi , and σi such that a0 =

k kν kν i , a1 = , . . . , ai = , ..., 2 2 2

σ0 = 2−1 (1 − θ )ρ, σ1 = 2−2 (1 − θ )ρ, . . . , σi = 2−(1+i) (1 − θ )ρ, . . . , and τ0 = ρ,

τ1 = ρ − σ0 ,

...,

τi = ρ −

i−1  j =0

σj,

....

224

Geometric Analysis

with ν = (2μ − 1)/μ. We also choose the sequences of si and vi , such that i 

s0 = T0 , s1 = T0 + 2−1 (T − T0 ), . . . si = T0 +

2−i (T − T0 ), . . . ,

j=1

and v0 = 2−1 (T − T0 ), v1 = 2−2 (T − T0 ), . . . , vi = 2−(1+i) (T − T0 ), . . . . Observe that limi→∞ ρi = θρ and limi→∞ si = T . Applying (19.14) to a = ai , τ = τi+1 , σ = σi , s = si , and v = vi , we have  −1 α−1+1/ν iα ) ν u2ai+1 ,τi+1 ,[si+1 ,T1 ] ≤ 21−α (k A)α C1 (ρ 2 CSD + (23+2i (1 − θ )−2 ρ −2 −1 1/ν + 22+i (T − T0 )−1 )(ρ 2 CSD )

1/kν i

u2ai ,τi ,[si ,T1 ]

with u2ai ,τi ,[si ,T1 ] =

T1

si

1/2ai

B p (τi )

u

2ai

.

Iterating this inequality, we conclude that u2ai+1 ,τi+1 ,[si+1 ,T1 ] ≤

i  -

−1 α−1+1/ν jα 21−α (k A)α C1 (ρ 2 CSD ) ν

j=0

+ (23+2 j (1 − θ )−2 ρ −2 −1 1/ν + 22+ j (T − T0 )−1 )(ρ 2 CSD )

1/kν j

× u2a0 ,τ0 ,[s0 ,T1 ] . On the other hand, we have the inequality lim ((T1 − T ) V (θρ))−1/2ai+1 u2ai+1 ,τi+1 ,[si+1 ,T1 ]

i→∞

≥ lim ((T1 − T ) V (θρ))−1/2ai+1 u2ai+1 ,θρ,[T,T1 ] i→∞

= u∞,θρ,[T,T1 ] ,

19

Parabolic Harnack inequality and regularity theory

225

where the right-hand side is defined to be the supremum of u over the set B p (θρ) × [T, T1 ]. Therefore, letting i → ∞, we conclude that ∞  -

u∞,θρ,[T,T1 ] ≤

−1 α−1+1/ν jα 21−α (k A)α C1 (ρ 2 CSD ) ν

j=0

+ (23+2 j (1 − θ )−2 ρ −2 −1 1/ν + 22+ j (T − T0 )−1 )(ρ 2 CSD )

1/kν j

uk,ρ,[T0 ,T1 ] . (19.15)

The product can be estimated by using the fact that ν = (2μ − 1)/μ > 1, hence we obtain the identity ∞ -

Bν

−j

= B ν/(ν−1) ,

j=0

and the fact that

∞

j=0

jμ− j is finite. Therefore we have

∞  -

−1 α−1+1/ν jα 21−α (k A)α C1 (ρ 2 CSD ) ν + (23+2 j (1 − θ )−2 ρ −2

j=0

+2

2+ j

≤

−1

(T − T0 )



−1

) ρ CSD 2

1/ν 1/kν j

∞  -

−1 α−1+1/ν 21−α (k A)α C1 (ρ 2 CSD ) + ((1 − θ )−2 ρ 2

j=0 −1 1/ν + (T − T0 )−1 )(ρ 2 CSD )

1/kν j

max{ν α , 4} j/kν

j

 −1 α −1 −(ν−1)/ν ≤ C2 (k A ρ 2 CSD ) (ρ 2 C SD ) + ((1 − θ )−2 ρ −2 −1 1/ν + (T − T0 )−1 )(ρ 2 CSD )

ν/k(ν−1)

,

226

Geometric Analysis

where C2 ≥ 0 depends only on k, μ, and q. Inequality (19.15) now takes the form −1 α ) u∞,θρ,[T,T1 ] ≤ C 2 (k A ρ 2 CSD

−1

+ (T − T0 )



CSD ρ2



ρ2 ) CSD

(ν−1)/ν

+ ((1 − θ )−2 ρ −2

1/ν ν/k(ν−1)

× (T1 − T0 )1/k –uk,ρ,[T0 ,T1 ]

(19.16)

for k ≥ 2, where –uk,ρ,[T0 ,T1 ] = (T1 − T0 )−1 V p−1 (ρ)



T1

1/k

B p (ρ)

T0

uk

.

We now claim that (19.16) holds for 1 < k < 2 also. To see this, we observe that the iteration process begins with a = k/2, and hence for k > 1, we have 2a − 1 > 0 in (19.2). Using the inequality 2

φ u 2a−1 ∇φ, ∇u ≥ −a −1 (2a − 1)−1

|∇φ|2 u 2a

− a(2a − 1)

φ 2 u 2a−2 |∇u|2

and combining it with (19.2), we conclude that −

φ 2 u 2a−1 u = 2δ

φ u 2a−1 ∇φ, ∇u + 2(1 − δ)

φ u 2a−1 ∇φ, ∇u

φ 2 u 2a−2 |∇u|2

+ (2a − 1) ≥ −2δ a

−1

−1

(2a − 1)

|∇φ|2 u 2a

+ 2(1 − δ)

φ u 2a−1 ∇φ, ∇u

+ (2a − 1 − δa(2a − 1))

φ 2 u 2a−2 |∇u|2 .

19

Parabolic Harnack inequality and regularity theory

227

Setting δ = (2a)−1 and using (19.3), we have 



1 2a − 1 + 2 a (2a − 1) 2a 2 2a − 1 + |∇(φ u a )|2 , 2a 2



φ 2 u 2a−1 u ≥ −

−

|∇φ|2 u 2a

hence

 2a − 1 1 φ fu + + |∇φ|2 u 2a a 2 (2a − 1) 2a 2 2a − 1 1 a 2 ≥ )| + φ 2 (u 2a )t |∇(φ u 2a 2 2a  1/μ (2a − 1)CSD 1 a 2μ (φ u ) ≥ + φ 2 (u 2a )t . 2a 2 ρ 2 2a 

2

2a

Using this inequality instead of (19.4) for values of 12 < a < 1, the same argument as above yields (19.16) for k > 1. For k ≤ 1, we begin with the inequality case k = 2. In that case, (19.16) yields  u∞,γ τ,[S,T1 ] ≤ C 2

τ2 2A CSD



τ2 + CSD

α 

CSD τ2

(ν−1)/ν

ν/2(ν−1)

1/ν ((1 − γ )

−2 −2

τ

+ (1 − ξ )

−1 −1

S

)

× (T1 − ξ S)1/2 –u2,τ,[ξ S,T1 ]  α   ρ2 CSD (ν−1)/ν 2A ≤ C2 CSD ρ2 −1/ν

+ C SD (1 − γ )−2 τ −2(ν−1)/ν 

ρ2 + CSD

ν/2(ν−1)

1/ν (1 − ξ ) k/2

−1 −1

S

1−k/2

× (T1 − ξ S)1/2 –uk,τ,[ξ S,T1 ] u∞,τ,[ξ S,T1 ] ,

(19.17)

228

Geometric Analysis

for any θρ ≤ τ ≤ ρ, 0 < γ < 1, T0 ≤ S ≤ T and 0 < ξ < 1. Let us choose the sequences of τi and γi to be τ−1 = θρ, τ0 = θρ + 2−1 (1 − θ)ρ, . . . , τi−1 = θρ + (1 − θ )ρ

i 

2− j , . . .

j=1

and γi τi = τi−1 . Also, the sequences Si and ξi are chosen to be S0 = T, S1 = (1 − 2−1 )(T − T0 ) + T0 , . . . , ⎞ ⎛ i  Si = ⎝1 − 2− j ⎠ (T − T0 ) + T0 , . . . j=1

and ξi Si = Si+1 . Applying (19.17) by setting τ = τi , γ = γi , S = Si , and ξ = ξi and iterating the inequality yields u∞,θρ,[T,T1 ] ≤

j -

 C2

i=0

ρ2 2A CSD

α 

−1/ν

CSD ρ2

−2(ν−1)/ν

+ C SD (1 − γi )−2 τi  +

ρ2 CSD

(ν−1)/ν

1/ν

ν/2(ν−1) (1−k/2)i

(Si − Si+1 )−1

× (T1 − Si+1 )(1−k/2) /2 i

(1−k/2) j+1 × u∞,ρ,[T0 ,T1 ]

j  -

k/2

–uk,τi ,[Si+1 ,T1 ]

i=0

We observe that since (1 − γi ) =

τi − τi−1 τi

≥ 2−(i+1) (1 − θ ),

(1−k/2)i

.

(19.18)

19

Parabolic Harnack inequality and regularity theory

229

we have −2(ν−1)/ν

(1 − γi )−2 τi

≤ 4(i+1) (1 − θ )−2 ρ −2(ν−1)/ν .

Also, since Si − Si+1 = 2−(i+1) (T − T0 ), (19.18) becomes

u∞,θρ,[T,T1 ] ≤

j -

 2A

C2

i=0

ρ2 C SD

α 

CSD ρ2

(ν−1)/ν

−1/ν

+ CSD (1 − θ )−2 ρ −2(ν−1)/ν −1/ν

(ν/2(ν−1)) (1−k/2)i

+ C SD ρ

×

j -

2/ν

−1

(T − T0 )

(1−k/2) j+1

2((i+1)ν/ν−1)(1−k/2) (T1 − T0 )1/k u∞,ρ,[T0 ,T1 ] i

i=0

×

j  -

k/2

–uk,τi ,[Si+1 ,T1 ]

(1−k/2)i

.

i=0

Letting j → ∞ and using the facts that ∞ -

2((i+1)ν/ν−1)(1−k/2) < ∞, i

i=0 (1−k/2) j+1

u∞,ρ,[T0 ,T1 ] → 1 as j → ∞, and –uk,τi ,[Si+1 ,T1 ] ≤ ((T1 − S1 ) V p (τ0 ))−1/2 uk,ρ,[T0 ,T1 ] k/2

k/2

 ≤

ρ  T1 − T0 Vp 2 2

−1/2

k/2

uk,ρ,[T0 ,T1 ] ,

230

Geometric Analysis

we obtain

 u∞,θρ,[T,T1 ] ≤ C 3

2A

ρ2 CSD

α 

CSD ρ2

(ν−1)/ν

−1/ν

+ CSD (1 − θ )−2 ρ −2(ν−1)/ν

ν/k(ν−1) −1/ν

+ C SD ρ 2/ν (T − T0 )−1 −1/k

× Vp

ρ  2

uk,ρ,[T0 ,T1 ] ,

proving the desired inequality for k ≤ 1.



To complete the proof of the Harnack inequality, we will need the following lemma. Lemma 19.2 Let g be a positive measurable function defined on a set D ⊂ M. Suppose {Dσ | 0 < σ ≤ 1} is a family of measurable subsets of D with the properties that Dσ  ⊂ Dσ for 0 < σ  ≤ σ ≤ 1 and D1 = D. Let us assume that there are constants C 4 > 0, γ > 0, 0 < δ < 1, and 0 < k0 < ∞ so that the inequality

1/k0 g

k0

 1/k−1/k0  ≤ C4 (σ − σ  )−γ V −1 (D)

Dσ 

1/k gk Dσ

is valid for all δ ≤ σ  ≤ σ ≤ 1 and 0 < k ≤ k0 . If there exists a constant C5 > 0 such that m({x ∈ D | ln g > λ}) ≤ C5 V (D) λ−1 for all λ > 0, then there exists a constant C6 > 0 depending only on γ , δ, C 4 and C5 , such that  g

k0

1/k0

≤ C6 V 1/k0 (D).

Dδ

Proof Let us normalize the volume of D to be V (D) = 1 and define the function  1/k0 k0 ψ(σ ) = ln g Dσ

19

Parabolic Harnack inequality and regularity theory

231

for δ ≤ σ < 1. We may assume that ψ(σ ) ≥ 2C5 , otherwise the lemma follows because of the monotonicity of ψ and by choosing C6 = exp(2C5 ). We first claim that ψ(σ  ) ≤ 34 ψ(σ ) + 8C42 C5 (σ − σ  )−2γ

(19.19)

for all σ  < σ. To see this, we only need to verify (19.19) for those σ  such that ψ(σ  ) ≥ 8C42 C5 (σ − σ  )−2γ .

(19.20)

Let us consider 

1/k gk

1/k

 =

Dσ ∩{ln g>ψ(σ )/2}

Dσ

 g k0

≤ Dσ



+ exp

gk

1/k

 +

Dσ ∩{ln g≤ψ(σ )/2}

gk

$  # ψ(σ ) 1/k−1/k0 m Dσ ∩ ln g > 2 

1/k0

ψ(σ ) 2

≤ exp(ψ(σ ))(2C5 ψ(σ )

−1 1/k−1/k0

)



 ψ(σ ) + exp , (19.21) 2

where we have used the fact that V (Dσ ) ≤ 1. Note that we may choose k ≤ k0 to satisfy 

1 1 − k k0

−1

=

  2 ψ(σ ) , ln ψ(σ ) 2C5

hence combining this identity with (19.21), we obtain 

1/k g Dσ

k

 ψ(σ ) . ≤ 2 exp 2 

232

Geometric Analysis

On the other hand, the assumption of the lemma asserts that   

 −γ 1/k−1/k0

ψ(σ ) ≤ ln (C4 (σ − σ )

)

g

1/k

k

Dσ

 ψ(σ ) 1 1 ln(2C4 (σ − σ  )−γ ) + − k k0 2 ⎛ ⎞ ψ(σ ) ln(2C4 (σ − σ  )−γ ) ⎝   ≤ + 1⎠ 2 ln ψ(σ ) 

≤

2C5

≤

3ψ(σ ) 4

because of (19.20) and the monotonicity of ψ. The claim (19.19) is then validated. For some β > 1, let us now choose a sequence of {σi } with σ0 = δ, and σi+1 = σi + β −i−1 (1 − δ)(β − 1) for all i ≥ 0. Inequality (19.19) asserts that ψ(σi ) ≤

3 4

ψ(σi+1 ) + 8C 42 C5 (1 − δ)−2γ (β − 1)−2γ β 2γ (i+1)

for all i ≥ 0. Iterating this inequality yields ψ(σ0 ) ≤

 j 3 4

ψ(σ j ) + 8C42 C5 (1 − δ)−2γ (β − 1)−2γ

j    i−1 3 4

β 2γ i .

i=1

Using the fact that σ j = (1 − δ)(β − 1) ∞, we conclude that

j

i=1 β

−i

≤ (1 − δ), and letting j →

ψ(σ0 ) ≤ 8C42 C5 (1 − δ)−2γ (β − 1)−2γ β 2γ

i ∞   3β 2γ i=0

4

.

By choosing 1 < β and 3β 2γ /4 < 1, we deduce that the right-hand side is finite and the lemma is proved.  Theorem 19.3 Let M be a complete manifold. Suppose that the geodesic ball B p (ρ) centered at p with radius ρ satisfies B p (ρ) ∩ ∂ M = ∅. Let u ≥ 0 be a function defined on B p (ρ) × [0, T ], satisfying the equation 

∂ − ∂t

 u = fu

19

Parabolic Harnack inequality and regularity theory

233

in the weak sense for some function f on B p (ρ) whose supremum norm is given by A =  f ∞,ρ,[0,T ] . Let ν = m/2 for m > 2, and let 1 < ν < ∞ be arbitrary when m = 2. If T ≥ ρ 2 , then for any 0 < θ < 1, there exists a constant C16 > 0, depending only on θ, CSD , C VD , and CP , such that  2 Aρ inf sup u ≤ C16 exp u. 2 B p (θρ)×[T −ρ 2 /4,T ] B p (θρ)×[T −3ρ 2 /4,T −ρ 2 /2] Proof Let us first prove the theorem for the case A = 0. We define the function w = − ln u, then it satisfies the equation w =

∂w + |∇w|2 . ∂t

Let us point out that in the argument that follows we only need the inequality w ≥

∂w + |∇w|2 , ∂t

hence for any compactly supported nonnegative smooth function φ ∈ Cc∞ (B p (ρ)), we have ∂ φ2 w ≤ φ 2 w − φ 2 |∇w|2 ∂t B p (ρ) B p (ρ) B p (ρ) = −2 φ ∇φ, ∇w − φ 2 |∇w|2 B p (ρ)

≤−

1 2

B p (ρ)

In particular, choosing φ to be ⎧ 1 ⎪ ⎪ ⎪ ⎨ ρ −r φ(r ) = (1 − δ)ρ ⎪ ⎪ ⎪ ⎩ 0

B p (ρ)





φ 2 |∇w|2 + 2

B p (ρ)

if

r ≤ δρ,

if

δρ ≤ r ≤ ρ,

if

ρ ≤ r,

|∇φ|2 .

we obtain 1 ∂ 2 φ w+ φ 2 |∇w|2 ≤ (1 − δ)−2 ρ −2 V p (ρ). ∂t B p (ρ) 2 B p (ρ)

(19.22)

234

Geometric Analysis

However, Theorem 18.3 asserts that there exists a constant C7 > 0, depending on CP , CVD , and δ, such that 2 2 −2 |∇w| φ ≥ C7 ρ |w − w| ¯ 2 φ2, B p (ρ)

B p (ρ)

where w¯ = Vφ−1 (B p (ρ)) with respect to the (19.22), we have

!

2 B p (ρ) w φ denotes measure φ 2 d V. Taking the

∂ ∂ w¯ = Vφ−1 (B p (ρ)) ∂t ∂t

the average of w over B p (ρ) above inequality together with

B p (ρ)

≤ −C 7 ρ −2 Vφ−1 (B p (ρ))

w φ2 B p (ρ)

|w − w| ¯ 2 φ2

V p (ρ) Vφ−1 (B p (ρ)) −2 −1 ≤ −C 7 ρ V p (ρ) |w − w| ¯ 2 + C8 ρ −2 , + (1 − δ)

−2

ρ

−2

B p (δρ)

(19.23)

because of V p (δρ) ≤ Vφ (B p (ρ)) ≤ V p (ρ). Let us define v(x, t) = w(x, t) − C8 (t − s  ) ρ −2 and v(t) ¯ = w(t) ¯ − C8 (t − s  ) ρ −2 for a fixed s  ∈ [T − ρ 2 /2, T − ρ 2 /4]. We can rewrite (19.23) as ∂ v¯ + C7 ρ −2 V p−1 (ρ) |v − v| ¯ 2 ≤ 0. ∂t B p (δρ) For any λ > 0, we define + λ (t) = {x ∈ B p (δρ) | v(x, t) > a + λ} and − λ (t) = {x ∈ B p (δρ) | v(x, t) < a − λ},

(19.24)

19

Parabolic Harnack inequality and regularity theory

235

¯  ). Since (∂/∂t)v¯ ≤ 0, we have where a = v(s ¯  ) = w(s v(x, t) − v(t) ¯ > a + λ − v(t) ¯ ≥λ  when x ∈ + λ (t) and t > s . Combining this inequality with (19.24), this implies that

 ∂ 2 ¯ V + v(t) ¯ + C 7 ρ −2 V p−1 (ρ) (a + λ − v(t)) λ (t) ≤ 0 ∂t and −C 7−1 ρ 2 V p (ρ)



 ∂  |a + λ − v| ¯ −1 ≥ V + λ (t) ∂t

for t > s  . Integrating over the interval (s  , T ) yields ¯ )|−1 ) C 7−1 ρ 2 V p (ρ)(λ−1 − |a + λ − v(T T ≥ V (+ λ (t) s

= m{(x, t) ∈ B p (δρ) × (s  , T ) | w(x, t) > a + λ + C8 (t − s  ) ρ −2 }, (19.25) where m{D} denotes the measure with respect to the product metric of the set D ⊂ M × R. If λ ≥ C8 (T − s  ) ρ −2 , then (19.25) implies C7−1 ρ 2 V p (ρ) λ−1 ≥ m{(x, t) ∈ B p (δρ) × (s  , T ) | w(x, t) > a + λ + C 8 (t − s  ) ρ −2 } ≥ m{(x, t) ∈ B p (δρ) × (s  , T ) | w(x, t) > a + 2λ}.

(19.26)

On the other hand, if 0 < λ < C8 (T − s  ) ρ −2 , then we can estimate m{(x, t) ∈ B p (δρ) × (s  , T ) | w(x, t) > a + 2λ} ≤ (T − s  ) V p (ρ) ≤ C8 (T − s  )2 ρ −2 λ−1 V p (ρ). However, using the fact that T − s ≤

ρ2 , 2

236

Geometric Analysis

together with (19.26), we conclude that m{(x, t) ∈ B p (δρ) × (s  , T ) | w(x, t) > a + λ} ≤ C 9 ρ 2 V p (ρ) λ−1 . (19.27) When t ≤ s  , again using (19.24), we have v(x, t) − v(t) ¯ ≤ a − λ − v(t) ¯ = −λ on − λ (t), hence

 ∂ 2 v¯ + C 7 ρ −2 V p−1 (ρ) (a − λ − v(t)) ¯ V − λ (t) ≤ 0 ∂t and 

∂ 2 (|a − λ − v| ¯ −1 ). V − λ (t) ≤ −C 7 ρ V p (ρ) ∂t Integrating from T − ρ 2 to s  we obtain m{(x, t) ∈ B p (δρ) × (T − ρ 2 , s  ) | w(x, t) < a − λ + C8 (t − s  ) ρ −2 } s

 = V − λ (t) dt T −ρ 2

¯ − ρ 2 )|−1 ≤ C7 ρ 2 V p (ρ) (λ−1 − |a − λ − v(T ≤ C7 ρ 2 V p (ρ) λ−1 , hence m{(x, t) ∈ B p (δρ) × (T − ρ 2 , s  ) | w(x, t) < a − λ} ≤ C7 ρ 2 V p (ρ) λ−1 . (19.28) We will now apply Lemma 19.2 to the function g = ea u. In particular, (19.28) asserts that m{(x, t) ∈ D | ln g > λ} ≤ C9 ρ 2 V p (ρ) λ−1 ≤ C 10 V (D) λ−1

19

Parabolic Harnack inequality and regularity theory

237

with D = B p (δρ) × (T − ρ 2 , s  ), where C10 depends on C9 , CVD , and δ. Let us define the subdomains Dσ ⊂ D by Dσ = B p (σ δρ) × (T − σρ 2 ), s  ). Obviously, Dσ  ⊂ Dσ for 14 < σ  ≤ σ ≤ 1 with D1 = D. Applying Lemma 19.1 to the domain Dσ  and Dσ we have g∞,σ  δρ,[T −σ  ρ 2 ,s  ] ≤ C 11 ((σ − σ  )δρ)−2 + (σ − σ  )−1 ρ −2 )ν/k(ν−1) −1/k

× (σ δρ)2/k(ν−1) V p

(σ δρ) gk,σ δρ,[T −σρ 2 ,s  ]

≤ C 11 (σ − σ  )−2ν/k(ν−1) (δρ)−2/k −1/k

× Vp

(σ δρ) gk,σ δρ,[T −σρ 2 ,s  ] ,

(19.29)

where C11 > 0 depends on k, ν = (2μ − 1)/μ, δ, CSD , CP and CVD . Note that for σ ≥ 14 , the volume doubling property implies that V (D) = ρ 2 V p (δρ) 2 ρ 2 V p (σ δρ) ≤ CVD

for k0 ≥ k, and (19.29) asserts that

1/k0 g

k0

 1/k−1/k0 ≤ C 12 (σ − σ  )−2k0 ν/(k0 −k)(ν−1) V (D)−1

Dσ 

1/k

 ×

gk

,

Dσ

where C 12 > 0 depends on k, ν, δ, CSD , CP and CVD . Lemma 19.1 now asserts that ⎛

⎞1/k0 ⎝— g k0 ⎠ ≤ C 13 ,

(19.30)

D3 4

where C13 > 0 depends on k, ν, δ, CSD , and CVD . On the other hand, applying (19.29) again, we have g∞,δρ/2,[T −ρ 2 /4,s  ] ≤ C 14 (δρ)

−2/k0

−1/k Vp 0



3δρ 4

 gk0 ,3δρ/4,[T −3ρ 2 /4,s  ] ,

238

Geometric Analysis

which when combined with (19.30) yields g∞,δρ/2,[T −ρ 2 /4,s  ] ≤ C15 e−a .

(19.31)

where C15 > 0 depends on ν, δ, C SD , C P and C V . Similarly, using (19.27), we apply Lemma 19.2 to the function g = e−a u −1 . As above, we use Lemma 19.1 to verify

1/k0



g −k0

 1/k−1/k0 ≤ C 12 (σ − σ  )−2k0 ν/(k0 −k)(ν−1) V (D)−1

Dσ 

 ×

g

−k

1/k ,

Dσ

hence the same argument implies that u −1 ∞,δρ/2,[s  +ρ 2 /4,T ] ≤ C 15 ea , which can be rewritten as e−a ≤ C 15

inf

B p (δρ/2)×[s  +ρ 2 /4,T ]

u.

(19.32)

The Harnack inequality follows by combining this with (19.31) and choosing θ = δ/2 and s  = T − ρ 2 /2. When f is not identically 0, we observe that   ∂ (e−At u) = f e−At u + Ae−At u − ∂t ≥0 and



∂ − ∂t



(e−At u −1 ) = f e−At u −1 − e−At |∇u|2 u −2 + Ae−At u −1 ≤ 0.

Hence we can apply (19.31) and (19.32) to the functions e−At u and e−At u −1 and obtain e−AT u∞,δρ/2,[T −ρ 2 /2,T ] ≤ C15 e−a

19

Parabolic Harnack inequality and regularity theory

and e−a ≤ C 15 e−A



2

T − ρ2

239

 inf

B p (δρ/2)×[T −ρ 2 /2,T ]

u,

respectively. In particular, we conclude that sup B p (δρ/2)×[T −ρ 2 /2,T ]

2 Aρ u ≤ C15 e

2 /2

inf

B p (δρ/2)×[T −ρ 2 /2,T ]

u.



The following corollary follows directly from Theorem 19.3. The case when M is quasi-isometric to Rm follows readily from the Nash–Moser theory. The general case when M is quasi-isometric to a complete manifold with nonnegative Ricci curvature was first proved by Saloff-Coste [SC] and Grigor’yan [G3]. Corollary 19.4 Let M be a manifold of dimension m. Suppose that ds 2 is a complete metric on M such that there is a point p ∈ M and, for all ρ, the quantities C P , CSD , and CVD are all bounded and positive independent 2 of ρ. Then for any metric ds (not necessarily continuous) on M which is uniformly equivalent to ds 2 , there does not exist any nonconstant positive 2 harmonic functions for the Laplacian with respect to ds . In particular, any manifold which is quasi-isometric to a complete manifold with nonnegative Ricci curvature does not admit a nonconstant positive harmonic function. Proof To see this, we first observe that CP , CSD , and C VD are quasiisometric invariants. Since a harmonic function is a stationary solution to the heat equation, Theorem 19.3 implies that any positive harmonic function u defined on M must satisfies the Harnack inequality sup u ≤ C

B p (ρ/4)

inf u

B p (ρ/4)

for any ρ > 0. On the other hand, since u is positive, by translation, we may assume that inf M u = 0. Hence, by taking ρ → ∞, we conclude that sup u ≤ C inf u = 0. M

Therefore, u must be identically 0.

M



The next corollary yields a H¨older regularity result for weak solutions. One 2 important point of this is that the metric ds need not be continuous as long as it is uniformly bounded with respect to some background metric ds 2 .

240

Geometric Analysis

Corollary 19.5 Let M be a complete manifold with metric ds 2 . Suppose ds is uniformly equivalent to ds 2 . Suppose u is a weak solution to the equation   ¯ − ∂ u = fu  ∂t

2

2

defined on B p (1) × [T − 1, T ] with respect to the metric ds . Assume that f is bounded on B p (1) × [T − 1, T ], then u must be H¨older continuous at the point ( p, T0 ) for all T − 1 < T0 ≤ T. Proof Let ρ be a constant such that 0 < ρ ≤ T0 − T + 1. We define s(ρ) = sup B p (ρ)×[T0 −ρ 2 ,T0 ] u and i(ρ) = inf B p (ρ)×[T0 −ρ 2 ,T0 ] u. Applying Theorem 19.3 to the functions s(ρ) − u and u − i(ρ), we have   ρ  s(ρ) − inf u ≤ C s(ρ) − s 2 B p (ρ/2)×[T0 −3ρ 2 /4,T0 −ρ 2 /2] and

  ρ  − i(ρ) . u − i(ρ) ≤ C i 2 B p (ρ/2)×[T0 −3ρ 2 /4,T0 −ρ 2 /2] sup

Adding the two inequalities yields       ρ  ρ 3ρ 2 ρ2 osc(ρ) + osc B p × [T0 − , T0 − ] ≤ C osc(ρ) − osc , 2 4 2 2 where osc(ρ) = s(ρ) − i(ρ) denotes the oscillation of u on B p (ρ) × [T0 − ρ 2 , T0 ]. This implies that ρ  osc ≤ γ osc(ρ) 2 for γ = (C − 1)/C < 1. Iterating this inequality gives osc(2−k ρ) ≤ γ k osc(ρ). This implies that u is H¨older continuous of exponent −log γ /log 2 in the space variable and H¨older continuous of exponent −log γ /log 4 in the time variable. 

20 Parabolicity

In view of the construction in Chapter 17, the existence and nonexistence of a positive Green’s function divides the class of complete manifolds into two categories. In general, the methods in dealing with function theory on these manifolds are different, hence it is important to understand the difference between the two categories. Definition 20.1 A complete manifold is said to be parabolic if it does not admit a positive Green’s function. Otherwise it is said to be nonparabolic. As pointed out in Theorem 17.3, a manifold is nonparabolic if and only if there exists a positive superharmonic function whose infimum is achieved at infinity. This property can be localized at any unbounded component at infinity. Definition 20.2 An end E with respect to a compact subset  ⊂ M is an unbounded connected component of M \. The number of ends with respect to , denoted by N (M), is the number of unbounded connected components of M \. It is obvious that if 1 ⊂ 2 , then N1 (M) ≤ N2 (M). Hence if {i } is a compact exhaustion of M, then Ni (M) is a monotonically nondecreasing sequence. If this sequence is bounded, then we say that M has finitely many ends. In this case, we denote the number of ends of M by N (M) = max Ni (M). i→∞

One can readily check that this is independent of the compact exhaustion {i }. In fact, it is also easy to see that there must be an i such that N (M) = Ni (M). Moreover, for any compact set  containing i , N (M) = N (M). Hence for 241

242

Geometric Analysis

all practical purposes, we may assume that M\B p (ρ0 ) has N (M) unbounded connect components, for some ρ0 . In general, when we say that E is an end we mean that it is an end with respect to some compact subset . In particular, its boundary ∂ E is given by ¯ ∂ ∩ E. Definition 20.3 An end E is said to be parabolic if it does not admit a positive harmonic function f satisfying f =1

on

∂E

and lim inf f (y) < 1,

y→E(∞)

where E(∞) denotes the infinity of E. Otherwise, E is said to be nonparabolic and the function f is said to be a barrier function of E. Observe that by subtraction and multiplication of constants, we may assume that the barrier function satisfies lim inf f (y) = 0.

y→E(∞)

With this notion, we can also count the number of nonparabolic (parabolic) ends as in the definition of N (M). Definition 20.4 Let {i } be a compact exhaustion of M. We let Ni0 (M) be the number of nonparabolic ends with respect to the compact set i . We say that M has finitely many nonparabolic ends if the sequence Ni0 (M) is bounded. In this case, we say that N 0 (M) = limi→∞ Ni0 (M) is the number of nonparabolic ends of M. Definition 20.5 Let {i } be a compact exhaustion of M. We let Ni (M) be the number of parabolic ends with respect to the compact   seti . We say that M has finitely many parabolic ends if the sequence Ni (M) is bounded. In this case, we say that N  (M) = limi→∞ Ni (M) is the number of parabolic ends of M. Note that if E is a nonparabolic end of M, then by extending f to be identically 1 on (M \)\ E, it can be used to construct a positive Green’s function on M. Hence, M is nonparabolic if and only if M has a nonparabolic end. Of course, it is possible for a nonparabolic manifold to have many parabolic ends. Let us also point out that E being nonparabolic is equivalent to saying that E has a positive Green’s function with the Neumann boundary condition. This

20

Parabolicity

243

can be seen by repeating the construction of the Green’s function in the previous chapter on a manifold with boundary by taking the Neumann boundary condition. One needs to check that the maximum principle arguments are still valid in this case because of the presence of ∂ E. However, the Hopf boundary lemma asserts that at a boundary maximum point of a harmonic function f , the outward pointing normal derivative must be positive, violating the Neumann boundary condition. For the purpose of geometric application, it is often important to obtain appropriate estimates on the barrier function. We will give a canonical method of constructing barrier functions which will help in obtaining estimates later [LT6]. Lemma 20.6 An end E with respect to the compact set B p (ρ0 ) is nonparabolic if and only if the sequence of positive harmonic functions { f i }, defined on E p (ρi ) = E ∩ B p (ρi ) for ρ0 < ρ1 < ρ2 < · · · → ∞, satisfying fi = 1

on

∂E

and fi = 0

on ∂ B p (ρi ) ∩ E,

converges uniformly on compact subsets of E ∪ ∂ E to a barrier function f . The barrier constructed in this manner is minimal among all barrier functions, and f has finite Dirichlet integral on E. Proof

If E is nonparabolic, then there exists a barrier function g satisfying g = 1 on

∂E

and lim inf g(y) = 0.

y→E(∞)

For each i, the maximum principle asserts that fi ≤ g

on

B p (ρi ) ∩ E

and the sequence of functions { f i } is monotonically increasing in i. Lemma 17.1 implies that f i must converge uniformly on compact subsets of E ∪ ∂ E to a nonnegative function f satisfying f ≤g

on

E

(20.1)

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Geometric Analysis

with the boundary conditions f =1

on

∂E

and lim inf f (y) = 0.

y→E(∞)

Moreover, f is harmonic on E and we claim that the convergence is Lipschitz up to the boundary ∂ E. To see this, it suffices to show that |∇ log fi |2 is also uniformly bounded on compact subsets of E ∪ ∂ E. Following the argument for the gradient estimate in Chapter 6, we can consider the function φ |∇ log fi |2 , where φ(r ) is the cutoff function satisfying the properties # 1 for r ≤ R0 , φ(r ) = 0 for r ≥ 2R0 , − C R0−1 ≤ φ  ≤ 0, and |φ  | ≤ C R0−2 . If the maximum point of this function occurs in the interior of E, then the same argument as in Theorem 6.1 will give the estimate of |∇ log f i |2 up to ∂ E. Therefore, we only need to show that we can still estimate |∇ log f i |2 if its maximum point occurs on ∂ E. If x0 ∈ ∂ E is the maximum point of φ |∇ log fi |2 , hence a maximum point of |∇ log f i |2 , then by setting h = log f i , the strong maximum principle asserts that ∂|∇h|2 (x0 ) < 0. ∂r On the other hand, since h = 0 on ∂ E, we can choose an orthonormal frame {e1 , . . . , em } at x 0 such that e1 = ∇h/|∇h| = −∂/∂r . Hence we have 0>

∂|∇h|2 ∂r

= −2h 1 h 11 = −2|∇h| h 11 .

(20.2)

20

Parabolicity

245

However, the Laplacian of h is given by −|∇h|2 = h = h 11 + H h 1 , where H is the mean curvature of ∂ E with respect to the outward normal vector e1 . Therefore combining the above inequality with (20.2), we have 0 > −|∇h| (−|∇h|2 − H h 1 ) = |∇h|3 + H |∇h|2 . This implies that the maximum point cannot occur on ∂ E if min∂ E H ≥ 0. If −H0 = min∂ E H is negative, then |∇h|2 (x0 ) ≤ H02 . Since x 0 is the maximum point for φ |∇h|2 , we obtain an estimate of |∇h|2 on ∂ E. Note that in this argument we assume that ∂ E is smooth so that H0 < ∞. If this is not the case, we can always smooth out ∂ E and the statement of the lemma is still valid for the smooth boundary. In particular, this proves that f is a barrier function and the minimal property of f follows from (20.1) because it applies to any barrier function g. The boundary conditions on f i and the fact that it is harmonic imply that

|∇ f i | = − 2

E p (ρi )

∂E

∂ fi ∂r

for each i. Hence for any fixed ρ < ρi ,

E p (ρ)

|∇ f i |2 ≤ −

∂E

∂ fi . ∂r

Since the right-hand side is uniformly bounded according to the above gradient estimate, by letting i → ∞ we conclude that E p (ρ)

|∇ f |2 ≤ C

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Geometric Analysis

for some constant 0 < C < ∞. In fact, since f is harmonic, we see that ∂f ∂f = ∂r ∂E ∂ B p (ρ)∩E ∂r ∂ fi = lim i→∞ ∂ B p (ρ)∩E ∂r ∂ fi , = lim i→∞ ∂ E ∂r ! hence C can be taken to be ∂ E ∂ f /∂r . Conversely, if the sequence f i converges uniformly on compact subsets of E to a barrier function f , then E is nonparabolic by definition.  If E is parabolic, the sequence fi will still converge to a harmonic function f because of Lemma 17.1. However, f will be identically 1. In this case, by renormalizing the sequence { f i }, we can still construct a harmonic function on E that reflects the parabolicity property of E. Lemma 20.7 Let E be a parabolic end with respect to B p (ρ0 ). Let ρi be an increasing sequence such that ρ0 < ρ1 < ρ2 < · · · < ρi → ∞. There exists a sequence of constants Ci → ∞ such that the sequence of positive harmonic functions gi defined on E p (ρi ) = B p (ρi ) ∩ E, satisfying gi = 0

on

∂E

and gi = Ci

on

∂ B p (ρi ) ∩ E,

has a convergent subsequence that converges uniformly on compact subsets of E ∪ ∂ E to a positive harmonic function g. Moreover, g will have the properties that g=0

on

∂E

and sup g = ∞. y∈E

Proof The fact that E is parabolic implies that the sequence f i in Lemma 17.1 converges to the constant function 1. Let us define gi = Ci (1 − f i ).

20

Parabolicity

247

Obviously gi will have the appropriate boundary conditions. If we set  −1 inf fi , Ci = 1 − ∂ B p (ρ1 )∩E

then sup∂ B p (ρ1 )∩E gi = 1. Since f i → 1, Ci → ∞. Also Lemma 17.1 implies that, after passing to a subsequence, gi → g on compact subset of E where g is a harmonic function. We will also call g a barrier function of E.  Using this characterization of parabolicity, a theorem of Royden [Ro] follows as a corollary. Corollary 20.8 (Royden) Parabolicity is a quasi-isometric invariant. Proof It suffices to show that if E is a nonparabolic end with respect to the metric ds 2 , then for any other metric ds12 which is uniformly equivalent to ds 2 , the end E remains nonparabolic with respect to ds12 . Let ∇1 , 1 , and ν1 be respectively the gradient, the Laplacian, and the unit outward normal to ∂ E with respect to the metric ds12 . According to Lemma 20.6, there exists a sequence of harmonic functions, { f i }, with respect to ds 2 , converging to a nonconstant harmonic function f . Moreover, they satisfy the property stated in Lemma 20.6. In particular, ∂ fi dA = |∇ f i |2 d V. (20.3) ∂ E ∂ν E p (ρi ) On the other hand, let h i be the harmonic function with respect to ds12 satisfying hi = 1

∂E

on

and h i = 0 on then we also have

∂E

∂h i d A1 = ∂ν1

∂ B p (ρi ) ∩ E, E p (ρi )

|∇1 h i |2 d V1 .

(20.4)

However, the fact that there exists a constant C > 1 such that C −1 ds 2 ≤ ds12 ≤ C ds 2 implies that



|∇1 h i | d V1 ≥ C 1 2

E p (ρi )

E p (ρi )

|∇h i |2 d V

(20.5)

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Geometric Analysis

for some constant C1 > 0. On the other hand, the Dirichlet integral minimizing property of harmonic functions implies that |∇h i |2 d V ≥ |∇ f i |2 d V. E p (ρi )

E p (ρi )

Hence combining this inequality with (20.3), (20.4), and (20.5) we conclude that ∂h i ∂ fi d A. d A1 ≥ C1 ∂ E ∂ν1 ∂ E ∂ν Since we know that f i → f by passing to a subsequence, and similarly h i → h, we have ∂h ∂f d A1 ≥ C 1 d A. ∂ E ∂ν1 ∂ E ∂ν The fact that f is harmonic and nonconstant implies that the right-hand side must be positive. On the other hand, we also know that h is harmonic and the fact that ∂h d A1 > 0 ∂ E ∂ν1 implies that h is nonconstant, hence it is a barrier function with respect to the metric ds12 .  The following theorem from [LT4] gives a necessary condition for a manifold to be nonparabolic. Theorem 20.9 (Li–Tam) Let M be a complete manifold. If M is nonparabolic, then for any point p ∈ M we must have ∞ dt < ∞, A p (t) 1 where A p (r ) denotes the area of ∂ B p (r ). Moreover, if G( p, y) is the minimal Green’s function, then r dt ≤ sup G( p, y) − inf G( p, y) y∈∂ B p (r ) 1 A p (t) y∈∂ B p (1) for all r > 1. Proof For p ∈ M, let G( p, y) be the minimal positive Green’s function with a pole at p. In view of the construction in Chapter 17, we may assume that G( p, y) is the limit of the sequence {G i ( p, y)}, where G i is the Dirichlet Green’s function defined on B p (ρi ).

20

Parabolicity

249

For any 1 < ρ < ρi , let si (1) =

sup

G i ( p, y)

inf

G i ( p, y).

y∈∂ B p (1)

and i i (ρ) =

y∈∂ B p (ρ)

Let f be the harmonic function defined on B p (ρ)\ B p (1) satisfying the boundary conditions f (y) = si (1)

∂ B p (1)

on

and f (y) = G i ( p, y)

on

∂ B p (ρ).

The maximum principle implies that f (y) ≥ G i ( p, y)

B p (ρ)\ B p (1).

on

In particular, we have ∂G i ∂f ≤ on ∂ B p (ρ). (20.6) ∂r ∂r On the other hand, the harmonicity of f and Stokes’ theorem imply that 0= f = Also, we observe that

B p (ρ)\B p (1)

∂ B p (ρ)

∂ B p (ρ)

∂f − ∂r

∂G i = ∂r

∂ B p (1)

∂f . ∂r

B p (ρ)

G i

= −1 and (20.6) implies that

∂ B p (1)

∂f ≤ −1. ∂r

(20.7)

Let us now consider a harmonic function h defined on B p (ρ)\ B p (1) satisfying the boundary conditions h(y) = si (1)

on

∂ B p (1)

250

Geometric Analysis

and h(y) = i i (ρ)

∂ B p (ρ).

on

Again the maximum principle implies that h(y) ≤ f (y)

B p (ρ)\ B p (1),

on

and ∂h ∂f ≤ ∂r ∂r

∂ B p (1).

on

Hence combining this inequality with (20.7), we obtain ∂h ∂h = ∂r ∂ B p (ρ) ∂ B p (1) ∂r ≤ −1.

(20.8)

If we define the function 

ρ

g(r ) = (si (1) − ii (ρ)) 1

dt A p (t)

−1

ρ

dt + ii (ρ), A p (t)

r

then g(r (y)) has the same boundary conditions as h(y). The Dirichlet integral minimizing property for harmonic functions implies that |∇h|2 ≤ |∇g|2 B p (ρ)\B p (1)

B p (ρ)\B p (1)

=

ρ



ρ

(si (1) − ii (ρ))

1

1

 = (si (1) − i i (ρ))

2

ρ

1

dt A p (t)

dt A p (t)

−1

−1

1 A p (r )

2

.

On the other hand, integration by parts and (20.8) yield ∂h ∂h 2 |∇h| = i i (ρ) − si (1) , ∂r B p (ρ)\B p (1) ∂ B p (ρ) ∂ B p (1) ∂r ≥ si (1) − ii (ρ) because si (1) ≥ si (ρ) ≥ ii (ρ) and si (r ) =

sup

y∈∂ B p (r )

G i ( p, y)

A p (r ) dr

(20.9)

20

Parabolicity

251

is a decreasing function of r . Combining this with (20.9) gives the estimate ρ dt ≤ si (1) − i i (ρ). A p (t) 1 Letting i → ∞ and using the fact that G is nonconstant, we arrive with the estimate ρ dt ≤ s(1) − i(ρ), 1 A p (t) where s(1) =

sup

G( p, y)

inf

G( p, y).

y∈∂ B p (1)

and i(ρ) =

y∈∂ B p (ρ)

Letting ρ → ∞, the finiteness of

∞ 1

dt A p (t) 

follows.

The following lemma gives a criterion for an end to be nonparabolic. It was first proved by Cao, Shen and Zhu in [CSZ] for minimal submanifolds. However, their arugment can be generalized to the following context. Lemma 20.10 Let E be an end of a complete Riemannian manifold. Suppose for some ν ≥ 1 and C > 0, E satisfies a Sobolev type inequality of the form 1/ν  2ν |u| ≤C |∇u|2 E

E

for all compactly supported functions u ∈ Hc1,2 (E) defined on E, then E must either have finite volume or be nonparabolic. Proof Let E be an end of M. For ρ sufficiently large, let us consider the set E(ρ) = E ∩ B p (ρ), where B p (ρ) is the geodesic ball of radius ρ in M centered at some point p ∈ M. Let us denote by r the distance function of M to the point p. Suppose the function f ρ is the solution to the equation  f ρ = 0 on

E(ρ),

252

Geometric Analysis

with boundary conditions fρ = 1

∂E

on

and f ρ = 0 on

E ∩ ∂ B p (ρ).

According to Lemma 20.6, f ρ converges to a nonconstant harmonic function f if and only if E is nonparabolic. For a fixed 0 < ρ0 < ρ such that E(ρ0 ) = ∅, let φ be a nonnegative cutoff function satisfying the properties that φ = 1 on φ=0

E(ρ)\ E(ρ0 ), on

∂ E,

and |∇φ| ≤ C 1 . Applying the inequality in the assumption and using the fact that fρ is harmonic, we obtain 1/ν  (φ f ρ )2ν E(ρ)



≤C E(ρ)

|∇(φ f ρ )|2





=C 

E(ρ)

=C

|∇φ|2 f ρ2 + 2 |∇φ|



E(ρ)

=C E(ρ)

E(ρ)

2

f ρ2

+

1 2



φ f ρ ∇φ, ∇ f ρ  + 

∇(φ ), ∇

E(ρ)

2



f ρ2



E(ρ)

φ 2 |∇ f ρ |2 

+

φ |∇ f ρ | 2

E(ρ)

2

|∇φ|2 f ρ2 .

In particular, for a fixed ρ1 satisfying ρ0 < ρ1 < ρ, we have  1/ν 2ν fρ ≤ C2 f ρ2 . E(ρ1 )\E(ρ0 )

E(ρ0 )

If E is parabolic, then the limiting function f is identically 1. Letting ρ → ∞, we obtain (V E (ρ1 ) − VE (ρ0 ))1/ν ≤ C VE (ρ0 ),

20

Parabolicity

253

where VE (ρ) denotes the volume of the set E(ρ). Since ρ1 > ρ0 is arbitrary, this implies that E must have finite volume and the theorem is proved.  Lemma 20.11 Let M be a complete Riemannian manifold. Given a geodesic ball B p (ρ0 ), suppose there exist constants ν > 1 and C > 0 such that a Sobolev type inequality of the form

1/ν B p (ρ0 )

|u|2ν

≤C

B p (ρ0 )

|∇u|2

is valid for all compactly supported functions u ∈ Hc1,2 (B p (ρ0 )). Then there exists a constant C1 > 0 depending only on C and ν such that 2ν/(ν−1)

V p (ρ0 ) ≥ C 1 ρ0

.

In particular, if an end E of M with respect to the compact set B p (ρ0 ) satisfies the Sobolev inequality  1/ν |u|2ν ≤C |∇u|2 E

E

for all compactly supported functions u ∈ Hc1,2 (E), then the end must have volume growth given by VE (ρ0 + ρ) ≥ C1 ρ 2ν/(ν−1) for all ρ > 0, where VE (ρ0 + ρ) denotes the volume of the set E p (ρ0 + ρ). Proof For a fixed point p ∈ M and function ⎧ 1 ⎪ ⎪ ⎪ ⎨ ρ  − r (x) u(x) = ⎪ ρ − ρ ⎪ ⎪ ⎩ 0

0 < ρ < ρ  < ρ0 , let us consider the on

B p (ρ),

on

B p (ρ  )\ B p (ρ),

on

M \ B p (ρ  ).

where r (x) is the distance function to the point p. Plugging this into the Sobolev inequality, we conclude that (20.10) V p (ρ) ≤ C(ρ  − ρ)−2 V p (ρ  ).   k 2−(i+1) ρ0 for k = 1, 2, . . . . Using ρ  = ρk−1 Let us define ρk = 1 − i=1 and ρ = ρk , (20.10) becomes 1/ν

1

V pν (ρk ) ≤ C4k+1 ρ0−2 V p (ρk−1 ).

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Geometric Analysis

This can be rewritten as

1/ν 2ν/(ν−1) ρ V p (ρk−1 ) V p (ρk ) k+1 −2 k−1 ≤ C4 ρ 0 2ν/(ν−1) 2/(ν−1) 2ν/(ν−1) ρk ρk ρk−1  k−1 −(i+1) 2ν/(ν−1) 1 − i=1 2 V p (ρk−1 ) = C4k+1  2/(ν−1) 2ν/(ν−1) . (20.11) k ρk−1 1 − i=1 2−(i+1) Using the estimate 1 2

≤1−

k 

2−(i+1) ≤ 1

i=1

for all 1 ≤ k, (20.11) yields

1/ν V p (ρk ) 2ν/(ν−1) ρk

≤ C1 4k+1

V p (ρk−1 ) 2ν/(ν−1)

ρk−1

,

where C1 > 0 depends on C and ν. Iterating this inequality k times, we obtain

ν −k

V p (ρk )

k

≤ C1

2ν/(ν−1) ρk

i=1 ν

−i

4

k

i=1 (i+1)ν

−i

V p (ρ0 ) 2ν/(ν−1)

ρ0

.

(20.12)

Observing that ρk → ρ0 /2 and ν −k → 0 as k → ∞, we conclude that the lefthand side of (20.12) tends to 1. Hence (20.12) becomes −

C1

∞

i=1 ν

−i

4−

∞

i=1 (i+1)ν

−i

2ν/(ν−1)

ρ0

≤ V p (ρ0 ).

The lower bound of the volume follows by observing that ∞  i=1

ν −i
1. If the Sobolev inequality is valid on an end E with respect to B p (ρ0 ), then we apply the above volume estimate to the ball Bx (ρ) with x ∈ ∂ B p (ρ0 + ρ) ∩ E. Hence we obtain VE (ρ0 + 2ρ) ≥ Vx (ρ) ≥ C1 ρ 2ν/(ν−1) , and the volume growth of E follows.



20

Parabolicity

255

Combining Lemma 20.10 and Lemma 20.11, we obtain the following corollary for the Sobolev type inequality with ν > 1. The case when ν = 1 is just the Dirichlet Poincar´e inequality and will be addressed separately in Chapter 22. In that case, it is possible to have a finite volume end given by a cusp. Corollary 20.12 Let E be an end of a complete Riemannian manifold. Suppose that for some ν > 1 and C > 0, E satisfies a Sobolev type inequality of the form 1/ν  2ν |u| ≤C |∇u|2 , E

for all compactly supported functions u ∈ be nonparabolic.

E

Hc1,2 (E)

defined on E, then E must

21 Harmonic functions and ends

In this chapter, we will construct harmonic functions defined on M by extending the barrier functions defined at all of the ends of M. This construction was first proved by Tam and Li in [LT1] for manifolds with nonnegative sectional curvature near infinity. They later gave a construction for arbitrary complete manifolds in [LT6]. In [STW], Sung, Tam, and Wang presented the construction in a more systematic manner. It is their version that is given here. Theorem 21.1 (Li–Tam and Sung–Tam–Wang) Let M be a complete manifold and let  be a smooth compact subdomain in M. Suppose g is a harmonic function defined on M \ which is smooth up to the boundary ∂ of . If M is nonparabolic, then there exists a harmonic function h defined on M and a constant C > 0 such that |g(x) − h(x)| ≤ C G( p, x), where G( p, x) is the minimal positive Green’s function and p ∈ . Moreover, the function g − h has a finite Dirichlet integral on M \. Proof Let p ∈  be a fixed point and let ρ0 > 0 be such that  ⊂ B p (ρ0 ). Let φ be a smooth nonnegative function satisfying # 0 on B p (ρ0 ), φ= 1 on M \ B p (2ρ0 ). If M is nonparabolic and G(x, y) is the minimal positive Green’s function on M, then we define the function h by h(x) = φ(x) g(x) + G(x, y) (φ g)(y) dy. M

256

21

Harmonic functions and ends

257

Since (φ g) = g = 0 on

M \ B p (2ρ0 ),

the function (φ g) has compact support and h is harmonic on M. For x ∈ / B p (4ρ0 ), we have |h(x) − g(x)| = |h(x) − φ(x) g(x)|       G(x, y) (φ g)(y) dy  =   B p (2ρ0 ) ≤ sup G(x, y) |(φ g)|(y) dy. B p (2ρ0 )

y∈B p (2ρ0 )

Since x ∈ / B p (4ρ0 ), G(x, y) as a function of y is harmonic on B p (4ρ0 ). Hence the local Harnack inequality asserts that G(x, y) ≤ C G(x, p) for y ∈ B p (2ρ0 ), where C is a constant depending only on the lower bound of the Ricci curvature of B p (4ρ0 ), m, and ρ0 . This proves that |h(x) − g(x)| ≤ C G( p, x) for all x ∈ / B p (4ρ0 ). Since G is positive, compactness implies that the same bound is valid for all x ∈ M by possibly adjusting the constant C. Note that from the construction of G in Chapter 17, the bound on the function |h − g| can be written as |h(x) − g(x)| ≤ C f (x)

on

M \ B p (4ρ0 ),

(21.1)

where f is the minimal barrier function of Chapter 20 satisfying the properties that f =1

on

∂ B p (4ρ0 ),

inf f (y) = 0,

y→∞

and

M\B p (4ρ0 )

|∇ f |2 < ∞.

For any ρ > 4ρ0 , let u ρ be the harmonic function defined on B p (ρ)\ B p (4ρ0 ) satisfying the boundary conditions uρ = h − g

on

∂ B p (4ρ0 )

(21.2)

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Geometric Analysis

and uρ = 0

∂ B p (ρ).

on

(21.3)

Since (21.1) implies that |u ρ − (h − g)| ≤ C f

∂ B p (4ρ0 ) ∪ ∂ B p (ρ),

on

the maximum principle asserts that |u ρ − (h − g)| ≤ C f

on

B p (ρ)\ B p (4ρ0 ).

Hence there exists a sequence of {ρi } such that u i − (h − g), with u i = u ρi , converges uniformly on compact subset of M \ B p (4ρ0 ) to a harmonic function u − (h − g). Moreover, u − (h − g) = 0

on

∂ B p (4ρ0 )

and |u − (h − g)| ≤ C f. In particular, the harmonic function f − C −1 (u − (h − g)) is nonnegative and has boundary value 1 on ∂ B p (4ρ0 ). Hence by the minimality property of f , f ≤ f − C −1 (u − (h − g)) and u − (h − g) ≤ 0. Running through the same argument using the function (h − g) − u, we conclude that (h − g) − u ≤ 0, hence h − g = u. In particular, the sequence of functions {u i } converges to h − g. However, since ∂u i 2 |∇u i | = − ui ∂r B p (ρi )\B p (4ρ0 ) ∂ B p (4ρ0 ) ∂u i =− (h − g) , ∂r ∂ B p (4ρ0 )

21

Harmonic functions and ends

259

we conclude that for any ρ > 4ρ0

B p (ρ)\B p (4ρ0 )

|∇u i |2 ≤ −

∂ B p (4ρ0 )

(h − g)

∂u i . ∂r

(h − g)

∂(h − g) . ∂r

Letting i → ∞ yields

B p (ρ)\B p (4ρ0 )

|∇(h − g)|2 ≤ −

∂ B p (4ρ0 )

Since ρ is arbitrary, this completes the proof of the theorem.



Theorem 21.2 (Li–Tam and Sung–Tam–Wang) Let M be a complete manifold and  be a smooth compact subdomain in M. Suppose g is a harmonic function defined on M \ which is smooth up to the boundary ∂ of . If M is parabolic, then there exists a harmonic function h defined on M such that |g − h| is bounded on M \ if ∂

∂g = 0, ∂ν

where ν is the outward unit normal to ∂. Moreover, |g − h| must have finite Dirichlet integral on M \. Proof Following an argument similar to that in the proof of Theorem 21.1, we define the harmonic function G(x, y) (φ g)(y) dy, h(x) = φ(x) g(x) + M

where G(x, y) is a symmetric Green’s function constructed in Chapter 17, and φ is given by # φ=

0 1

on on

B p (ρ0 ), M \ B p (2ρ0 ).

Again when for x ∈ / B p (4ρ0 ), we have |h(x) − g(x)| = |h(x) − φ(x) g(x)|       = G(x, y) (φ g)(y) dy   B p (2ρ0 ) 

260

Geometric Analysis       (G(x, y) − G(x, p)) (φ g)(y) dy  ≤   B p (2ρ0 )       (φ g)(y) dy  . (21.4) + G(x, p)   B p (2ρ0 )

However, since



B p (2ρ0 )

(φ g)(y) dy =

∂ B p (2ρ0 )

=

∂

∂g ∂r

∂g ∂ν

=0 and

      (G(x, y) − G(x, p)) (φ g)(y) dy    B p (2ρ0 )  ≤ sup |G(x, y) − G(x, p)| |(φ g)(y)| dy, B p (2ρ0 )

y∈B p (2ρ0 )

(21.4) becomes |h(x) − g(x)| ≤ C

sup

y∈B p (2ρ0 )

|G(x, y) − G(x, p)|.

On the other hand, from the construction of G, we know that G(y, x) − G( p, x) is a bounded harmonic function on M \ B p (4ρ0 ) for y ∈ B p (2ρ0 ). This implies that |h − g| is bounded on M \. To see that h − g has a finite Dirichlet integral, we follow a similar argument to that in the proof of Theorem 21.1. By solving the Dirichlet boundary problem (21.2) and (21.3), we obtain a harmonic function u with the properties that u − (h − g) = 0 on

∂ B p (4ρ0 )

and |u − (h − g)| ≤ C. Unless u − (g − h) is identically 0, then either sup

M\B p (8ρ0 )

u − (g − h) = α

21

Harmonic functions and ends

261

or inf

M\B p (8ρ0 )

u − (g − h) = −α

for some α > 0. In the first case, the function f = α − u + (h − g) is a nonconstant positive harmonic function whose infimum is achieved at infinity. In the second case, the function f = u − (h − g) − α has the same property. This contradicts the assumption that M is parabolic, and we must have u = h − g. The rest of the argument is exactly the same as in the proof of Theorem 21.1.  There is one more property resulting from the above constructions that is important to point out. Recall that when M is nonparabolic the minimal Green’s function G is given by the limit of a sequence of Dirichlet Green’s function defined on a compact exhaustion. Let us assume that the compact exhaustion is given by {B p (ρi )}. Similarly, for a parabolic manifold, the constructed Green’s function can be obtained by taking the limit of G i − ai where ai is a sequence of constants. In either case, if we define the sequence of harmonic functions h i (x) = φ(x) g(x) + G i (x, y) (φ g)(y) dy B p (ρi )

for ρi ≥ 4ρ0 , then h i will solve the Dirichlet problem h i = 0 on

B p (ρi )

(21.5)

∂ B p (ρi ).

(21.6)

and hi = g

on

In the nonparabolic case, obviously h i → h, since G i → G. In the parabolic case, this is also true because ai (φ g)(y) dy = 0, B p (ρi )

hence

h i (x) = φ(x) g(x) +

B p (ρi )

(G i (x, y) − ai ) (φ g)(y) dy,

which converges to h. This implies that the harmonic function constructed in Theorem 21.1 and Theorem 21.2 can be obtained by taking limits of harmonic

262

Geometric Analysis

functions h i solving the Dirichlet problem (21.5) and (21.6). By the maximum principle, we conclude that inf g ≤ h i (x) ≤ sup g

∂ B p (ρi )

∂ B p (ρi )

for all x ∈ B p (ρi ). Taking the limit as i → ∞, we obtain lim inf g ≤ h(x) ≤ lim sup g. y→∞

y→∞

We are now ready to construct harmonic functions that reflect the geometry and topology of a complete manifold. Theorem 21.3 (Li–Tam) Let M be a complete manifold that is nonparabolic. There exist spaces of harmonic functions K0 (M) and K (M), with (possibly infinite) dimensions given by k 0 (M) and k  (M), respectively, such that k 0 (M) = N 0 (M) and k  (M) = N  (M). In particular, k 0 (M) + k  (M) = N (M). Moreover, K0 (M) is a subspace of the space of bounded harmonic functions with a finite Dirichlet integral on M, and K (M) is spanned by a set of positive harmonic functions. Proof The assumption that M is nonparabolic implies that N 0 (M) ≥ 1. If N 0 (M) = 1, then the space K0 (M) is given by the constant functions. Let us now assume that N 0 (M) ≥ 2 and E 1 , . . . , E Nρ0 (M) is the set of all nonparabolic ends with respect to the compact set B p (ρ). It suffices to show that we can construct Nρ0 (M) many linearly independent bounded harmonic functions with finite Dirichlet integral. Since ρ is arbitrary, this will prove that K0 (M) exists. For each 1 ≤ i ≤ Nρ0 (M), let us define the harmonic function ψi on M \ B p (ρ) given by # 1 on E i , ψi = 0 on M \(B p (ρ) ∪ E i ). By Theorem 21.1, there exists a harmonic function h i defined on M such that h i − ψi is bounded and has a finite Dirichlet integral on M \ B p (ρ). By the remark following Theorem 21.2, h i must also be bounded between 0 and 1

21

Harmonic functions and ends

263

and has a finite Dirichlet integral on M. Moreover, on any nonparabolic end E k , Theorem 21.1 also asserts that |ψi − h i | ≤ C f k

on

Ek ,

where fk isthe minimal barrier function of E k . Applying this to E i , it implies that if x ij is a sequence of points in E i with x ij → E i (∞) as j → ∞,     such that f i x ij → 0, then h i x ij → 1. Similarly, this also shows that for   any sequence x αj ⊂ E α with x αj → E α (∞) for α = i as j → ∞ such that     f α x αj → 0, then h i x αj → 0. Obviously, this construction yields Nρ0 (M) bounded harmonic functions {h i } with finite Dirichlet integrals and satisfying   lim h i x ij = 1 j→∞

and

  lim h i x αj = 0

j→∞

for

α = i.

The linear independence of this set of functions follows and K0 is constructed. Now let E 1 , . . . E N  (M) be the set of parabolic ends with respect to B p (ρ). ρ

We will construct a set of Nρ (M) linearly independent positive harmonic functions. Again, since ρ is arbitrary, this will prove that a space K exists with the properties that dim K = k  (M) = N  (M) and K is spanned by positive harmonic functions. Therefore K(M) = K0 (M) + K (M) and the theorem follows. Modifying the above argument, for each Ei let gi be a barrier function for  E i given by Lemma 20.7. Let us define the function ψi by # ψi =

gi 0

on on

E i ,

 M \ B p (ρ) ∪ E i .

Using Theorem 21.1 again, there exists a harmonic function h i with the property that |ψi − h i | ≤ C

on

E α for α = i

and |ψi − h i | ≤ C f k

on

Ek .

264

Geometric Analysis

In particular, since gi (y) → ∞ as y → E i (∞), we conclude that lim sup h i (y) = ∞.

y→E i (∞)

Also h i is bounded on M \ E i . Note that according to the remark preceding Theorem 21.2, h i must be positive since it can be viewed as a limit of a sequence of positive harmonic functions satisfying (21.5) and (21.6). Moreover, since there is a nonparabolic end, inf M h i = 0. Clearly, the functions in the set {h i } are linearly independent and they span the space Kρ (M). This completes the theorem.  A similar construction also gives a corresponding theorem for parabolic manifolds. Theorem 21.4 (Li–Tam) Let M be a complete manifold that is parabolic. There exists a space of harmonic functions K(M), with (possibly infinite) dimension given by k(M), such that k(M) = N (M). Moreover, K(M) is spanned by a set of harmonic functions which are bounded from either above or below when restricted on each end of M. Proof Let E i for 1 ≤ i ≤ Nρ (M) be the set of parabolic ends with respect to B p (ρ). Following the same notation as before, we let gi be the barrier functions for the ends. Note that for each i, from the construction gi achieves its minimum value on ∂ E i . Therefore, by the maximum principle, we conclude that ∂gi = αi ∂ E i ∂r for some αi > 0. Let us define the function ψi by ⎧ ⎪ on E 1 , ⎨ g1 α1 ψi = − αi gi on E i ,

 ⎪ ⎩0 on M \ E 1 ∪ E i . Clearly,

∂ B p (ρ)

∂ψi = 0, ∂r

and we can apply Theorem 21.2 to obtain a harmonic function h i on M with the property that |ψi − h i | is bounded and has finite Dirichlet integral. In

21

Harmonic functions and ends

265

particular, the function h i has the properties that lim sup h i (y) = ∞,

y→E 1 (∞)

lim inf h i (y) = −∞,

y→E i (∞)

and |h i (y)| ≤ C

on

 M − B p (ρ) ∪ E 1 ∪ E i .

Moreover, since the barrier functions {gi } are positive, h i must be bounded from either above or below on each end. Clearly, the set of functions {h i } together with a constant function forms a linear independent set and spans the  space Kρ (M). This completes the proof of the theorem. Combining Theorem 21.4 with Theorem 7.4, we obtain the following finiteness theorem. Theorem 21.5 (Li–Tam) Let M be an m-dimensional complete noncompact Riemannian manifold without boundary. Suppose that the Ricci curvature of M is nonnegative on M \ B p (1) for some unit geodesic ball centered at p ∈ M. Let us assume that the lower bound of the Ricci curvature on B p (1) is given by Ri j ≥ −(m − 1)R for some constant R ≥ 0. Then M must have finitely many ends. Moreover, there exists a constant C(m, R) > 0 depending only on m and R such that N (M) ≤ C(m, R). Proof If M is nonparabolic, we define the space of harmonic functions 4 H (M) = K0 (M) K (M) as constructed in Theorem 21.3. Since functions in K0 (M) are all bounded harmonic functions and any finite dimensional subspace of K (M) is spanned by harmonic functions that are positive and each tends to infinity at each end, their linear combination must be either bounded above or bounded below on each end. Hence we can apply Theorem 7.4 to estimate dim H (M), therefore obtaining the bound N (M) ≤ C(m, R) by combining with Theorem 21.3. If M is parabolic, we simply combine Theorem 21.4 and Theorem 7.4 to obtain the estimate on N (M). 

266

Geometric Analysis

We would like to point out that when M has nonnegative Ricci curvature everywhere, this argument also recovers the splitting theorem (Theorem 4.4) of Cheeger and Gromoll [CG2] . The interested reader may note that the estimate on the number of ends actually holds on manifolds whose Ricci curvature need not be nonnegative at infinity. In fact, it was proved by Li and Tam [LT1] that if we assume that the Ricci curvature of M is bounded from below by Ri j (x) ≥ −(m − 1)k(r (x)), for some nonincreasing function k(r ) ≥ 0 satisfying the property ∞ k(r ) r m−1 dr < ∞, 1

then N (M) is finite and can be estimated. However, the proof is more involved and might not substantially add to our overall educational purpose.

22 Manifolds with positive spectrum

In this chapter, we will consider manifolds whose spectrum of the Laplacian, Spec(), acting on L 2 functions is bounded from below by a positive number. If we denote μ1 (M) = inf Spec() to be the infimum of the spectrum of , it can be characterized by ! |∇ f |2 M ! inf , μ1 (M) = 2 f ∈Hc1,2 (M) M f where the infimum is taken over all compactly supported functions in the Sobolev space Hc1,2 (M). In particular, μ1 (M) = inf μ1 (i ) i→∞

for any compact exhaustion {i }, where μ1 (i ) is the Dirichlet first eigenvalue of  on i . In 1975, Cheng and Yau [CgY] gave a necessary condition for a complete manifold to have μ1 (M) > 0 by showing that if M has polynomial volume growth, then μ1 (M) = 0. Another important quantity of the spectrum is the infimum of the essential spectrum μe (M) of . It has the property that μe (M) ≥ μ1 (M) and given any  > 0 there exists a compact set  ⊂ M such that μe (M) ≤ μ1 ( ) +  267

268

Geometric Analysis

for any compact set  ⊂ M \. In view of this remark, most of the statements proved in this chapter with the assumption of μ1 (E) > 0 on an end can be stated as μe (M) > 0. Throughout this chapter, we will assume that E is an end of M with respect to the compact set B p (ρ0 ). We also assume that the infimum of the Dirichlet spectrum of  on E is positive. In particular, ! |∇ f |2 0 < μ1 (E) ≤ E! 2 E f for any compactly supported function f ∈ Hc1,2 (E) defined on E. Our goal is to give decay L 2 -estimates on the harmonic functions f ∈ K0 (M) constructed in Theorem 21.1. This is also equivalent to estimating the barrier functions on a nonparabolic end, and hence the Green’s function on a nonparabolic manifold. Note that by taking ρ0 sufficiently large, the condition μ1 (E) > 0 is guaranteed if μe (M) > 0. In 1981, Brooks [Br] improved Cheng and Yau’s theorem and showed that if M has infinite volume and if we denote the volume entropy, τ (M), by τ (M) = lim sup ρ→∞

log V p (ρ) , ρ

then μe (M) ≤

τ 2 (M) . 4

In a paper of Li and Wang [LW5], the authors proved a sharp estimate of the barrier functions. As one of the consequences, this implies an estimate on the volume growth (Corollary 22.6) and can be viewed as an improvement of Brooks’ result. Theorem 22.1 (Li–Wang) Let M be a complete Riemannian manifold. Suppose E is an end of M with respect to B p (ρ0 ) such that μ1 (E) > μ for some constant μ ≥ 0. Let f be a nonnegative function defined on E satisfying the differential inequality  f ≥ −μ f. If f satisfies the growth condition E(ρ)

f 2 e−2ar = o(ρ)

22 as ρ → ∞, with a =

√

Manifolds with positive spectrum

269

μ1 (E) − μ, then it must satisfy the decay estimate

f 2 ≤ C(a)(1 + (ρ − ρ0 )−1 ) e−2aρ

E(ρ0 +1)\E(ρ0 )

E(ρ+1)\E(ρ)

e2ar f 2

for some constant C(a) > 0 depending on a and for all ρ ≥ 2(ρ0 + 1), where E(ρ) = B p (ρ) ∩ E. Proof We will first prove that for any 0 < δ < 1, there exists a constant 0 < C < ∞ such that e2δar f 2 ≤ C. E

Indeed, let φ(r (x)) be a nonnegative cutoff function with support on E with r (x) being the geodesic distance to the fixed point p. Then for any function h(r (x)), integration by parts yields |∇(φ eh f )|2 E





=



|∇(φ e )| f + h 2



E

=

(φ e ) |∇ f | + 2

2

h 2



E

|∇(φ e )| f + h 2



φ e

2

E

2 2h



E

|∇(φ eh )|2 f 2 +

=

E

≤

1 |∇ f | + 2



E

∇(φ 2 e2h ), ∇( f 2 )

1 2





E

φ e2h ∇φ, ∇h f 2 E





φ 2 |∇h|2 e2h f 2 + μ E

φ 2 e2h ( f 2 ) E

φ 2 e2h f 2



+

E

E

|∇φ|2 e2h f 2 + 2 E

E

2

φ 2 e2h |∇ f |2 −

|∇(φ eh )|2 f 2 + μ

=

φ eh f ∇(φ eh ), ∇ f 

2

φ 2 e2h f 2 . E

On the other hand, using the variational principle for μ1 (E), we have



μ1 (E)

φ 2 e2h f 2 ≤ E

|∇(φ eh f )|2 , E

(22.1)

270

Geometric Analysis

hence (22.1) becomes 2 φ 2 e2h f 2 a E





≤



|∇φ|2 e2h f 2 + 2 E

φ e2h ∇φ, ∇h f 2 + E

φ 2 |∇h|2 e2h f 2 . E

(22.2) Let us now choose ⎧ r (x) − ρ0 ⎪ ⎪ ⎨ 1 φ(r (x)) = ⎪ ρ −1 (2ρ − r (x)) ⎪ ⎩ 0 and h(r ) =

on on on on

E(ρ0 + 1)\ E(ρ0 ), E(ρ)\ E(ρ0 + 1), E(2ρ)\ E(ρ), E \ E(2ρ)

⎧ ⎨ δar

for

r ≤ A/(1 + δ)a,

⎩ A − ar

for

r ≥ A/(1 + δ)a

for some fixed constant A > (ρ0 + 1)(1 + δ)a. When ρ ≥ A/(1 + δ)a, we see that ⎧ ⎨ δ 2 a 2 for r ≤ A/(1 + δ)a, 2 |∇h| = ⎩ a2 for r ≥ A/(1 + δ)a and ⎧ ⎨ δa ∇φ, ∇h = ρ −1 a ⎩ 0

on E(ρ0 + 1)\ E(ρ0 ), on E(2ρ)\ E(ρ), otherwise.

Substituting the above inequality into (22.2), we obtain φ 2 e2h f 2 ≤ e2h f 2 + ρ −2 a2 E

E(ρ0 +1)\E(ρ0 )



+ 2δa

E(ρ0 +1)\E(ρ0 )

E(2ρ)\E(ρ)

e2h f 2 + 2ρ −1 a

×

E(A((1+δ)a)−1 )\E(ρ0 )

E(2ρ)\E(A((1+δ)a)−1 )



e2h f 2 E(2ρ)\E(ρ)

+ δ2 a2

e2h f 2

φ 2 e2h f 2 + a 2

φ 2 e2h f 2 .

22

Manifolds with positive spectrum

This can be rewritten as 2 a

E(A((1+δ)a)−1 )\E(ρ0 +1)

271

e2h f 2



≤ a2 ≤

E(A((1+δ)a)−1 )

E(ρ0 +1)\E(ρ0 )

φ 2 e2h f 2

e2h f 2 + ρ −2

e2h f 2 E(2ρ)\E(ρ)



+ 2δa

E(ρ0 +1)\E(ρ0 )

e

2h

2

f + 2ρ

−1



e2h f 2

a E(2ρ)\E(ρ)

+δ a

2 2 E(A((1+δ)a)−1 )\E(ρ

0)

φ 2 e2h f 2 ,

hence (1 − δ 2 )a 2

E(A((1+δ)a)−1 )\E(ρ



≤ (δ 2 a 2 + 2δa + 1) + ρ −2



0 +1)

e2h f 2

E(ρ0 +1)\E(ρ0 )

e2h f 2

e2h f 2 + 2ρ −1 a E(2ρ)\E(ρ)

e2h f 2 . E(2ρ)\E(ρ)

The definition of h and the assumption on the growth estimate on f imply that the last two terms on the right-hand side tend to 0 as ρ → ∞, and we obtain the estimate e2δar f 2 (1 − δ 2 )a 2 E(A((1+δ)a)−1 )\E(ρ0 +1)



≤ (δ 2 a 2 + 2δa + 1)

E(ρ0 +1)\E(ρ0 )

e2δar f 2 .

Since the right-hand side is independent of A, by letting A → ∞ we conclude that e2δar f 2 ≤ C1 , (22.3) E\E(ρ0 +1)

with δ 2 a 2 + 2δa + 1 C1 = (1 − δ 2 )a 2

E(ρ0 +1)\E(ρ0 )

e2δar f 2 .

272

Geometric Analysis

Our next step is to improve this estimate by setting h = ar in the preceding argument. Note that (22.2) asserts that φ e2ar ∇φ, ∇r  f 2 − 2a

E

≤

|∇φ|2 e2ar f 2 . E

For ρ0 < ρ1 < ρ, let us now choose φ to be ⎧ r (x) − ρ0 ⎪ ⎪ on ⎨ ρ1 − ρ0 φ(x) = ⎪ ρ − r (x) ⎪ ⎩ on ρ − ρ1

E(ρ1 )\ E(ρ0 ), E(ρ)\ E(ρ1 ).

We conclude that 2a (ρ − r ) e2ar f 2 (ρ − ρ1 )2 E(ρ)\E(ρ1 ) 1 1 2ar 2 ≤ e f + e2ar f 2 (ρ1 − ρ0 )2 E(ρ1 )\E(ρ0 ) (ρ − ρ1 )2 E(ρ)\E(ρ1 ) 2a + (r − ρ0 ) e2ar f 2 . (ρ1 − ρ0 )2 E(ρ1 )\E(ρ0 ) On the other hand, for any 0 < t < ρ − ρ1 , since 2at e2ar f 2 (ρ − ρ1 )2 E(ρ−t)\E(ρ1 ) 2a ≤ (ρ − r ) e2ar f 2 , (ρ − ρ1 )2 E(ρ)\E(ρ1 ) we deduce that

2at e2ar f 2 (ρ − ρ1 )2 E(ρ−t)\E(ρ1 )   2a 1 ≤ + e2ar f 2 ρ1 − ρ0 (ρ1 − ρ0 )2 E(ρ1 )\E(ρ0 ) 1 + e2ar f 2 . (ρ − ρ1 )2 E(ρ)\E(ρ1 )

Observe that if we take ρ1 = ρ0 + 1, t = a −1 , and set e2ar f 2 , g(ρ) = E(ρ)\E(ρ0 +1)

(22.4)

22

Manifolds with positive spectrum

273

then (22.4) can be written as g(ρ − a −1 ) ≤ C2 ρ 2 + where 2a + 1 C2 = 2

1 2

g(ρ),

E(ρ0 +1)\E(ρ0 )

e2ar f 2

is independent of ρ. Iterating this inequality, for any positive integer k and ρ ≥ 1, we obtain g(ρ) ≤ C 2

k  (ρ + ia −1 )2 i=1

≤ C2 ρ 2

2i−1

+ 2−k g(ρ + ka −1 )

∞  (1 + ia −1 )2

2i−1

i=1

+ 2−k g(ρ + ka −1 )

≤ C 3 ρ + 2−k g(ρ + ka −1 ), 2

where C 3 = C2

∞  (1 + ia −1 )2

2i−1

i=1

.

However, our previous estimate (22.3) implies that g(ρ + ka −1 ) = e2ar f 2 E(ρ+ka −1 )\E(ρ0 +1)

≤ e2a(ρ+ka

−1 )(1−δ)

≤ C e2a(ρ+ka



E(ρ+ka −1 )\E(ρ0 +1) −1 )(1−δ)

e2δar f 2

.

Hence, 2−k g(ρ + ka −1 ) → 0 as k → ∞ by choosing 2(1 − δ) < ln 2. This proves the estimate e2ar f 2 ≤ C 3 ρ 2 E(ρ)\E(ρ0 +1)

for all ρ ≥ ρ0 + 1.

(22.5)

274

Geometric Analysis

Using inequality (22.4) again and choosing ρ1 = ρ0 + 1 and t = ρ/2 this time, we conclude that e2ar f 2 aρ E(ρ/2)\E(ρ0 +1)



≤ (2a + 1)(ρ − ρ0 − 1)



2 E(ρ0 +1)\E(ρ0 )

e

2ar

f + 2

E(ρ)\E(ρ0 +1)

e2ar f 2 .

Applying the estimate (22.5) to the second term on the right-hand side, we have e2ar f 2 ≤ C5 ρ, E(ρ/2)\E(ρ0 +1)

where C5 = (2a + 1)a

−1

∞

(ρ − ρ0 − 1)2  (1 + ia −1 )2 + 2i ρ2 i=1

×

E(ρ0 +1)\E(ρ0 )

e2ar f 2 .

Therefore, for ρ ≥ 2(ρ0 + 1), we have 2ar 2 e f ≤ C(a) ρ E(ρ)

E(ρ0 +1)\E(ρ0 )

e2ar f 2 ,

(22.6)

where C(a) denotes a constant depending only on a. We are now ready to prove the lemma by using (22.6). Setting t = 2a −1 and ρ1 = ρ − 4a −1 in (22.4), we obtain e2ar f 2 E(ρ−2a −1 )\E(ρ−4a −1 )



≤

8

+



4

a(ρ − ρ0 − 4a −1 ) a 2 (ρ − ρ0 − 4a −1 )2 1 + e2ar f 2 . 4 E(ρ)\E(ρ−4a −1 )

E(ρ−4a −1 )\E(ρ

0)

e2ar f 2

According to (22.6), the first term of the right-hand side is bounded by −1 −1 C(a)(1 + (ρ − ρ0 − 4a ) ) e2ar f 2 E(ρ0 +1)\E(ρ0 )

22

Manifolds with positive spectrum

275

for ρ ≥ 2(ρ0 + 1). Hence, by renaming ρ, the above inequality can be rewritten as e2ar f 2 ≤ C(a)(1 + (ρ − ρ0 )−1 ) E(ρ+2a −1 )\E(ρ)



×

E(ρ0 +1)\E(ρ0 )

e

2ar

f + 2

1 3

E(ρ+4a −1 )\E(ρ+2a −1 )

e2ar f 2 .

Iterating this inequality k times, we arrive at E(ρ+2a −1 )\E(ρ)

e2ar f 2 ≤ C(a)(1 + (ρ − ρ0 )−1 )

k−1  i=0

×

3−i + 3−k

E(ρ+2a −1 (k+1))\E(ρ+2a −1 k)

e2ar f 2 .

However, using (22.6) again, we conclude that the second term is bounded by 3−k e2ar f 2 E(ρ+2(k+1))\E(ρ+2k)

≤ C(a) 3−k (ρ + 2(k + 1))

E(ρ0 +1)\E(ρ0 )

e2ar f 2 ,

which tends to 0 as k → ∞. Hence e2ar f 2 ≤ C(a)(1 + (ρ − ρ0 )−1 ) E(ρ+2a −1 )\E(ρ)

for ρ + 4a −1 ≥ 2(ρ0 + 1), and the theorem follows when a ≤ 2. If a ≥ 2, we simply sum the above estimate [a/2] times by dividing the interval [ρ, ρ + 1] into [a/2] components.  The following proposition is helpful in finding a lower bound for μ1 (M). Proposition 22.2 Let M be a compact Riemannian manifold with smooth boundary ∂ M. If there exists a positive function f defined on M satisfying  f ≤ −μf, then the first Dirichlet eigenvalue, μ1 (M), of M must satisfy μ1 (M) ≥ μ. In particular, if M is complete, noncompact, and without boundary, and the positive function f described above is defined on M, then μ1 (M) ≥ μ.

276

Geometric Analysis

Proof Let us first assume that M is compact with smooth boundary ∂ M. Let u be the first eigenfunction satisfying u = −μ1 (M) u

on

M

and u = 0 on

∂ M.

We may assume that u ≥ 0 on M, and the regularity of u asserts that u > 0 in the interior of M. Integration by parts yields uf ≥ u f − f u (μ1 (M) − μ) M



M

∂f = u − ∂ M ∂ν

M



∂M

f

∂u ∂ν

≥ 0, where ν is the outward unit normal of M. Hence μ1 (M) ≥ μ and the first part of the proposition is proved. When M is complete, noncompact, and without boundary, we apply the previous argument to any compact smooth subdomain D ⊂ M and obtain μ1 (D) ≥ μ. The second part of the proposition follows from the fact that inf μ1 (D) = μ1 (M).

D⊂M



We will now show that the hypothesis of Theorem 22.1 is the best possible. Indeed, if we consider the hyperbolic space form of constant −1 sectional curvature, then the metric in terms of polar coordinates is given by 2 2 2 2 dsH m = dr + sinh t dsSm−1 .

The volume growth is given by



r

V p (r ) = αm−1

sinhm−1 t dt ∼ C e(m−1)r

0

and (m − 1)2 4 by Cheng’s theorem (Corollary 6.4). In fact, if we consider μ1 (Hm ) ≤

β(x) = lim (t − r (γ (t), x)) t→∞

(22.7)

22

Manifolds with positive spectrum

277

to be the Buseman function with respect to some geodesic ray γ , then following the computation in Chapter 4, we conclude that β(x) = − lim r (γ (t), x) t→∞

= −(m − 1). Defining the positive function  h(x) = exp

 m−1 β(x) , 2

a direct computation implies that h = −

(m − 1)2 h. 4

Proposition 22.2 implies that μ1 (Hm ) ≥ (m − 1)2 /4, and hence when combined with (22.7) yields μ1 (Hm ) = (m − 1)2 /4. Let us now consider any nonconstant bounded harmonic function f , then √ f 2 e−2 μ1 r = O(ρ). B p (ρ)

Of course, if the conclusion of Theorem 22.1 is valid, then f will be in L 2 (Hm ), which implies that f is identically constant by Yau’s theorem (Lemma 7.1). However, it is known that Hm has an infinite dimensional space of bounded harmonic functions. This implies that the hypothesis of Theorem 22.1 cannot be relaxed. Corollary 22.3 Let M be a complete Riemannian manifold. Suppose E is an end of M such that μ1 (E) > 0. Then for any harmonic function f ∈ K0 (M), there exists a constant a such that f − a must be in L 2 (E). Moreover, the function f − a must satisfy the decay estimate   " ( f − a)2 ≤ C exp −2ρ μ1 (E) E(ρ+1)\E(ρ)

for some constant C > 0 depending on f , μ1 (E), and m, where E(ρ) = B p (ρ) ∩ E. Proof It suffices to prove the corollary for those functions f constructed in Theorem 21.1 because the decay property is preserved under linear combinations. Following the remark preceding Theorem 21.2, for a nonparabolic

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Geometric Analysis

end E 1 , let f ρ be a sequence of harmonic functions by solving the Dirichlet boundary problem  fρ = 0

on

B p (ρ),

f ρ = 1 on

∂ B p (ρ) ∩ E 1

fρ = 0

∂ B p (ρ)\ E 1 .

and on

A subsequence of this sequence converges to f ∈ K0 (M) uniformly on compact subsets of M. For any fixed end E, since f ρ has the boundary value either 0 or 1 on ∂ E(ρ), by considering either the function f ρ or 1 − f ρ , we may assume that f ρ has the boundary value 0 on ∂ B p (ρ) ∩ E. Let us define the function gρ by # f ρ (x) on E(ρ), gρ (x) = 0 on E \ E(ρ). Clearly, gρ is a nonnegative subharmonic function defined on E. Moreover, since gρ has compact support, the hypothesis of Theorem 22.1 is satisfied and the decay estimate holds for gρ . The corollary follows by taking ρ → ∞.  We point out that Corollary 22.3 also holds for any function f with a = 0 provided that f is the limit of a sequence of harmonic functions fρ on E(ρ) satisfying f ρ = 0 on ∂ E(ρ) regardless of their boundary values on ∂ E. In particular, when applying Corollary 22.3 to the Green’s function, we obtain the following sharp decay estimate. Corollary 22.4 Let M be a complete manifold with μ1 (M) > 0. Then the minimal positive Green’s function G( p, ·) with a pole at p ∈ M must satisfy the decay estimate   " G 2 ( p, x) d x ≤ C exp −2ρ μ1 (M) B p (ρ+1)\B p (ρ)

for ρ ≥ 1. In the case when M = Hm , (17.8) asserts that the Green’s function is given by ∞ dt G( p, x) = , r (x) A p (t)

22

Manifolds with positive spectrum

279

where A p (t) = αm−1 sinh(m−1) t is the area of the boundary of the geodesic ball of radius t centered at p ∈ Hm . One computes readily that G 2 ( p, x) d x ∼ C exp(−(m − 1) ρ). B p (ρ+1)\B p (ρ)

√ Since μ1 (Hm ) = (m − 1)2 /4, the quantity 2 μ1 (Hm ) is exactly (m − 1), indicating the sharpness of Corollary 22.4. Applying Corollary 22.3, we obtain volume estimates for those ends with positive spectrum. As pointed out in the introduction, these estimates are sharp. The sharp growth estimate is realized by the hyperbolic space Hm , while the sharp decay estimate is realized by a hyperbolic cusp (see Example 21.1). To state our estimate, let us denote the volume of the set E(ρ) by VE (ρ) and the volume of the end E by V (E). Theorem 22.5 (Li–Wang) Let E be an end of complete manifold M with μ1 (E) > 0. (1) If E is a parabolic end, then E must have exponential volume decay given by VE (ρ + 1) − VE (ρ) ≤ C (1 + (ρ − ρ0 )−1 ) (VE (ρ0 + 1) − VE (ρ0 ))   " × exp −2(ρ − ρ0 ) μ1 (E) for some constant C > 0 depending on the μ1 (E). In particular, we have V (E) − VE (ρ) ≤ C (VE (ρ0 + 1) − VE (ρ0 ))   " × exp −2(ρ − ρ0 ) μ1 (E) . (2) If E is a nonparabolic end, then E must have exponential volume growth given by  "  VE (ρ) ≥ C exp 2ρ μ1 (E) for all ρ ≥ ρ0 + 1 and for some constant C > 0 depending on the end E. Proof Let fρ be the harmonic function on E(ρ) with f ρ = 1 on ∂ E and f ρ = 0 on ∂ E(ρ). The assumption that E is parabolic implies that f ρ converges to f = 1 as ρ → ∞. Hence the estimate from Theorem 22.1 yields the first

280

Geometric Analysis

estimate of the theorem. Letting ρ = ρ + i for i = 0, 1, . . . and summing over i, we conclude that V (E) − VE (ρ) ≤ C (VE (ρ0 + 1) − VE (ρ0 ))

∞ 

(1 + (ρ + i − ρ0 )−1 )

i=0

 × exp −2(ρ + i − ρ0 ) μ1 (E) 

"

  " ≤ C (VE (ρ0 + 1) − VE (ρ0 )) exp −2(ρ − ρ0 ) μ1 (E) . This proves the volume decay estimate for the case of parabolic ends. If E is nonparabolic, then f ρ converges to a nonconstant harmonic function f on E. Thus, we conclude that there exists a positive constant C such that for r ≥ ρ0 C=

∂E

=

∂f ∂ν

∂ B p (r )∩E

∂f ∂ν

≤

≤

∂ B p (r )∩E 1/2 A E (r )

|∇ f |

1/2

∂ B p (r )∩E

|∇ f |2

,

or equivalently C ≤ A E (r )

∂ B p (r )∩E

|∇ f |2 .

Integrating the preceding inequality with respect to r from ρ to ρ + 1 and using Corollary 22.3 and Lemma 7.1, we obtain

ρ+1 ρ

1 dr ≤ C A E (r )

|∇ f |2 E(ρ+1)\E(ρ)

  " ≤ C exp −2ρ μ1 (E) .

22

Manifolds with positive spectrum

281

Therefore, 1≤

ρ+1 ρ

A E (r ) dr

ρ

ρ+1

1 dr A E (r )

  " ≤ C exp −2ρ μ1 (E) (VE (ρ + 1) − VE (ρ))   " ≤ C exp −2ρ μ1 (E) VE (ρ + 1). Since ρ is arbitrary, we conclude that

 "  V E (ρ) ≥ C exp 2ρ μ1 (E)

by adjusting the constant C, and the theorem is proved.



Corollary 22.6 Let M be a complete manifold with infinite volume. Suppose the essential spectrum of M has a positive lower bound, i.e., μe (M) > 0, then for any  > 0 there exists C > 0, depending on , such that  "  V p (ρ) ≥ C exp 2ρ μe (M) −  . Proof For  > 0, let ρ0 be sufficiently large that μ1 (M \ B p (ρ0 )) +  ≥ μe (M). Theorem 22.5 and the infinite volume assumption on M assert that M must have at least one infinite volume end E with respect to B p (ρ0 ). In fact, the volume growth of this end must satisfy  "  VE (ρ) ≥ C exp 2ρ μe (M) −  , hence

 "  V p (ρ) ≥ C exp 2ρ μe (M) −  .



Our last corollary concerns L q harmonic functions on an end with positive bottom spectrum. Corollary 22.7 Let E be an end of M with μ1 (E) > 0. Let Lq (E) be the space of L q harmonic functions on E. If u ∈ Lq (E) with q ≥ 2, then u must be bounded and it must satisfy the estimate   " u 2 ≤ C exp −2ρ μ1 (E) . E(ρ+1)\E(ρ)

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Geometric Analysis

If u ∈ Lq (E) with 1 < q < 2, then the same conclusion is true provided that the volume growth of E is bounded by   " 2q ρ μ1 (E) . VE (ρ) ≤ C exp 2−q Proof Let u ∈ Lq (E) be an L q harmonic function. Define f ρ to be the harmonic function on E(ρ) satisfying fρ = 0

on

∂ B p (ρ) ∩ E

and fρ = u

on

∂ E.

Clearly, the maximum principle asserts that a subsequence of f ρ as ρ → ∞ converges to a function f ∈ L∞ (E) with u = f on ∂ E. Moreover, by Corollary 22.3, f satisfies the estimate   " f 2 ≤ C exp −2ρ μ1 (E) . (22.8) E(ρ+1)\E(ρ)

If q ≥ 2, the boundedness of f and (22.8) imply that f ∈ Lq (E). In particular, the function u − f is in Lq (E) with 0 boundary condition on ∂ E. Applying the uniqueness theorem of Yau [Y2] (Lemma 7.1) for L q harmonic functions, we conclude that u = f. For 1 < q < 2, the Schwarz inequality, (22.8), and the volume growth bound give q/2  fq ≤ f2 (VE (ρ + 1) − VE (ρ))(2−q)/2 E(ρ+1)\E(ρ)

E(ρ+1)\E(ρ)

   "  " ≤ C exp −q ρ μ1 (E) exp q ρ μ1 (E) ≤ C,

implying that the L q -norm of f is at most of linear growth. Again, by applying the argument of Lemma 7.1 to the subharmonic function g = | f − u| defined on E with boundary condition g=0

on

∂ E,

we conclude that g = 0. To see this, let φ be a cutoff function satisfying # 1 on E(ρ), φ= 0 on E \ E(2ρ),

22

Manifolds with positive spectrum

283

and |∇φ| ≤ C ρ −1

on

E(2ρ) \ E(ρ).

Integration by parts yields φ 2 g q−1 g 0≤ E





= −2

φ g q−1 ∇φ, ∇g − (q − 1) E

φ 2 g q−2 |∇g|2 . E

On the other hand, applying the Schwarz inequality q −1 2 q−1 2 q−2 2 −2 φ g ∇φ, ∇g ≤ φ g |∇g| + |∇φ|2 g q , 2 q −1 E E E we conclude that

φ g 2

E

q−2

4 |∇g| ≤ (q − 1)2

|∇φ|2 g q .

2

E

Using the definition of φ, this implies that C g q−2 |∇g|2 ≤ 2 gq . ρ E(ρ) E(2ρ)\E(ρ) The growth estimate on the L q -norm of f and the fact that u ∈ L q imply that the right-hand side tends to 0 as ρ → ∞. Hence g must be identically constant. The boundary condition of g asserts that it must be identically 0, and f = u. In both cases, since f = u, the function u must satisfy (22.7). This concludes the corollary. 

23 Manifolds with Ricci curvature bounded from below

In this chapter, we will assume that M m has Ricci curvature bounded from below by −(m − 1)R for some constant R > 0. After normalizing, we may assume that R = 1. Note that the Bishop volume comparison theorem asserts that for any x ∈ M, the ratio of the volumes of geodesic balls Bx (ρ1 ) and Bx (ρ2 ) for ρ1 < ρ2 must satisfy Vx (ρ2 ) V¯ (ρ2 ) ≤ , Vx (ρ1 ) V¯ (ρ1 )

(23.1)

where V¯ (ρ) is the volume of a geodesic ball of radius ρ in the m-dimensional hyperbolic space form Hm of constant −1 curvature. In particular, by taking x = p, ρ1 = 0, and ρ2 = ρ this implies that V p (ρ) ≤ C3 exp((m − 1)ρ)

(23.2)

for sufficiently large ρ. On the other hand, if we let x ∈ ∂ B p (ρ), ρ1 = 1, and ρ2 = ρ + 1, (23.1) implies Vx (1) ≥ C4 Vx (ρ + 1) exp(−(m − 1)ρ) ≥ C4 V p (1) exp(−(m − 1)ρ).

(23.3)

Hence we have the following volume estimate. Proposition 23.1 Let M m be a complete manifold with Ricci curvature bounded from below by Ri j ≥ −(m − 1). The volume growth of M must satisfy the upper bound V p (ρ) ≤ V¯ (ρ) ≤ C3 exp((m − 1)ρ) 284

23

Manifolds with Ricci curvature bounded from below

285

for some constant C3 > 0 depending only on m. Also the volume cannot decay faster than exponentially and must satisfy the estimate Vx (1) ≥ C 4 V p (1) exp(−(m − 1)r (x)), where r (x) denotes the distance from p to x. In particular, V p (ρ + 1) − V p (ρ − 1) ≥ C4 V p (1) exp(−(m − 1)ρ). Note that, in view of Theorem 22.5, if we also assume that μ1 (M) > 0, then one has both upper and lower control of the volume growth for a parabolic end and also for a nonparabolic end. Recall that by Cheng’s theorem (Corollary 6.4), under the Ricci curvature assumption μ1 (M) ≤

(m − 1)2 . 4

While Hm achieves equality, it is not the only manifold with this property. In fact, let us consider the following example. Example 23.2 Let M m = R × N m−1 be the complete manifold with the warped product metric 2 ds M = dt 2 + exp(2t) ds N2 .

According to the computation in (A.3) and (A.4) (see Appendix A), by setting f (t) = exp(t), the Ricci curvature on M is given by R1 j = −(m − 1)δ1 j and ˜ αβ − (m − 1)δαβ , Rαβ = exp(−2t) R ˜ αβ is the Ricci tensor on N and e1 = ∂/∂t. In particular, if the Ricci where R curvature of N is nonnegative, then Ri j ≥ −(m − 1). Moreover, N is Ricci flat if and only if M is Einstein with Ri j = −(m − 1).

286

Geometric Analysis

Let us consider the function g(t) = exp(−αt) for a constant (m − 1)/2 ≤ α ≤ m − 1. One computes that g =

dg d2g + (m − 1) 2 dt dt

= α 2 g − (m − 1)α g = −α(m − 1 − α)g and |∇g|2 = α 2 g 2 . If we let α = (m − 1)/2, then the function   (m − 1)t g(t) = exp − 2 satisfies the equation g = −

(m − 1)2 g. 4

Since g is positive, Proposition 22.2 implies that μ1 (M) ≥ (m − 1)2 /4. Using the upper bound of Cheng (Corollary 6.4), we conclude that μ1 (M) = (m − 1)2 /4. It turns out that Example 23.2 gives the only class of manifolds with dimension m ≥ 3 that have more than one end on which Cheng’s inequality is achieved. There is another important example that is relevant to the issues being considered in this chapter. Example 23.3 For m ≥ 3, let M m = R × N m−1 be the product manifold endowed with the warped product metric 2 = dt 2 + cosh2 (t) ds N2 , ds M

where N is a compact manifold with Ricci curvature bounded from below by ˜ αβ ≥ −(m − 2). R Following the notation and the computation in Appendix A, by setting f (t) = cosh(t), the Ricci curvature on M is given by R1 j = −(m − 1)δ1 j

23 and

Manifolds with Ricci curvature bounded from below

287

  ˜ αβ − 1 + (m − 2) tanh2 (t) δαβ . Rαβ = cosh−2 (t) R

˜ αβ ≥ −(m − 2). Hence Ri j ≥ −(m − 1) because of the assumption that R Moreover, M is Einstein with Ri j = −(m − 1) if and only if N is Einstein ˜ αβ = −(m − 2). with R Note that if we define g(t) = cosh−(m−2) (t), then   2 sinh(t) ∂ ∂ + (m − 1) cosh−(m−2) t g = cosh(t) ∂t ∂t 2 = −(m − 2) cosh−(m−2) (t) = −(m − 2)g(t). Proposition 22.2 implies that μ1 (M) ≥ (m − 2). If we compute the L 2 -norm of g on the set [−t, t] × N , then t g2 = V (N ) coshm−1 (t) cosh−2(m−2) (t) dt [−t,t]×N

−t



= V (N )

t −t

cosh3−m (t) dt.

This implies that g ∈ L 2 (M) for m > 3, and g is an eigenfunction, hence μ1 (M) ≤ m − 2. When m = 3, the L 2 -norm of g over [−t, t] × N grows linearly in t. Although g is not an eigenfunction, it is still sufficient to show that μ1 (M) ≤ m − 2, as will be seen in the next theorem. In any event, we have μ1 (M) = m − 2. The following improved version of the Bochner formula for harmonic functions was first proved by Yau [Y1]. It is by now standard in the literature. Lemma 23.4: (Yau) Let M m be a complete manifold whose Ricci curvature is bounded by Ri j (x) ≥ −(m − 1)k(x) for some function k(x). Suppse f is a harmonic function defined on M, then |∇ f |q ≥ −q(m − 1)k(x) |∇ f |q   m 1 + q − 2 |∇ f |−q |∇(|∇ f |q )|2 + q m−1

288

Geometric Analysis

for all q > 0. In particular, if q ≥ (m − 2)/(m − 1), then |∇ f |q ≥ −q(m − 1)k(x) |∇ f |q . Proof Let {e1 , . . . , em } be an orthonormal frame near a point x ∈ M so that |∇ f | e1 = ∇ f at x. We will follow a similar computation to that in the proof of Lemma 5.6. Since f is harmonic, a direct computation and the commutation formula yield |∇ f |2 ≥ 2 f i2j − 2(m − 1)k(x) |∇ f |2 . Also, just like (5.13) and (5.14), we have |∇|∇ f |2 |2 = 4|∇ f |2

m 

f 12j

j=1

and 2 f i2j ≥ f 11 +2

m 

2 f 1α +

α=2

( f − f 11 )2 m−1

m  2 f1 j . m−1 m

≥

j=1

Combining the above three inequalities, we obtain |∇ f |2 ≥ =

m |∇ f |−2 |∇|∇ f |2 |2 − 2(m − 1)k(x) |∇ f |2 2(m − 1) 2m |∇|∇ f ||2 − 2(m − 1)k(x) |∇ f |2 . (m − 1)

The lemma follows directly from this inequality.



Theorem 23.5 (Li–Wang) Let M m be a complete Riemannian manifold of dimension m ≥ 3. Suppose Ri j ≥ −(m − 1) and μ1 (M) ≥ m − 2, then either: (1) M has only one end with infinite volume; or (2) M is the warped product given by Example 23.3.

23

Manifolds with Ricci curvature bounded from below

289

Proof Assume that M has more than one infinite volume end. According to Theorem 21.3, there exists a nonconstant function f ∈ K0 (M) given by the construction. Let g be the function defined by g = |∇ f |(m−2)/(m−1) . Lemma 23.4 asserts that g ≥ −(m − 2)g. We now claim that the function g must satisfy the integral condition g 2 ≤ C ρ. B p (2ρ)\B p (ρ)

To see this, let us apply H¨older’s inequality and get g2 B p (2ρ)\B p (ρ)

≤



B p (2ρ)\B p (ρ)

 " exp 2r μ1 (M) |∇ f |2

×



B p (2ρ)\B p (ρ)

exp −2(m − 2)r

"

(m−2)/(m−1)

 1/(m−1)

μ1 (M)

.

(23.4)

Using the lower bound on the Ricci curvature, the volume comparison theorem asserts that A p (r ) ≤ C exp((m − 1)r ), for some constant C > 0 depending only on m alone. The Fubini theorem now yields   " exp −2(m − 2)r μ1 (M) B p (2ρ)\B p (ρ)



≤C

2ρ

ρ

=C

2ρ ρ

  " exp −2(m − 2)r μ1 (M) exp((m − 1)r ) dr   " exp (m − 1)r − 2(m − 2)r μ1 (M) dr.

(23.5)

Using the lower bound of μ1 (M), we conclude that the right-hand side of (23.5) is at most linear in ρ when m = 3, and exponentially decays to 0 when m ≥ 4.

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Geometric Analysis

On the other hand, recall that the decay estimate in Theorem 22.1 implies that

  " ( f − a)2 ≤ C exp −2ρ μ1 (M) , E(ρ+1)\E(ρ)

where a is the asymptotic value of f at infinity of an end E given by either 1 or 0. Combining this inequality with Lemma 7.1 by choosing α = 1, ρ1 = ρ, ρ2 = ρ + 1, ρ3 = ρ + 2, and ρ4 = ρ + 3, we have

  " |∇ f |2 ≤ 8C exp −2ρ μ1 (M) . E(ρ+2)\E(ρ+1)

In particular, there exists a constant C 1 > 0 independent of ρ such that

 "  exp 2r μ1 (M) |∇ f |2 ≤ C 1 E(ρ+2)\E(ρ+1)

and

 "  exp 2r μ1 (M) |∇ f |2 ≤ C1 ρ.

B p (2ρ)\B p (ρ)

Applying this and (23.5) to (23.4), we conclude that B p (2ρ)\B p (ρ)

g 2 ≤ C2 ρ

for the case when n = 3, and B p (2ρ)\B p (ρ)

g2 → 0

when n ≥ 4. This proves our claim on the L 2 estimate of g. To complete our proof of the theorem, we consider φ to be a nonnegative compactly supported function on M, then





|∇(φ g)|2 = M



|∇φ|2 g 2 + 2 M

φ g ∇φ, ∇g + M

φ 2 |∇g|2 . (23.6) M

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Manifolds with Ricci curvature bounded from below

291

The second term on the right-hand side can be written as φ g ∇φ, ∇g = 12 ∇(φ 2 ), ∇(g 2 ) 2 M

M

=−

φ 2 g g −

M

φ 2 |∇g|2 M



= (m − 2)



φ g − 2

φ 2 |∇g|2

2

M

−

M

φ 2 g (g + (m − 2) g).

(23.7)

M

Combining (23.6) with (23.7) and the variational charaterization of μ1 (M) ≥ m − 2, this implies that (m − 2) φ2 g2 ≤ |∇(φ g)|2 M

M





= (m − 2)

φ g + 2

M

−

|∇φ|2 g 2

2

M

φ 2 g (g + (m − 2) g) , M

hence we have



φ 2 g (g + (m − 2) g) ≤ M

|∇φ|2 g 2 .

(23.8)

M

For ρ > 0, let us choose φ to satisfy the properties that # 1 on B p (ρ), φ= 0 on M \ B p (2ρ), and |∇φ| ≤ C ρ −1

on

B p (2ρ)\ B p (ρ)

for some constant C > 0. Then the right-hand side of (23.8) can be estimated by |∇φ|2 g 2 ≤ C ρ −2 g2 . M

B p (2ρ)\B p (ρ)

By the L 2 estimate of g, this tends to 0 as ρ → ∞. Therefore taking the above inequality together with (23.8), we conclude that either g must be identically 0 or it must satisfy g = −(m − 2) g.

292

Geometric Analysis

If M has more than one infinite volume end, then by Theorem 21.3 there must exist a nonconstant f , hence g = 0. So all the inequalities used in deriving Lemma 23.4 become equalities. We can now argue to conclude that M = R × N with the warped product metric ds 2 = dt 2 + cosh2 (t) ds N2 for some compact manifold N with Ricci ˜ ≥ −(m − 2). curvature satisfying R Indeed, since  f = 0, the Hessian of f must be of the form ⎛ −(m − 1)μ 0 ⎜ 0 μ ⎜ ⎜ 0 0 ( fi j ) = ⎜ ⎜ . .. . ⎝ . .

0 0 μ .. .

⎞ ··· 0 ··· 0 ⎟ ⎟ ··· 0 ⎟ ⎟. ⎟ .. . ⎠

0 0 ··· μ

0

The fact that f 1α = 0 for all α = 1 implies that |∇ f | is identically constant along the level set of f. In particular, the level sets of |∇ f | and f coincide. Moreover, μ δαβ = f αβ = h αβ f 1 with (h αβ ) being the second fundamental form of the level set of f with respect to e1 . Hence f 11 = −H f 1 ,

(23.9)

where H is the mean curvature of the level set of f with respect to e1 . Applying the same computation to the function g, we obtain −(m − 2) g = g = g11 + H g1 . On the other hand, since g = |∇ f |(m−2)/(m−1) , we have   g1 = |∇ f |(m−2)/(m−1) 1

m−2 = |∇ f |−m/(m−1) f i f i1 m−1 m − 2 −1/(m−1) = f 11 . f m−1 1

(23.10)

23

Manifolds with Ricci curvature bounded from below

293

Hence, combining this identity with (23.9), we conclude that H = − f 1−1 f 11 =−

m−1 g1 g −1 . m−2

(23.11)

Substituting this into (23.10) yields g11 −

m−1 (g1 )2 g −1 + (m − 2) g = 0. m−2

Setting u = g −1/(m−2) = |∇ f |−1/(m−1) , this differential equation becomes u 11 − u = 0. Viewing this as an ordinary differential equation along the integral curve generated by the vector field e1 , one concludes that u(t) = A exp(t) + B exp(−t). Since u must be nonnegative, A and B must be nonnegative. Moreover, ∇ f = 0. Note that M is assumed to have at least two infinite volume ends. We claim that any fixed level set N of |∇ f | must be compact. Indeed, by the facts that f has no critical points and that the level set of f coincides with the level set of |∇ f |, M must be topologically the product R × N . If N is noncompact, then M has only one end, hence N must be compact. If both A and B are nonzero, then the function u must have its minimum along N ; hence by reparameterizing, we may assume N is given by t = 0. Moreover, by scaling f , we may also assume that N = {|∇ f | = 1}. Therefore, 0 = u  (0) = A − B and 1 = u(0) = A + B. This implies that u(t) = cosh(t) and g(t) = cosh−(m−2) (t).

294

Geometric Analysis

Using (23.11), we conclude that H (t) = (m − 1) tanh(t) and − → I I αβ (t) = δαβ tanh(t). This implies that the metric on M = R × N must be of the form ds M = dt 2 + cosh2 (t) ds N2 as claimed. If either A or B is 0, say B = 0, then u(t) = A exp(t). In this case, after normalizing A = 1, we have g(t) = exp (−(m − 2)t). Using (23.11), we conclude that H (t) = (m − 1) and h αβ (t) = δαβ . This implies that the metric on M = R × N must be of the form 2 ds M = dt 2 + exp(2t) ds N2 .

However, this implies that M has only one infinite volume end which contradicts our assumption. This completes the proof of the theorem. The curvature restriction on N follows by direct computation and from the curvature assumption on M.  Theorem 23.6 (Li–Wang) Let M m be a complete m-dimensional manifold with m ≥ 3. Suppose that Ri j ≥ −(m − 1) and μ1 (M) ≥

(m − 1)2 . 4

23

Manifolds with Ricci curvature bounded from below

295

Then: (1) M has only one end; (2) M = R × N with the warped product metric 2 = dt 2 + exp(2t) ds N2 , ds M

where N is a compact manifold with nonnegative Ricci curvature; or (3) M is of dimension 3 and M 3 = R × N 2 with the warped product metric 2 ds M = dt 2 + cosh2 (t) ds N2 ,

where N is a compact surface with Gaussian curvature bounded from below by −1. Proof Assuming that M has more than one end, then Theorem 23.5 implies that there must only be one nonparabolic end unless m = 3, in which case m − 2 = (m − 1)2 /4, accounting for case (3) in our theorem. We may now assume that M has only one nonparabolic end E 1 and at least one parabolic end E 2 . Theorem 21.3 implies that there exists a nonconstant positive harmonic function f ∈ K (M) defined on M with the property that lim supx→E 2 (∞) f (x) = ∞. The gradient estimate of Theorem 6.1 asserts that |∇ f |2 ≤ (m − 1)2 f 2 . Combining this estimate with the fact that f is harmonic we obtain 1  f 1/2 = − f −3/2 |∇ f |2 4 ≤−

(m − 1)2 1/2 f . 4

(23.12)

If we write h = f 1/2 , then for any nonnegative cutoff function φ we have |∇(φ h)|2 = |∇φ|2 h 2 − φ 2 h h M



M

M

|∇φ|2 h 2 +

≤ M





−

φ h 2

M

(m − 1)2 4

φ2 h2 M

 (m − 1)2 h + h . 4

On the other hand, the assumption μ1 (M) ≥ (m − 1)2 /4 implies that (m − 1)2 φ2 h2 ≤ |∇(φ h)|2 . 4 M M

296

Geometric Analysis

Hence, we conclude that   (m − 1)2 φ2 h |∇φ|2 h 2 . h + h ≤ 4 M M

(23.13)

Integrating the gradient estimate of Theorem 6.1 along geodesics, we deduce that f must satisfy the growth estimate f (x) ≤ C exp((m − 1)r (x)), where r (x) is the geodesic distance from x to a fixed point p ∈ M. In particular, the above inequality when restricted on the parabolic end E2 together with the volume estimate of Theorem 22.5 imply f ≤ C ρ. (23.14) E 2 (ρ)

On the other hand, Corollary 22.3 asserts that on E1 , the function f must satisfy the decay estimate f 2 ≤ C exp(−(m − 1)ρ) E 1 (ρ+1)\E 1 (ρ)

for ρ sufficiently large. In particular, Schwarz inequality implies that   (m − 1) 1/2 f ≤ C exp − ρ VE 1 (ρ + 1), 2 E 1 (ρ+1)\E 1 (ρ) where VE 1 (r ) denotes the volume of E 1 (r ). Combining this with the volume estimate of Proposition 23.1, we conclude that f ≤C E 1 (ρ+1)\E 1 (ρ)

for some constant C independent of ρ. In particular, f ≤ C ρ, E 1 (ρ)

and taking this together with (23.14) we obtain the growth estimate f ≤ C ρ. B p (ρ)

Choosing φ to be

⎧ 1 ⎪ ⎪ ⎨ 2ρ − r φ= ⎪ ⎪ ⎩ ρ 0

on

B p (ρ),

on

B p (2ρ)\ B p (ρ),

on

M \ B p (2ρ)

(23.15)

23

Manifolds with Ricci curvature bounded from below

297

in (23.13), we conclude that the right-hand side is given by |∇φ|2 h 2 = ρ −2 h2. B p (2ρ)\B p (ρ)

M

Inequality (23.15) implies that |∇φ|2 h 2 → 0 M

as ρ → ∞. Hence we obtain h = −

(m − 1)2 h 4

and the inequalities in deriving the estimate h ≥ −

(m − 1)2 h 4

all become equalities. In particular, |∇ f | = (m − 1) f and |∇(log f )|2 = (m − 1)2 ,

(23.16)

hence the inequalities used to prove Theorem 6.1 must all become equalities. More specifically, (log f )1α = 0 for all 2 ≤ α ≤ m and (log f )αβ = −

δαβ |∇(log f )|2 m−1

= −(m − 1)δαβ for all 2 ≤ α, β ≤ m. On the other hand, since e1 is the unit normal to the level set of log f, the second fundamental form (h αβ ) of the level set is given by (log f )αβ = (log f )1 h αβ = (m − 1)h αβ , hence implying that h αβ = −δαβ . Moreover, (23.16) also implies that if we set t = log( f )/(m − 1), then t must be the distance function between the level sets of f , and hence also

298

Geometric Analysis

for the level set of log f. The fact that h αβ = −δαβ implies that the metric on M can be written as 2 = dt 2 + exp(−2t) ds N2 . ds M

Since M has two ends, N must be compact. A direct computation shows ˜ i j ≥ 0. This proves the that the condition Ri j ≥ −(m − 1) implies that R theorem. 

24 Manifolds with finite volume

In this chapter, we assume that M m is a complete manifold with finite volume. We assume that the Ricci curvature is bounded from below by Ri j ≥ −(m − 1). Since the constant functions are L 2 harmonic functions, this implies that 0 is an eigenvalue for the L 2 -spectrum of the Laplacian. We define the quantity λ1 (M) by the Rayleigh quotient ! λ1 (M) =

inf !

φ∈H 1,2 (M),

M

φ=0

M !

|∇φ|2 , 2 Mφ

where the infimum! is taken over all functions φ in the Sobolev space H 1,2 (M) satisfying M φ = 0. This plays the role of a generalized first nonzero Neumann eigenvalue, although λ1 (M) might not necessarily be an eigenvalue. Note that λ1 (M) ≤ max{μ1 (1 ), μ1 (2 )}, for any two disjoint domains 1 and 2 of M, where μ1 (1 ) and μ1 (2 ) are their first Dirichlet eigenvalues, respectively. In particular, we have λ1 (M) ≤ μe (M), the greatest lower bound for the essential spectrum of M. Therefore, according to Cheng’s theorem [Cg1] (Corollary 6.4), one always has λ1 (M) ≤

(m − 1)2 . 4

The main purpose of this chapter is to prove the following theorem. 299

300

Geometric Analysis

Theorem 24.1 Let M m be a complete Riemannian manifold with Ricci curvature bounded from below by Ri j ≥ −(m − 1). Assume that M has a finite volume given by V (M) and (m − 1)2 . 4 Then there exists a constant C(m) > 0, depending only on m, such that     V (M) V (M) 2 ln N (M) ≤ C(m) +1 , V p (1) V p (1) λ1 (M) ≥

where V p (1) denotes the volume of the unit ball centered at any point p ∈ M. The assumption λ1 (M) ≥ (m − 1)2 /4 implies that 0 is the only eigenvalue below (m − 1)2 /4, so the spectrum of the M satisfies Spec(M) ⊂ {0} ∪ [(m − 1)2 /4, ∞). In the special case when m = 2, our estimate is less effective than using the Cohn–Vossen–Hartman formula (see [LT4]). Indeed, the assumption that K ≥ −1 implies that the negative part of the Gaussian curvature defined by # 0 if K > 0, K− = −K if K ≤ 0 is at most 1. In particular,

M

K − ≤ V (M)

and M has finite total curvature. Hartman’s theorem then implies that M must be conformally equivalent to a compact Riemannian surface of genus g with N (M) punctures. Moreover, since M has finite volume, the Cohn–Vossen– Hartman formula (see [LT4]) asserts that −V (M) ≤ K M

= 2π χ (M). In particular, using the fact that the Euler characteristic is given by χ (M) = 2 − 2g − N (M), we conclude that N (M) ≤ 2 − 2g + (2π )−1 V (M).

24

Manifolds with finite volume

301

This indicates that the dependency on V (M) is better than the one provided by Theorem 24.1, and the value of the theorem lies in the case when m ≥ 3. Note that when M = Hm /  is a hyperbolic manifold, with its universal covering given by the hyperbolic m-space Hm , the L 2 -spectrum of the Laplacian on Hm is the interval [(m − 1)2 /4, ∞). In this special case, the assumption that λ1 (M) ≥ (m − 1)2 /4 can be expressed as λ1 (M) = μ1 (Hm ), where μ1 (Hm ) is the greatest lower bound of the spectrum of  on Hm . With this point of view, it is also possible to prove a theorem analogous to Theorem 24.1 for locally symmetric spaces of finite volume. However, stronger results are available by using Margulis’ thick–thin decomposition. For any finite volume, locally symmetric space M = X/ , where X is a symmetric space and  a discrete subgroup of the isometry group of X, the number of ends of M is always bounded by N (M) ≤ C(X ) V (M), where the constant C depends on X and V (M) is the volume of M. The interested reader should refer to [Ge] for more details. Theorem 24.2 Let M m be a complete Riemannian manifold with Ricci curvature bounded from below by Ri j ≥ −(m − 1). Assume that M has a finite volume given by V (M), and μ1 (M \ B p (ρ0 )) ≥

(m − 1)2 . 4

Then there exists a constant C(m) > 0 depending only on m, such that the number of ends of M satisfies N (M) ≤ C(m) V (M) V p−1 (1) exp((m − 1)ρ0 ), where V p (1) denotes the volume of the unit ball centered at point p ∈ M. Proof According to Theorem 22.5, if we let V p (ρ) be the volume of the geodesic ball B p (ρ), then for all ρ ≥ 2(ρ0 + 1) we have V p (ρ + 2) − V p (ρ) ≤ C (1 + (ρ − ρ0 )−1 ) exp(−(m − 1)(ρ − ρ0 )) × (V p (ρ0 + 1) − V p (ρ0 )).

(24.1)

On the other hand, Proposition 23.1 asserts that if y ∈ ∂ B p (ρ + 1), then Vy (1) ≥ C1−1 exp(−(m − 1)ρ) V p (1).

(24.2)

Obviously, if Nρ (M) denotes the number of ends with respect to B p (ρ), then there exists Nρ (M) number of points {yi ∈ ∂ B p (ρ)} such that B yi (1) ∩

302

Geometric Analysis

B y j (1) = ∅ for i = j. In particular applying (24.2) to each of the yi and combining the result with (24.1), we conclude that Nρ (M) C1−1 exp(−(m − 1)ρ) V p (1) ≤

Nρ (M)



Vyi (1)

i=1

≤ V p (ρ + 2) − V p (ρ) ≤ C (1 + (ρ − ρ0 )−1 ) exp(−(m − 1) × (ρ − ρ0 ))(V p (ρ0 + 1) − V p (ρ0 )). This implies that Nρ (M) ≤ C C1 (1 + (ρ − ρ0 )−1 ) × exp((m − 1)ρ0 ) (V p (ρ0 + 1) − V p (ρ0 )) V p−1 (1). Letting ρ → ∞, we conclude that the number of ends N (M) of M is bounded by N (M) ≤ C C 1 exp((m − 1)ρ0 ) (V p (ρ0 + 1) − V p (ρ0 )) V p−1 (1)

(24.3) 

and the result follows.

We remark that if M has finitely many eigenvalues 0 < μ1 ≤ μ2 ≤ · · · ≤ μk below (m − 1)2 /4, then it is easy to see there exists ρ0 > 0 such that μ1 (M \ B p (ρ0 )) ≥

(m − 1)2 . 4

In particular, Theorem 24.1 implies that M must have finitely many ends. However, the estimate of the number of ends is not effective as it is unclear to us at this moment how to control the size of ρ0 in terms of the eigenvalues below (m − 1)2 /4. On the other hand, we will demonstrate below that ρ0 can be effectively controlled if k = 1. The following lemma allows us to estimate μ1 (B p (ρ)) of a geodesic ball centered at p with radius ρ in terms of the volume of the ball. Note that we do not need to impose any curvature assumptions on M. Lemma 24.3 Let M be a complete Riemannian manifold. Then for any 0 < δ < 1, ρ > 2, and p ∈ M, we have   2   V p (ρ) 12 1 μ1 (B p (ρ)) ≤ 2 + ln ln . V p (1) 1−δ 4δ (ρ − 1)2

24

Manifolds with finite volume

303

Proof For the ease of notation, we will use μ1 to denote μ1 (B p (ρ)). We may assume μ1 ≥ 4ρ −2 as otherwise the conclusion automatically holds true. Let r (x) denote the distance function to a fixed point p ∈ M. The variational characterization of μ1 (B p (ρ)) implies that

 √ φ 2 exp −2δ μ1 r

μ1 M



√  |∇(φ exp −δ μ1 r |2

≤

M



= M

√



√ |∇φ| exp −2δ μ1 r − 2δ μ1 2



 √ φ exp −2δ μ1 r ∇φ, ∇r  M





+ δ 2 μ1

√

φ 2 exp −2δ μ1 r



M

for any nonnegative Lipschitz function φ with support in B p (ρ). In particular, for ρ > 2, if we choose   ⎧ − 12 ⎪ ⎪ ρ − μ , 1 on B ⎪ p 1 ⎪ ⎨   − 12 φ= √ ρ − μ , μ (ρ − r ) on B (ρ)\ B ⎪ 1 p p 1 ⎪ ⎪ ⎪ ⎩ 0 on M \ B p (ρ), then we have

√  (1 − δ 2 ) μ1 exp −2δ μ1 V p (1)

 √ 2 φ 2 exp −2δ μ1 r ≤ (1 − δ ) μ1 M



 √ √ |∇φ|2 exp −2δ μ1 r − 2δ μ1

= M





 √ φ exp −2δ μ1 r ∇φ, ∇r  M

√  ≤ (1 + 2δ) μ1 exp −2δ ρ μ1 − 1 V p (ρ). Therefore, 

√ μ1 (ρ − 1) ≤ exp 2δ

V p (ρ) 3e2 2(1 − δ) V p (1)

304

Geometric Analysis

and 2δ

√ μ1 (ρ − 1) ≤ ln



12 1−δ



 + ln

 V p (ρ) . V p (1)

The lemma follows by rewriting this inequality.



As a corollary to this lemma, one recovers Brook’s theorem asserting that the greatest lower bound of the spectrum of any complete manifold M can be estimated in terms of its volume entropy. This result was proved in Corollary 22.6 with a different argument. Corollary 24.4 Let M be a complete manifold and μ1 (M) be the bottom of the spectrum. Then   ln V p (ρ) 2 1 . lim inf μ1 (M) ≤ 4 ρ→∞ ρ Proof Note that μ1 (M) = limρ→∞ μ1 (B p (ρ)). Now the result follows by first letting ρ go to infinity and then letting δ go to 1 in the estimate of Lemma 24.3.  We are now ready to prove Theorem 24.1. Proof let

Proof of Theorem 24.1 Let p ∈ M be a fixed point. For any 0 < δ < 1, 1 ρ0 = (m − 1)δ

     12 V (M) ln + ln + 1. 1−δ V p (1)

Then according to Lemma 24.3, μ1 (B p (ρ0 )) ≤

(m − 1)2 . 4

(24.4)

We now observe that μ1 (M \ B p (ρ0 )) ≥

(m − 1)2 . 4

Indeed, if (24.4) is valid and also μ1 (M \ B p (ρ0 )) < (m − 1)2 /4, then the variational principle implies that λ1 (M) < (m − 1)2 /4, contradicting our assumption. By Theorem 24.2, we have N (M) ≤ C(m) V (M) V p−1 (1) exp((m − 1)ρ0 ).

24

Manifolds with finite volume

305

If V (M) > eV p (1), then the claimed estimate follows by plugging in the value of ρ0 and setting δ =1−

1 ln(V (M) V p−1 (1))

.

On the other hand, if V (M) ≤ eV p (1), then ρ0 is bounded from above and hence N (M) ≤ C(m) implying the estimate again.



As pointed out earlier, the key ingredients in the proof of Theorem 24.1 rely on the decay estimate of the volume (24.2) given by the upper bound of the greatest lower bound of the spectrum of the model manifold hyperbolic space Hm . Similar theorems for K¨ahler manifolds and quaternionic K¨ahler manifolds also follow by using the corresponding comparison results from [LW8] and [KLZ], respectively.

25 Stability of minimal hypersurfaces in a 3-manifold

In this and the next chapter, we will apply the theory of harmonic functions to the study of complete minimal hypersurfaces in a complete manifold with nonnegative curvature. Let N m+1 be a complete manifold with nonnegative Ricci curvature. − → Suppose M m is a complete minimal hypersurface in N . If | I I |2 denotes the N is the square of the length of the second fundamental form of M and Rνν Ricci curvature of N in the direction of the unit normal ν to M, then M being stable in N is characterized by the stability inequality (1.4) →2 2 − 2 N ψ |I I | + ψ Rνν ≤ |∇ψ|2 (25.1) M

M

M

Hc1,2 (M).

for any compactly supported function ψ ∈ Geometrically, the stability inequality is derived from the second variation formula for the volume functional under normal variations. Hence a stable minimal hypersurface is not only a critical point of the volume functional but its second derivative is nonnegative with respect to any normal variations. The elliptic operator associated with the stability inequality is given by − → N . L =  + | I I |2 + Rνν The stability of M is equivalent to the fact that the operator −L is nonnegative. We say that M has finite index when the operator −L has only finitely many negative eigenvalues. This has the geometric interpretation that there is only a finite dimensional space of normal variations violating the stability inequality. The study of stable minimal hypersurfaces can be viewed as an effort to prove a generalized Bernstein’s theorem. Bernstein first established that an entire minimal graph in R3 must be a plane. Recall that a minimal graph is a minimal hypersurface which is given by a graph of a function defined on R2 . 306

25

Stability of minimal hypersurfaces in a 3-manifold

307

The validity of Bernstein’s theorem in higher dimensions was established for the entire minimal graph in Rm+1 for m ≤ 7 by Simons [S], and many other authors, such as Fleming [Fl], Almgren [A], and De Giorgi [De], for the lower dimensional cases. Counterexamples for m ≥ 8 were found by Bombieri, De Giorgi, and Guisti [BDG]. Since entire minimal graphs are area minimizing, a natural question to ask is if a Bernstein type theorem is valid for stable minimal hypersurfaces in Rm+1 . In 1979, do Carmo and Peng [dCP] proved that a complete, stable, minimally immersed hypersurface M in R3 must be planar. At the same time, Fischer-Colbrie and Schoen [FCS] independently showed that a complete, stable, minimally immersed hypersurface M in a complete three-dimensional manifold N with nonnegative scalar curvature must be either conformally a plane R2 or conformally a cylinder R × S1 . For the special case when N is R3 , they also proved that M must be planar. In 1984, Gulliver [Gu1] studied a yet larger class of submanifolds in R3 . He proved that a complete, oriented, minimally immersed hypersurface with finite index in R3 must have finite total curvature. In particular, applying Huber’s theorem, one concludes that the hypersurface must be conformally equivalent to a compact Riemann surface with finitely many punctures. The same result was also independently proved by Fischer-Colbrie in [FC]. In addition, she also proved that a complete, oriented, minimally immersed hypersurface with finite index in a complete three-dimensional manifold with nonnegative scalar curvature must be conformally equivalent to a compact Riemann surface with finite punctures. Shortly after, Gulliver [Gu2] improved the result of Fischer-Colbrie and proved that if the ambient manifold has nonnegative scalar curvature, then a minimal hypersurface with finite index must have quadratic area growth and finite topological type, and the square of the length of the second fundamental form must be integrable. Indeed, a complete surface with quadratic area growth and finite topological type must be conformally equivalent to a compact Riemann surface with finitely many punctures. In 1997, Cao, Shen, and Zhu [CSZ] considered the high dimensional cases of the theorem of do Carmo–Peng and Fischer-Colbrie–Schoen. They proved that a complete, oriented, stable, minimally immersed hypersurface M n in Rn+1 must have only one end. This theorem was generalized by Li and Wang [LW4] when they showed that a complete, oriented, minimally immersed hypersurface M n in Rn+1 with finite index must have finitely many ends. In a later paper [LW7], Li and Wang also generalized their theorem to minimal hypersurfaces with finite index in a complete manifold with nonnegative sectional curvature. We will present the relationship between harmonic functions and the stability of minimal hypersurfaces in this chapter. Some two-dimensional results

308

Geometric Analysis

will be proved using this technique, while the higher dimensional results will be presented in the next chapter. Note that when the ambient manifold is Euclidean space and n ≥ 3, then M is nonparabolic by applying Corollary 24.2. The key issue is the validity of the Sobolev inequality proved by Michael and Simon [MS] in the form (m−2)/m  2m/(m−2) |u| ≤C |∇u|2 M

M

Rn .

on any minimal submanifolds of However, when N is only assumed to have nonnegative Ricci (or sectional) curvature, then M can be parabolic as in the case of the cylinder M = R × P in N = R2 × P. The next theorem states that the case in which M is parabolic is a very special situation. Theorem 25.1 Let M m be a complete, minimally immersed, stable, hypersurface in a manifold, N m+1 , with nonnegative Ricci curvature. If M is parabolic, then M must be totally geodesic in N . Moreover, the Ricci curvature Rνν of N in the normal direction also vanishes, and M must have nonnegative scalar curvature. Proof According to Lemma 24.3 and the discussion thereafter, the assumption that M is parabolic can be characterized by the following construction. For a fixed p ∈ M, a given ρ > 0, and a sequence ρi > ρ with ρi → ∞, let f i be a sequence of harmonic functions satisfying  f i = 0 on

B p (ρi )\ B p (ρ),

with boundary conditions fi = 1

on

∂ B p (ρ)

f i = 0 on

∂ B p (ρi ).

and

Then the manifold M is parabolic if and only if fi converges uniformly on compact subsets of M \ B p (ρ) to the constant function 1. If this is the case, since ∂ fi ∂ fi 2 |∇ f i | = fi fi − ∂r ∂r B p (ρi )\B p (ρ) ∂ B p (ρi ) ∂ B p (ρ) ∂ fi , =− ∂ B p (ρ) ∂r the fact that f i → 1 implies that the right-hand side must tend to 0 as ρi → ∞.

25

Stability of minimal hypersurfaces in a 3-manifold

309

To prove the proposition, we consider the compactly supported function ⎧ ⎨1 ψ = fi ⎩ 0

on on on

B p (ρ), B p (ρi )\ B p (ρ), M \ B p (ρi ).

Using this as a test function in the stability inequality (25.1), we conclude that − → N | I I |2 + Rνν ≤ |∇ f i |2 . B p (ρ)

B p (ρ)

B p (ρi )\B p (ρ)

Since the right-hand side vanishes as i → ∞, we have − → N | I I |2 + Rνν = 0, B p (ρ)

B p (ρ)

− → N vanish identically on B (ρ). Due to the fact that ρ is hence | I I |2 and Rνν p − → N on M. The nonnegativity arbitrary, this implies the vanishing of I I and Rνν of the scalar curvature of M follows by applying the Gauss curvature equation and using the assumption that N has nonnegative Ricci curvature.  Theorem 25.1 reduces our study of stable minimal hypersurfaces to the nonparabolic case. In this case, we will recall the lemma of Schoen and Yau [SY1]. Lemma 25.2 (Schoen–Yau) Let M m be a complete, minimally immersed, stable, hypersurface in N m+1 . Let R N and K N be the Ricci curvature and the sectional curvature of N , respectively. Suppose u is a harmonic function defined on M. Then the inequality 1 →2 2 2 2 − 2 N |∇φ| |∇u| ≥ φ | I I | |∇u| + φ 2 Rνν |∇u|2 m M M M m  1 2 N 2 + φ K (e1 , eα ) |∇u| + φ 2 |∇|∇u||2 m−1 M M α=1

holds for any compactly supported, nonnegative function φ ∈ Hc1,2 (M). Proof

The Bochner formula of Lemma 23.4 asserts that |∇u|2 ≥ 2Ri j u i u j +

m |∇u|−2 |∇|∇u|2 |2 . 2(m − 1)

(25.2)

310

Geometric Analysis

If {e1 , e2 , . . . , em } is an orthonormal frame of M and (h i j ) is the second fundamental form of M, then the Gauss curvature equation asserts that R11 =

m 

K N (e1 , eα ) +

α=2

=

m 

m 

h 11 h αα −

α=2

m 

h 21α

α=2

K N (e1 , eα ) − h 211 −

α=2

m 

h 21α ,

α=2

where we have used the assumption that M is minimal and K N (e1 , eα ) is the sectional curvature for the 2-plane section spanned by e1 and eα . On the other hand, using the inequality m  − →2 = h i2j |I I | i, j=1

≥ h 211 +

m  α=2

h 2αα + 2

m

α=2 h αα

≥ h 211 +

m h 211 + m−1

h 21α

α=2

2

+2

m−1



≥

m 

m 

m 

h 21α

α=2

h 21α ,

α=2

we conclude that R11 ≥

m 

K N (e1 , eα ) −

α=2

m−1 − → | I I |2 . m

Choosing the orthonormal frame so that ∇u = |∇u| e1 , this implies that Ri j u i u j ≥

m 

K N (e1 , eα ) |∇u|2 −

α=2

m−1 − → | I I |2 |∇u|2 . m

Substituting this into (25.2) yields |∇u| ≥

m  α=2

K N (e1 , eα ) |∇u| −

m−1 − |∇|∇u||2 → | I I |2 |∇u| + . (25.3) m (m − 1)|∇u|

25

Stability of minimal hypersurfaces in a 3-manifold

311

By setting ψ = φ|∇u|, where φ is a nonnegative compactly supported function on M, the stability inequality (25.1) implies that − → N φ 2 | I I |2 |∇u|2 + φ 2 Rνν |∇u|2 M

M



≤



|∇φ| |∇u| + 2 2

φ |∇u| ∇φ, ∇|∇u|

2

M

M



+

φ 2 |∇|∇u||2 M

=



|∇φ|2 |∇u|2 − M

φ 2 |∇u| |∇u|. M

Combining this inequality with (25.3), we conclude that 1 m



− → φ | I I |2 |∇u|2 +



2

M

+

m  α=1 M

≤

M

N φ 2 Rνν |∇u|2

φ 2 K N (e1 , eα ) |∇u|2 +

1 m−1

φ 2 |∇|∇u||2 M

|∇φ|2 |∇u|2 .



M

As a corollary of Theorem 25.1 and Lemma 25.2, we readily recover the theorem of Fischer-Colbrie and Schoen [FCS]. This theorem was also independently proved by do Carmo and Peng [dCP] for the special case when N = R3 . In the case of minimal surface in a three-dimensional manifold, if we let S N be the scalar curvature of N and K be the Gaussian curvature of M, then using the Gauss curvature equation the stability inequality can be written as →2 1 2 − 2 N 2 1 ψ |I I | + 2 ψ S − ψ K ≤ |∇ψ|2 (25.4) 2 M

M

M

M

for any nonnegative, compactly supported function ψ ∈ Hc1,2 (M). Theorem 25.3 (Fischer-Colbrie–Schoen) Let M 2 be an oriented, complete, stable, minimal hypersurface in a complete manifold N 3 with nonnegative scalar curvature. Then M must be conformally equivalent to either the complex plane C or the cylinder R × S1 . If M is conformally equivalent to the cylinder and has finite total curvature, then it must be totally geodesic and the scalar curvature of N along M must be identically 0.

312

Geometric Analysis

Proof Let M˜ be the universal covering of M. By the uniformization theorem, M˜ must be conformally equivalent to either the unit disk D2 or the complex plane C. We claim that M˜ cannot be conformally equivalent to D2 . To see this, we first observe that the stability inequality still holds on M˜ − → ˜ In fact, if ψ is a nonnegative, by lifting the functions | I I |2 and S N to M. ˜ compactly supported function on M and π : M˜ → M is the covering map, then the function ψ¯ defined by  ψ 2 (x) ˜ ψ¯ 2 (x) = x∈π ˜ −1 (x)

is a nonnegative, compactly supported function on M. The stability inequality on M asserts that →2 1 2 − 2 N 1 ψ | I I | + ψ S − ψ2 K 2 2 M˜

=

M˜

1 2

− → ψ¯ 2 | I I |2 + M



M˜

1 2

ψ¯ 2 S N − M



ψ¯ 2 K M

¯ 2. |∇ ψ|

≤ M

On the other hand, the Schwarz inequality implies that  2       ¯ ψ| ¯ 2= ψ∇ψ |ψ∇   x∈π  −1 ˜ (x)   ≤ ψ2 |∇ψ|2 x∈π ˜ −1 (x)

= ψ¯ 2

x∈π ˜ −1 (x)



|∇ψ|2 .

x∈π ˜ −1 (x)

Therefore we conclude that

¯ 2≤ |∇ ψ| M

M˜

|∇ψ|2

˜ If M˜ is conformally equivalent to and the stability inequality is valid on M. 2 D , the invariance of the Laplace operator in dimension 2 asserts that there exist nonconstant, bounded harmonic functions with a finite Dirichlet integral ˜ Moreover, if u is such a harmonic function, then Lemma 7.3 asserts that on M. |∇u| ≥ K |∇u| +

|∇|∇u|| |∇u|

25

Stability of minimal hypersurfaces in a 3-manifold

313

and the proof of Lemma 25.2 applied to the stability inequality (25.4) on M˜ implies that →2 2 − 2 1 1 φ | I I | |∇u| + φ 2 S N |∇u|2 2 2 M˜

+



M˜

φ 2 |∇|∇u||2 ≤

for any nonnegative, compactly supported choosing ⎧ 1 on ⎪ ⎪ ⎨ 2ρ − r on φ= ⎪ ρ ⎪ ⎩ 0 on

M˜

M˜

|∇φ|2 |∇u|2

(25.5)

˜ However, function φ ∈ Hc1,2 ( M). B p (ρ), B p (2ρ), M˜ \ B p (2ρ),

the right-hand side becomes 1 |∇φ|2 |∇u|2 = 2 |∇u|2 . ρ ˜ M B p (2ρ)\B p (ρ) The fact that u has a finite Dirichlet integral implies that this tends to 0 − → as ρ → ∞. Therefore, (25.5) asserts the vanishing of I I and |∇|∇u||. In particular, |∇u| must be identically constant and M˜ has a finite volume because u has a finite Dirichlet integral. This contradicts the assumption and M˜ must be conformally equivalent to C. Using the uniformization theorem again, we conclude that M must be conformally equivalent to either the complex plane C or the cylinder R × S1 . This proves the first part of the theorem. If M is conformally equivalent to R × S1 and has finite total curvature, then applying the proof of Theorem 25.1 to the stability inequality (25.4) we conclude that − →2 1 1 N | I I | + S − K ≤ 0. (25.6) 2 2 M

M

M

In particular, the assumption that N has nonnegative scalar curvature implies that K ≥ 0. M

The Cohn-Vossen inequality [CV] then asserts that 2π χ (M) ≥ K ≥ 0. M

314

Geometric Analysis

! Since M is the cylinder, we conclude that M K = 0 and (25.6) asserts that − →  | I I |2 and S N are both identically 0 on M as claimed. Combining this theorem with Theorem 25.1, we obtained the following corollary, which was also proved in [FCS]. Corollary 25.4 (Fischer-Colbrie–Schoen) Let M 2 be an oriented, complete, stable, minimal hypersurface in a complete manifold N 3 with nonnegative Ricci curvature. Then M must be totally geodesic in N and the Ricci curvature of N in the normal direction to M must be identically zero along M. Moreover, M is either (1) conformally equivalent to the complex plane C; or (2) isometrically the cylinder R × S1 . Moreover, if N = R3 , then M must be planar in R3 .

26 Stability of minimal hypersurfaces in a higher dimensional manifold

Let us now assume that N m+1 is a complete manifold with nonnegative sectional curvature. For a fixed point p ∈ N , let γ : [0, ∞) → N be a normal geodesic ray emanating from p. Recall that the Buseman function, βγ , with respect to γ is defined by βγ (x) = lim (t − d(x, γ (t)). t→∞

We define the Buseman function, β, with respect to the point p by β(x) = sup βγ (x). γ

It was proved in [CG1] that β(x) is a convex exhaustion function on N . Suppose M m is a minimally immersed hypersurface in N m+1 , then a direct computation implies that the restriction of β on M is subharmonic with respect to the induced metric. The following lemma is useful for the construction of harmonic functions on a manifold possessing subharmonic functions with certain properties. Since the manifold is only assumed to be complete, it is likely that there are other applications of this lemma. Lemma 26.1 (Li–Wang) Let M m be a complete manifold, and E be an end of M with respect to B p (1). Suppose g is a subharmonic function defined on E with the property that its maximum is not achieved on ∂ E. Let us define E(ρ) = B p (ρ) ∩ E and s(ρ) = sup∂ B p (ρ)∩E g. For any sequence {ρi } with ρi → ∞, there exists a subsequence, which we also denote by {ρi }, and a sequence of 315

316

Geometric Analysis

positive constants {Ai } such that the sequence of solutions {u i } to the boundary value problem u i = 0

on

E(ρi ),

ui = 0

on

∂ E,

and u i = Ai

on

∂ E(ρi )\∂ E

converges on compact subsets of E to a positive harmonic function u with boundary value u=0

on

∂ E.

Moreover, the sequence { Ai } satisfies the bound 0 < Ai ≤ C s(ρi ) for some constant 0 < C < ∞, and E(ρi )

|∇u i |2 = Ai .

Proof Lemma 20.7 asserts that there exists a sequence ρi → ∞, a sequence of constants Ci → ∞, and a sequence of positive functions f i satisfying  fi = 0 on fi = 0

on

E(ρi ), ∂ E,

and f i = Ci

on

∂ B p (ρi ) ∩ E,

which converges to a positive harmonic function f. Moreover, ∂ fi ∂ fi |∇ f i |2 = fi fi − ∂r ∂r E(ρi ) ∂ B p (ρi )∩E ∂E ∂ fi = Ci ∂ B p (ρi )∩E ∂r ∂ fi = Ci . ∂ E ∂r

26

Minimal hypersurface stability in a higher dimensional manifold

317

Let us define  ui =

−1

∂ fi ∂r

∂E

fi ,

then  u i = Ci

∂E

∂ fi ∂r

−1 on

∂ B p (ρi ) ∩ E.

The sequence u i will also converge to the harmonic function  u=

∂E

∂f ∂r

−1 f.

We now claim ! that there exists a constant 0 < C < ∞ such that the sequence { Ai = Ci ( ∂ E ∂ fi /∂r )−1 } satisfies lim sup i→∞

Ai = C. s(ρi )

Indeed, the maximum principle asserts that Ai (g(x) − s(1)) ≤ u i (x) s(ρi )

(26.1)

on E(ρi ) since this inequality is satisfied on ∂ E(ρi ). The assumption that g does not achieve its maximum on ∂ E and the maximum principle imply that s(ρ) > s(1) for some ρ > 1. For this particular ρ, let x ∈ ∂ E(ρ)\∂ E such that g(x) = s(ρ). Applying (26.1) and letting i → ∞, we conclude that lim sup i→∞

Ai u(x) ≤ , s(ρi ) s(ρ) − s(1) < ∞,

confirming the existence of C. Integrating by parts yields |∇u i |2 = E(ρi )

∂ E(ρi )\∂ E

ui



= Ai

∂ E(ρi )\∂ E

∂u i ∂r ∂u i . ∂r

318

Geometric Analysis

On the other hand, since



0= = we conclude that

E(ρi )

u i

∂ E(ρi )\∂ E

∂u i − ∂r

∂E



E(ρi )

|∇u i |2 = Ai = Ai .

∂E

∂u i , ∂r ∂u i ∂r 

Theorem 26.2 (Li–Wang) Let M m be a complete, stable, minimally immersed hypersurface in N m+1 . Suppose N is a complete manifold with nonnegative sectional curvature. If M is parabolic, then it must be totally geodesic and have nonnegative sectional curvature. In particular, M either has only one end, or M = R × P with the product metric, where P is compact with nonnegative sectional curvature. If M is nonparabolic, then it must only have one nonparabolic end. In this case, any parabolic end of M must be contained in a bounded subset of N . Proof If M is parabolic, Theorem 25.1 implies that M must be totally geodesic and the Ricci curvature of N in the normal direction must vanish along M. In particular, the Gauss curvature equation asserts that M has nonnegative sectional curvature. Moreover, if M has more than one end, then Theorem 4.4 implies that M must be isometrically R × P for some compact manifold P with nonnegative sectional curvature. From this point on, we may assume that M is nonparabolic and has at least two ends. Suppose E and F are two ends of M with respect to the compact set B p (ρ0 ). The nonparabolicity of M implies that at least one of the ends is nonparabolic, hence we may assume E is a nonparabolic end. Let us assume either that F is nonparabolic, or that the Buseman function β when restricted to F is unbounded, hence satisfying the hypothesis of Lemma 26.1. Note that if M is properly immersed in N , then β is unbounded on each end of M. Using the Buseman function β in the role of g in Lemma 26.1, we observe that because |∇β| ≤ 1, s(ρ) has at most linear growth. In particular, there exists a sequence of harmonic functions {u i } defined on F(ρi ) that converges to a positive harmonic

26

Minimal hypersurface stability in a higher dimensional manifold

319

function u defined on F. Moreover, they satisfy ui = 0 and



∂F

on

∂u i = 1. ∂r

∂F

Similarly, because E is assumed to be nonparabolic, by passing to a subsequence if necessary, there exists a sequence of harmonic functions {vi } defined on E(ρi ) that converges to a bounded harmonic function v defined on E. The sequence also satisfies vi = 0 on and

∂E

∂vi = ∂r

∂E

E(ρi )

|∇vi |2 = 1.

Let us define a sequence of functions ki on B p (ρi ) by ⎧ on E(ρi ), ⎨ vi (x) ki (x) = −u i (x) on F(ρi ), ⎩ 0 on B p (ρi )\(E(ρi ) ∪ F(ρi )). Obviously, ki is harmonic on E(ρi ) ∪ F(ρi ) and ∂ki ∂vi ∂u i = − B p (1) ∂r ∂ E ∂r ∂ F ∂r = 0. Let wi be the solution to the boundary value problem wi = 0

B p (ρi )

on

and wi = ki

on

∂ B p (ρi ).

The fact that wi minimizes the Dirichlet integral implies that 2 |∇wi | ≤ |∇ki |2 B p (ρi )



B p (ρi )

=



|∇vi | +

|∇u i |2

2

Ei

≤ 2C ρi .

Fi

320

Geometric Analysis

On the other hand, according to Theorem function ⎧ on ⎨ v(x) k(x) = −u(x) on ⎩ 0 on

21.2, since ki converges to the E, F, M \(E ∪ F),

there exists a constant C1 > 0 independent of i such that a subsequence of {wi } converges to a harmonic function w on M satisfying |w − k| ≤ C 1 . Applying Lemma 25.2, we have − → φ 2 |∇wi |2 | I I |2 + m φ 2 |∇|∇wi ||2 (m − 1) M

M



≤ m(m − 1)

|∇φ|2 |∇wi |2

(26.2)

M

for any nonnegative function φ ∈ Hc1,2 (M) supported on B p (ρi ). In particular, for any fixed ρ0 < ρ < ρi , if we set ⎧ ⎨1 on B p (ρ), φ(x) = ρi − r (x) on B p (ρi )\ B p (ρ), ⎩ ρi − ρ then we have



(m − 1)

B p (ρ)

− → |∇wi |2 | I I |2 + m



B p (ρ)

− → φ 2 |∇wi |2 | I I |2 + m

≤ (m − 1)

|∇|∇wi ||2



M

φ 2 |∇|∇wi ||2 M



≤ m(m − 1)

|∇φ|2 |∇wi |2 M

m(m − 1) ≤ (ρi − ρ)2 ≤



B p (ρi )

|∇wi |2

C ρi . (ρi − ρ)2

(26.3)

Letting i → ∞, this implies that B p (ρ)

− → |∇w|2 | I I |2 = 0

26

Minimal hypersurface stability in a higher dimensional manifold

321

and B p (ρ)

|∇|∇w||2 = 0.

Since ρ is arbitrary, we conclude that |∇w| is constant. Because w is a − → nonconstant harmonic function, this means that |∇w| = 0, hence | I I |2 must be identically 0 and M is a totally geodesic submanifold. The Gauss curvature equation then asserts that M must have nonnegative sectional curvature also. The assumption that M has at least two ends and Theorem 4.4 together imply that M must split isometrically into R × P, where P must be a compact manifold with nonnegative sectional curvature. However, this contradicts the assumption that M is nonparabolic, hence M must have only one end.  Corollary 26.3 Let M m be a complete, properly immersed, stable, minimal hypersurface in N m+1 . Suppose N is a complete manifold with nonnegative sectional curvature. Then either (1) M has only one end; or (2) M = R × P with the product metric, where P is compact with nonnegative sectional curvature, and M is totally geodesic in N . We will now consider the case when the minimal hypersurface has finite index. The following lemma proved by Fischer-Colbrie [FC] gives an effective way to use the finite index assumption. Lemma 26.4 (Fischer-Colbrie) Let M be a complete manifold and V be a locally bounded function defined on M. Suppose the spectrum, Spec(−L), of the elliptic operator L =  + V (x) on the negative axis is given by only finitely many negative eigenvalues, then there exists a compact set D such that −L is nonnegative on M \ D. Proof Let  ⊂ M be any compact subset of M. The variational principle asserts that the lowest eigenvalue of −L is given by ! μ1 (L , ) = inf

! −  V φ2 , 2 φ

2  |∇φ|!

322

Geometric Analysis

where the infimum is taken over all compactly supported φ ∈ Hc1,2 (). In particular, since |∇φ|2 − V φ 2 ≥ μ1 () φ 2 − sup V φ2, 









where μ1 () is the first Dirichlet eigenvalue of the Laplacian on , we conclude that μ1 (L , ) ≥ μ1 () − sup V. 

(26.4)

For a fixed p ∈ M, let us consider the geodesic ball B p (ρ). Inequality (26.4) asserts that μ1 (L , B p (ρ)) ≥ 0 for ρ sufficiently small because μ1 (B p (ρ)) → ∞ as ρ → ∞. Let us define ρ1 = 2 sup{ρ | μ1 (L , B p (ρ)) ≥ 0}. If ρ1 = ∞, then −L is nonnegative on M and the lemma is proved. Hence we may assume that ρ1 < ∞ and define ρ2 = 2 sup{ρ | μ1 (L , B p (ρ)\ B p (ρ1 )) ≥ 0}. Again, if ρ2 = ∞, then −L is nonnegative on M \ B p (ρ1 ). Inductively we can define ρi = 2 sup{ρ | μ1 (L , B p (ρ)\ B p (ρi−1 )) ≥ 0}. We claim that some ρi must be infinite, hence −L is nonnegative on M \ B p (ρi−1 ). If not, we get a infinite sequence of ρi → ∞. Moreover the definition of ρi and the monotonicity of μ1 imply that −L is negative on all the annuli B p (ρi )\ B p (ρi−1 ). Let f i be the eigenfunction on B p (ρi )\ B p (ρi−1 ) such that L f i = −μ1 (L , B p (ρi )\ B p (ρi−1 )) f i , where μ1 (L , B p (ρi )\ B p (ρi−1 )) < 0. Since the set of { fi } has disjoint support, they span an infinite-dimensional subspace of Hc1,2 (M) such that −L is negative. This contradicts the assumption that −L has only finitely many negative eigenvalues and the lemma is proved.  Theorem 26.5 (Li–Wang) Let M m be a complete, minimally immersed hypersurface with finite index in N m+1 . Suppose N is a complete manifold with nonnegative sectional curvature. Let N (M) be the number of ends of M that are either nonparabolic, or parabolic but not contained in any bounded subset

26

Minimal hypersurface stability in a higher dimensional manifold

323

of N , then there exists a constant C > 0 depending on a compact set of the M such that N (M) ≤ C. Proof Lemma 26.4 asserts that the assumption that M has finite index implies that there exists a compact subset D such that M \ D is stable. In particular, the stability inequality (25.1) holds for any compactly supported function ψ defined on M \ D. Following the argument in the proof of Theorem 26.2, for each pair of such ends E and F, the sequence of harmonic functions {wi } converges to a harmonic function w on M satisfying (26.2) with φ being a compact support function on M \ D. Moreover, Theorem 23.4 asserts that the space of harmonic functions K = {w} constructed above will have dimension equal to the number of ends. Hence, it suffices to estimate the dimension of K. Note that for any w ∈ K, we may assume that it can be approximated by a sequence of harmonic functions wi whose Dirichlet integral has at most linear growth, as indicated by Lemma 26.1. Let ρ0 be sufficiently large such that D ⊂ B p (ρ0 ) and set φ to be

φ(x) =

⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ r (x) − ρ0 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ρi − r (x) ρi − ρ0 − 2

on

B p (ρ0 ),

on

B p (ρ0 + 1)\ B p (ρ0 ),

on

B p (ρ0 + 2)\ B p (ρ0 + 1),

on

B p (ρi )\ B p (ρ0 + 2)

for ρ0 + 2 < ρi . Following the argument for (26.3), we conclude that

− → φ |∇wi | | I I |2 + m 2

(m − 1) M

φ 2 |∇|∇wi ||2 M



≤ m(m − 1)

|∇φ|2 |∇wi |2

≤ m(m − 1)



2

M

B p (ρ0 +1)\B p (ρ0 )

m(m − 1) + (ρi − ρ0 − 2)2 ≤ m(m − 1)

|∇wi |2

B p (ρi )\B p (ρ)

B p (ρ0 +1)\B p (ρ0 )

|∇wi |2

|∇wi |2 +

C ρi . (ρi − ρ0 − 2)2

324

Geometric Analysis

Letting i → ∞, we obtain the estimate →2 2 2 − φ |∇w| | I I | + m φ 2 |∇|∇w||2 (m − 1) M

M



≤ m(m − 1)

|∇w|2 .

B p (ρ0 +1)\B p (ρ0 )

(26.5)

On the other hand, the Poincar´e inequality for mixed boundary conditions asserts that there is a constant C > 0 depending on the set B p (ρ0 + 2)\ B p (ρ0 ) such that 2 f ≤ |∇ f |2 C B p (ρ0 +2)\B p (ρ0 )

B p (ρ0 +2)\B p (ρ0 )

for any f ∈ H 1,2 (B p (ρ0 + 2)\ B p (ρ0 )) with the boundary condition f = 0 on

∂ B p (ρ0 ).

Applying this to the function φ |∇w|, we conclude that φ 2 |∇w|2 C B p (ρ0 +2)\B p (ρ0 )



≤

B p (ρ0 +2)\B p (ρ0 )

|∇(φ |∇w|)|2



≤2 ≤2



B p (ρ0 +1)\B p (ρ0 )

B p (ρ0 +1)\B p (ρ0 )

|∇φ|2 |∇w|2 + 2 |∇w|2 + 2

B p (ρ0 +2)\B p (ρ0 )

B p (ρ0 +2)\B p (ρ0 )

φ 2 |∇|∇w||2

φ 2 |∇|∇w||2 .

Combining this inequality with (26.5), we have C |∇w|2 B p (ρ0 +2)\B p (ρ0 +1)



≤ C1 ≤

B p (ρ0 +2)\B p (ρ0 )

B p (ρ0 +1)

φ 2 |∇w|2

|∇w|2 .

Since the function |∇w| satisfies the differential inequality − → |∇w| ≥ −| I I |2 |∇w|,

(26.6)

26

Minimal hypersurface stability in a higher dimensional manifold

325

the local mean value inequality (Corollary 14.8) asserts that there exists a constant C > 0 depending on the set B p (ρ0 + 2), such that if x ∈ B p (ρ0 + 1), then 2 |∇w|2 . |∇w| (x) ≤ C Bx (1)

Together with (26.5), this implies sup

B p (ρ0 +1)



|∇w| ≤ C 2

B p (ρ0 +1)

|∇w|2 .

The dimension estimate on K then follows from Lemma 7.3 and the same argument as in the proof of Theorem 7.4.  We would like to remark that the properness assumption of M in N in Corollary 26.3 is only used to ensure that the Buseman functions satisfy the hypothesis of Lemma 26.1 on a parabolic end. In the event that the end is not properly immersed, it is still possible that one can find a Buseman function satisfying the property that its maximum is not achieved on ∂ E. For example, if the end E is not contained in a compact set of N , then the Buseman function is not be bounded on E and the hypothesis of Lemma 26.1 is met. If this is the case, the proof of Corollary 26.3 is still valid for that end. When N = Rm+1 , we can also recover the theorems in [CSZ] and [LW4]. Corollary 26.6 (Cao–Shen–Zhu) Let M m be a minimal hypersurface in Rm+1 with m ≥ 3. If M is stable, then M must have only one end. Proof According to Corollary 26.3 and the above remark, we only need to rule out the case when M has a parabolic end. Indeed, by a Sobolev inequality of Michael and Simon [MS] and Corollary 20.12, any end E must be nonparablic, and the corollary follows.  The above argument also proved the finite index case. Corollary 26.7 (Li–Wang) Let M m be a minimally immersed hypersurface in Rm+1 with m ≥ 3. If M has finite index, then there exists a constant C > 0 depending on M such that N (M) ≤ C.

27 Linear growth harmonic functions

In this and the next two chapters, we will consider polynomial growth harmonic functions on a complete manifold. Recall that any polynomial growth harmonic function in Rm is necessarily a polynomial with respect to the variables in rectangular coordinates. Hence the space of polynomial growth harmonic functions of at most order d is given by the space of harmonic polynomials of degree at most d. In particular, these spaces are all finite dimensional (see Appendix B). Definition 27.1 Let Hd (M) be the vector space of all polynomial growth harmonic functions defined on M of order at most d, i.e., Hd (M) = { f |  f = 0, and | f (x)| ≤ C r d (x) for some constant C > 0}, where r (x) is the distance from x to a fixed point p ∈ M. We also denote the dimension of the vector space Hd (M) by h d (M). Using this notation, one computes (see Appendix B) that



m+d −1 m+d −2 d m + . h (R ) = d d −1 On the other hand, Cheng’s theorem (Corollary 6.6) asserts that if M has nonnegative Ricci curvature, then h d (M) = 1 for all d < 1. In view of this, Yau conjectured that h d (M) must be finite dimensional if M has nonnegative Ricci curvature. In 1989, Li and Tam proved that h 1 (M) ≤ m + 1 326

27

Linear growth harmonic functions

327

if M has nonnegative Ricci curvature. Note that this estimate is sharp and it is achieved by Rm . In fact, Li [L7] showed that if M is K¨ahler and has nonnegative holomorphic bisectional curvature, then equality holds if and only if M = Cn with 2n = m. The real case of this theorem was later proved by Cheeger, Colding, and Minicozzi [CCM]. They showed that if M has nonnegative Ricci curvature and h 1 (M) = m + 1, then M = Rm . We will first present the theorem of Li and Tam [LT3], which is a finer version of the statement mentioned above. The proof relies on Lemma 16.4, which can be viewed as a sharp mean value inequality at infinity Theorem 27.2 (Li–Tam) Let M be a complete manifold with nonnegative Ricci curvature. Suppose the volume growth of M satisfies lim sup ρ −n V p (ρ) < ∞ ρ→∞

for some n ≤ m, then h 1 (M) ≤ n + 1 ≤ m + 1. Proof For any f ∈ H1 (M), the curvature assumption, Lemma 3.4, and Corollary 6.6 imply that |∇ f |2 is a bounded subharmonic function on M. In particular, Lemma 16.4 asserts that |∇ f |2 = sup |∇ f |2 . (27.1) lim V p−1 (ρ) ρ→∞

B p (ρ)

M

For a fixed point p ∈ M, let us define the subspace of H1 (M) given by H = { f | f ( p) = 0, f ∈ H1 (M)}, and a bilinear form D¯ on H by ¯ f, g) = lim V p−1 (ρ) D( ∇ f, ∇g. ρ→∞

B p (ρ)

Observe that the fact that f − f ( p) ∈ H for any f ∈ H1 (M) implies that the codimension of H ⊂ H1 (M) is 1. Note that D¯ is an inner product on H because of (27.1). Given any finite dimensional subspace H of H with dim H = k, let { f 1 , . . . , f k } be an orthonormal basis of H with respect to this inner product. Note that if we define the function F(x) by F 2 (x) =

k  i=1

fi2 (x),

328

Geometric Analysis

then it is independent of orthogonal transformation on the set { fi }. Similarly to the argument in Lemma 7.3, for each x ∈ M, we may assume that f i (x) = 0 for all i = 1. At this point, we see that F 2 (x) = f 12 (x) and F(x) ∇ F(x) = f 1 (x) ∇ f 1 (x). Hence |∇ F|(x) ≤ |∇ f 1 |(x) ≤ 1

(27.2)

because sup M |∇ f 1 |2 = 1 due to (27.1). Also, integrating along a geodesic and using the fact that F( p) = 0, we have F(x) ≤ r (x).

(27.3)

Applying (27.2) to the integral 2

k  i=1

B p (ρ)

|∇ f i |2 =

B p (ρ)

(F 2 )



=2 ≤2

∂ B p (ρ)

∂ B p (ρ)

F

∂F ∂r

F

(27.4)

and combining this with (27.3), we obtain V p−1 (ρ)

k  i=1

B p (ρ)

|∇ f i |2 ≤

ρ A p (ρ) . V p (ρ)

(27.5)

The fact that { fi } are orthonormal with respect to the inner product D¯ implies that for any  > 0, there exists ρ sufficiently large such that k −  ≤ V p−1 (ρ)

k  i=1

B p (ρ)

|∇ f i |2 .

Hence the above inequality together with (27.5) yield A p (ρ) k− ≤ ρ V p (ρ)

27

Linear growth harmonic functions

329

for ρ ≥ ρ0 with ρ0 sufficiently large. Integrating this inequality from ρ0 to ρ gives  k− V p (ρ) ρ ≤ . ρ0 V p (ρ0 ) This implies that k −  ≤ n, and the theorem follows since  is arbitrary.



We are now ready to consider the equality case when h 1 (M) = m + 1 for the above theorem. First we need the following lemmas. Lemma 27.3 Let M m be a complete manifold with nonnegative Ricci curvature. If h 1 (M) = m + 1, then the function F(x) =

m 

1/2 fi2 (x)

,

i=1

defined in the proof of Theorem 27.2, must satisfy lim

x→∞

Proof

F(x) = 1. r (x)

Inequality (27.7) asserts that m  |∇ f i |2 ≤ V p−1 (ρ) B p (ρ) i=1

ρ V p (ρ)

∂ B p (ρ)

F . ρ

(27.6)

On the other hand, since Ri j ≥ 0, the Laplacian comparison theorem asserts that r 2 ≤ 2m, hence r 2 2mV p (ρ) ≥ B p (ρ)

= 2ρ A p (ρ). Combining this estimate with (27.6), we obtain m  −1 2 −1 |∇ f i | ≤ m A p (ρ) V p (ρ) B p (ρ) i=1

Letting ρ → ∞ and using (27.3), we conclude that F −1 = 1. lim A (ρ) ρ→∞ p ∂ B p (ρ) ρ

∂ B p (ρ)

F . ρ

(27.7)

330

Geometric Analysis

For any given δ > 0, let m δ (ρ) be the (m − 1)-dimensional measure of the set  $ #  F(x)  0 and a sequence of {xi } with xi → ∞ such that F(xi ) r −1 (xi ) ≤ 1 − α. For any y ∈ Bxi (α r (xi )/4), inequality (27.2) asserts that F(y) r −1 (y) ≤ F(xi ) r −1 (y) + r (xi , y) r −1 (y) α r (xi ) r −1 (y), 4

≤ (1 − α)r (xi ) r −1 (y) +

where r (xi , y) is the distance between xi and y. However, since r (y) ≥ (1 − α/4)r (xi ), this implies that F(y) r −1 (y) ≤ 1 −

2α . 4−α

Setting δ = 2α/(4 − α) and using (27.8), we see that for any  > 0, by taking i sufficiently large, we have Vxi (α r (xi )/4) ≤

r (xi )+α r (x i )/4

r (xi )−α r (xi )/4



≤

m δ (r ) dr

r (xi )+α r (xi )/4

r (xi )−α r (xi )/4

A p (r ) dr

  α r (xi ) ≤  V p r (xi ) + . 4 On the other hand, the volume comparison theorem asserts that     α r (xi ) α n Vxi Vxi (3r (x i )) ≥ 4 12  α n ≥ V p (2r (x i )). 12

(27.9)

27

Linear growth harmonic functions

331

Combining the volume estimate with (27.9), we conclude that  α n ≤ . 12 Since  is arbitrary, this give a contradiction, and the lemma is proved.



Using the orthonormal basis { f 1 , . . . , f m } of H = { f ∈ H1 (M) | f ( p) = 0}, we define the map  : M → Rm by (x) = ( f 1 , . . . , f m ). Lemma 27.4 Suppose X : B p (ρ) → Rm is a function defined on B p (ρ) with |X (x)| = 1 for all x ∈ B p (ρ). For any subset A ⊂ B p (ρ) and any θ > 0, let us define the set C θ,A = {v ∈ S(A) | |d(v) − X (π(v))| < θ }, where S(A) = {v ∈ Tq (M) | q ∈ A, |v| = 1} is the unit sphere bundle over A. Then for any 0 <  ≤ θ/4, we can take ρ sufficiently large, such that   V¯ (θ − 2) V (A) V (C θ,A ) ≥ − , V (S(B p (ρ)) V p (ρ) V (Sm−1 ) where V¯ (θ ) denotes the (m − 1)-dimensional measure of the set given by {v ∈ Sm−1 | |v − v0 | ≤ θ } for some fixed vector v0 ∈ Sm−1 . Proof

For any  > 0, let us define the set

D = {y ∈ B p (ρ) | |v, w − d(v), d(w)| < , for all v, w ∈ Sy (M)}. The fact that { f i } form an orthonormal basis with respect to the inner product D¯ ρ implies that lim V p−1 (ρ) |∇ fi , ∇ f j  − δi j | = 0 ρ→∞

B p (ρ)

for all 1 ≤ i, j ≤ m, hence V (D) ≥1− V p (ρ)

332

Geometric Analysis

for ρ sufficiently large. This implies that V (A ∩ D) ≥ V (A) −  V p (ρ). For any fixed y ∈ A ∩ D, and an orthonormal basis {ei } of Ty (M), the set {d(ei )} forms a basis for d(Ty (M)) if  is sufficiently small. This means m 2 m with i=1 ai = 1 such that that there exists a set of numbers {ai }i=1   m      ai d(ei ) < .  X (y) −   i=1

Clearly, the set defined by   3 . m      ai ei  < θ − 2, with y ∈ D ∩ A P = v ∈ S y (M)  v −   i=1

satisfies P ⊂ Cθ,A . Therefore, V (Cθ,A ) ≥ V (P) = V¯ (θ − 2) V (D ∩ A) ≥ V¯ (θ − 2)(V (A) − V p (ρ)), 

and this implies the lemma. We are now ready to consider the equality case in Theorem 27.2.

Theorem 27.5 (Cheeger–Colding–Minicozzi) Let M m be a complete manifold with nonnegative Ricci curvature. If h 1 (M) = m + 1, then M = Rm . Proof According to the Bishop volume comparison theorem, to show that M = Rm it suffices to show that for any  > 0 the volume of geodesic balls satisfies V p (ρ) ≥ (1 − ) ωm ρ m ,

27

Linear growth harmonic functions

333

for all ρ sufficiently large. Indeed, this implies that lim ρ −m V p (ρ) = ωm

ρ→∞

and the equality case of the volume comparison implies that M = Rm . For any  > 0, let us define A to be a subset of Rm by A = B0 ((1 − )ρ)\(B p (ρ)). Let us choose a set of open balls B1 = {Bxi (ρi )} using the following procedure. For any x ∈ A, let ρx be the supremum of those values r such that Bx (r ) ⊂ A. In particular, ∂ Bx (ρx ) ∩ ∂ A = ∅. If ∂ Bx (ρx ) ∩ (B p (ρ)) = ∅, then we say that Bx (ρx ) ∈ B1 . If ∂ Bx (ρx ) ∩ (B p (ρ)) = ∅, then there must be a point y ∈ ∂ Bx (ρx ) ∩ B0 ((1 − )ρ) and x must be contained in the line segment γ (t) joining y to 0. Suppose x = γ (t0 ) for some 0 < t0 < (1 − )ρ. Let us consider the balls Bγ (t) (t) centered at γ (t) of radius t for t ≥ t0 . It is clear that y ∈ ∂ Bγ (t) (t) and x ∈ Bγ (t) (t). Let t1 be the first t > t0 such that Bγ (t1 ) (t1 ) ∩ (B p (ρ)) = ∅. Note that t1 < (1 − )ρ/2 must exist since 0 = ( p) and Bγ (t1 ) (t1 ) ⊂ B0 ((1 − )ρ). In particular, Bγ (t1 ) (t1 ) ⊂ A, and we include Bγ (t1 ) (t1 ) in our collection B1 . Clearly, B1 covers A and ρx ≤ (1 − )ρ/2 for all Bx (ρx ) ∈ B1 . We will choose the set {Bxi (ρi )} among the collection B1 inductively. Let Bx1 (ρ1 ) = Bx1 (ρx1 ) be chosen such that ρx1 ≥ 12 supx∈A ρx . For a fixed α > 1, we define B2 = {Bx (ρx ) ∈ B1 | Bx (α ρx ) ∩ Bx1 (α ρ1 ) = ∅}. Let us choose Bx2 (ρ2 ) = Bx2 (ρx2 ) such that ρx2 ≥ Bi+1

sup Bx (ρx )∈B2 ρx . Inductively, we define ⎫ ⎛ ⎞  i ⎬   = Bx (ρx ) ∈ B1  Bx (α ρx ) ∩ ⎝ Bx j (α ρ j )⎠ = ∅ . ⎩ ⎭ 1 2

⎧ ⎨

j=1

If Bi+1 = ∅, then {Bx j (ρ j )}ij=1 is our collection. If Bi+1 = ∅, then we choose Bxi+1 (ρi+1 ) = Bxi+1 (ρxi+1 ) such that ρxi+1 ≥ 12 sup Bx (ρx )∈Bi+1 ρx . In any case, we will get a collection B = {Bx j (ρ j )} whether it is finite or not. This collection will have the properties that Bxi (Ri ) ∩ (B p (ρ)) = ∅,

(27.10)

334

Geometric Analysis

and Bxi (α ρi ) ∩ Bx j (α ρ j ) = ∅

for i = j.

(27.11)

We now claim that A ⊂ ∪B Bx (5α ρx ).

(27.12)

To see this, suppose y ∈ A\∪B Bx (5α ρx ), then there exists Bx (ρx ) ∈ B1 such that y ∈ Bx (ρx ) and Bx (α ρx ) ∩ (∪B Bx (α ρx )) = ∅, otherwise the inductive procedure will continue. Let i be the smallest such number so that Bx (α ρx ) ∩ Bxi (α ρi ) = ∅. This implies that Bx (ρx ) ∈ Bi and hence ρi ≥ 12 ρx . Therefore, we conclude that y ∈ Bxi (5α ρi ), which is a contradiction, and hence (27.12) is valid. Let D be any relatively compact subdomain D ⊂ A such that 2V (D) ≥ V (A). Since D is compact, there exists a finite subcover {Bxi (ρi )} of the cover B such that D ⊂ ∪i Bxi (5α ρi ). In particular, this implies that V (D) ≤



Vxi (5α ρi )

i

= (5α)m



Vxi (ρi ).

i

Hence the finite collection {Bxi (ρi )} not only satisfies (27.10) and (27.11), it also satisfies  V (A) ≤ 2(5α)m Vxi (ρi ). (27.13) i

27

Linear growth harmonic functions

335

Let yi ∈ ∂ Bxi (ρi ) ∩ (B p (ρ)) and pi ∈ B p (ρ) such that ( pi ) = yi . Since |∇ f i |2 ≤ 1, we have |d(v)| ≤

√

m|v|

for any tangent vector v. Hence, we have  √    −1 ρi ⊂ B yi (2−1 ρi )  B pi 2 m ⊂ Bxi ((2−1 + 1)ρi ). For any θ > 0, let us define  #   ρi Cθ,i = v ∈ S B pi √ 2 m

(27.14)

 $    d(v) + ∇r x ((π(v))) < θ , i 

where r xi is the distance function to xi , and Cθ = ∪i Cθ,i . Note that by taking α = 32 , (27.14) and (27.11) imply that √ √ (B pi ((2 m)−1 ρi ) ∩ (B p j ((2 m)−1 ρ j ) = ∅ and Cθ,i ∩ Cθ, j = ∅ √ √ for i = j. Also B pi (ρi /2 m) ⊂ B p ((1 + (4 m)−1 ) ρ) since ρi ≤ ρ/2, and after applying Lemma 27.4, we conclude that V (Cθ ) √ V (S(B p ((1 + (4 m)−1 ) ρ)))

  √ −1 ρi V¯ (θ/2)  −1 V pi V p ((1 + (4 m) ) ρ) −  . (27.15) ≥ √ V (Sm−1 ) 2 m i

On the other hand, the volume comparison theorem asserts that     √ √ ρi ρi V p−1 ((1 + (4 m)−1 )ρ) ≥ V pi V p−1 √ √ V pi ((2 + (4 m)−1 ) ρ) i 2 m 2 m m  ρi . ≥ √ (4 m + 2−1 ) ρ

336

Geometric Analysis

Therefore combining the above inequality with (27.15), we have  m   V¯ (θ/2) V (Cθ ) i ρi −  . ≥ √ −1 √ V (Sm−1 ) ((4 m + 2−1 )ρ)m V (S(B p ((1 + (4 m) ) ρ)) However, (27.13) asserts that  m  15 ωm ρim ≥ V (A), 2 2 i

hence   V (Cθ ) V (A) V¯ (θ/2) C −  ≥ √ −1 1 V (Sm−1 ) ρm V (S(B p ((1 + (4 m) ) ρ)))

(27.16)

for some constant C1 > 0 depending only on m. √ Let v ∈ S(B pi (ρi /2 m)) be a unit tangent vector. We let γv be the normal geodesic emanating in the direction of v, i.e., γv (0) = v. If γv (ρi ) ∈ B p (ρ), / A. If γv (ρi ) ∈ / B p (ρ), then because of Lemma 27.3, for ρ then (γv (ρi )) ∈ sufficiently large, ρ(γv (ρi )) > (1 − )r (γv (ρi )), hence (γv (ρi )) ∈ / B0 ((1 − )ρ) and (γv (ρi )) ∈ / A √ also. Note that since γv (0) ∈ B pi (ρi /2 m) and (27.14) imply that (γv (0)) ∈ Bxi (3ρi /2)\ Bxi (ρi /2), we conclude that ρ  i . (γv (0)) − ρi ∇r xi ((γv (0)) ∈ Bxi 2 / A, we have Taking this together with the fact that (γv (ρi )) ∈   −1 1   2 < − ∇r xi ((π(v))) − ρi ((γv (ρi ) − (γv (0))) √ for all v ∈ S(B pi (ρi /2 m)). Thus for θ ≤ we obtain 1 4

1 4

√ and v ∈ Cθ ∩ S(B pi (ρi /2 m)),

< |d(v) − ρi−1 ((γv (ρi )) − (γv (0))|.

27 Hence we have  V (C θ ) ≤ 4 i

Linear growth harmonic functions



337

−1  |d(v) − ρi ((γv (ρi )) − (γv (0))|.

√

C θ ∩S B pi (ρi /2 m )

Combining this with (27.16), we obtain the estimate   1 V¯ (θ/2) V (A) C1 m −  ≤ 4V −1 (S(B p ((1 + 2m − 2 ) ρ))) m−1 ρ V (S )  × √ C θ ∩S(B pi (ρi /2 m))

i

× |d(v) − ρi−1 ((γv (ρi )) − (γv (0))|. (27.17) We now claim that for any  > 0, we can choose ρ sufficiently large such that the right-hand side of (27.17) is at most . To see this, we consider |d(v) − ρi−1 ((γv (ρi )) − (γv (0))|   ρi   d −1  ((γv (t))) dt  = d(v) − ρi 0 dt   ρi  −1

    d(v) − d γv (t) dt  = ρi 0

  ρ t   i

  d

−1  = ρi  d γv (s) ds dt  0 ds 0 ρi ρi

 |Hess() γv (s), γv (s) | ds dt ≤ ρi−1 0



ρi

≤ 0

0

 |Hess() γv (s), γv (s) | ds.

Therefore, we have 1

V −1 (S(B p ((1 + 2m − 2 ) ρ)))

 i

C θ ∩S(B pi (ρi ))

× |d(v) − ρi−1 ((γv (ρi )) − (γv (0))| −1 −1 − 12 ≤ 2 ρ V (B p ((1 + 2m ) ρ))

B p ((1+2m

− 12

On the other hand, the Bochner formula implies that |∇ f |2 ≥ 2 f i2j

) ρ)

|Hess()|.

(27.18)

338

Geometric Analysis

for any harmonic function on M. Let φ(t) be a cutoff function satisfying the properties . 1 for t ≤ ρ, φ(t) = 0 for t ≥ 2ρ, with 0 ≤ φ(t) ≤ 1, −C ρ −1 ≤ φ  (t) ≤ 0, and |φ  (t)| ≤ C ρ −2 . Taking the

 2  12 and integrating by parts, we composition with the function F(x) = i fi obtain 2 fi2j ≤ φ(F) fi2j B p ((1−)ρ)



M

≤

φ(F) (|∇ f |2 − 1)

M

≤

φ (F) (|∇ f |2 − 1)

M

(φ  (F) F + φ  (F)|∇ F|2 )(|∇ f |2 − 1).

≤

(27.19)

M

The fact that F 2 = 2



i

|∇ f i |2 ≥ 2|∇ F|2 implies that F ≥ 0.

Also 2FF + 2|∇ F|2 ≤ F 2 ≤ 2m implies that F ≤ m F −1 . By taking f = fi for i = 1, . . . , m and using the fact that |∇ f i | ≤ 1, (27.18) can be estimated by 2 −1 −1 2 −2 2 f jk ≤ Cρ F (1 − |∇ f | ) + Cρ (1 − |∇ f |2 ) B p ((1−)ρ)

F≤2ρ

≤ C ρ −2

F≤2ρ



B p ((2+)ρ)

(1 − |∇ f |2 )

≤  C ρ −2 V p ((2 + )ρ)   2+ m 2 ≤  C ρ V p ((1 − )ρ) . 1−

27

Linear growth harmonic functions

339

This implies that

1/2



B p (ρ)

|Hess()| ≤

B p (ρ)

|Hess()|

2

1/2

V p (ρ)

≤  C ρ −1 V p (ρ). Combining this estimate with (27.17) and (27.18), we conclude that V (A) ≤  C ρ m , hence implying V ((B p (ρ))) ≥ V0 ((1 − )ρ) − V (A) ≥ (1 − )m V0 (ρ) −  Cρ m .

(27.20)

On the other hand, since |d| ≤ 1, we have V p (ρ) ≥ V ((B p (ρ)). Therefore combining this inequality with (27.19), we conclude that V p (ρ) ≥ (1 − ) V0 (ρ) for sufficiently large ρ. This proves the theorem.



28 Polynomial growth harmonic functions

Yau’s conjecture was first proved by Colding and Minicozzi in 1996 [CM1]. Later that year they announced [CM2] the dimension estimate of the form h d (M) ≤ C1 d log CVD /log 2 , with C 1 depending on the Neumann Poincar´e inequality (equivalent to CP (M) > 0 in Definition 18.2) and the volume doubling constant (see Definition 18.1). In particular, it gives a sharp order estimate of the form h d (M) ≤ C1 d m−1 for manifolds with nonnegative Ricci curvature. The complete proofs of these announcements were published in [CM3, CM4] in 1997. Meanwhile, in 1997, Li [L8] gave a much simplified proof of this estimate which holds for a larger class of manifolds. In this paper, Li only required the manifold to satisfy a volume comparison condition (Vν ) (see Definition 28.1) and a mean value inequality (M) (see Definition 28.2). Note that the volume comparison condition and the volume doubling condition can easily be seen to be equivalent. However, the mean value inequality is weaker than the Neumann Poincar´e inequality. Moreover, the author’s argument can be applied to sections of vector bundles (Theorem 28.4). When Theorem 28.4 is applied to manifolds with nonnegative Ricci curvature (Corollary 28.5), it recovers the sharp order estimate of Colding–Minicozzi. In 1998, Colding and Minicozzi [CM5, CM6] also published a proof for the dimension estimate using the mean value inequality and volume doubling much in the spirit of [L8]. Finally, it is important to note that in [L8] the author proved that finite dimensionality and estimates of h d (M) actually hold for an even more general class of manifolds (Theorem 28.7), namely those satisfying a weak mean value inequality and 340

28

Polynomial growth harmonic functions

341

with polynomial volume growth. However, in this case, the estimate is not as sharp as Theorem 28.4. Definition 28.1 A manifold is said to satisfy a volume comparison condition (Vμ ) for some μ > 1, if there exists a constant CV > 0, such that for any point x ∈ M and any real numbers 0 < ρ1 ≤ ρ2 < ∞ the volume of the geodesic balls centered at x satisfies the inequality  μ ρ2 Vx (ρ2 ) ≤ CV Vx (ρ1 ) . ρ1 Definition 28.2 A manifold is said to satisfy a mean value inequality (M) if there exists a constant CM > 0 such that for any x ∈ M, ρ > 0 and any nonnegative subharmonic function f defined on M it must satisfy f 2 (y) dy. f 2 (x) ≤ CM Vx−1 (ρ) Bx (ρ)

Note that if M has nonnegative Ricci curvature, then M satisfies condition (Vm ) with C V = 1, by the Bishop volume comparison theorem. Moreover, condition (M) is also valid on such a manifold because of Theorem 7.2. Lemma 28.3 (Li) Let K be a k-dimensional linear space of sections of a vector bundle E over M. Assume that M has polynomial volume growth at most of order μ, i.e., V p (ρ) ≤ C ρ μ for p ∈ M and ρ → ∞. Suppose each section u ∈ K has polynomial growth of at most degree d such that |u|(x) ≤ C r d (x), where r (x) is the geodesic distance to the fixed point p ∈ M. For any β > 1, k δ > 0, and ρ0 > 0, there exists ρ > ρ0 such that if {u i }i=1 ! is an orthonormal basis of K with respect to the inner product Aβρ (u, v) = B p (βρ) u, v, then k  i=1

B p (ρ)

|u i |2 ≥ k β −(2d+μ+δ) .

Proof For each ρ > !0, let Aρ be the nonnegative bilinear form defined on K given by Aρ (u, v) = B p (ρ) u, v. Let us denote the trace of the bilinear form Aρ with respect to Aρ  by trρ  Aρ . Similarly, let detρ  Aρ be the determinant of

342

Geometric Analysis

Aρ with respect to Aρ  . Assuming that the lemma is false, then for all ρ > ρ0 , we have trβρ Aρ < k β −(2d+μ+δ) . On the other hand, the arithmetic-geometric mean asserts that

detβρ Aρ

1 k

≤ k −1 trβρ Aρ .

This implies that detβρ Aρ ≤ β −k (2d+μ+δ) for all ρ > ρ0 . Setting ρ = β i ρ for i = 0, 1, . . . , j − 1 and iterating this inequality j times yields detβ j ρ Aρ ≤ β − jk(2d+μ+δ) .

(28.1)

k However, for a fixed Aρ -orthonormal basis {u i }i=1 of K , the assumptions on K and on the volume growth imply that there exists a constant C > 0 depending on K , such that   2d+μ |u i |2 ≤ C 1 + ρ1 B p (ρ1 )

for all 1 ≤ i ≤ k and for all ρ1 ≥ ρ. In particular, this implies that detρ Aβ j ρ ≤ k! C β jk(2d+μ) ρ k(2d+μ) . This contradicts (28.1) as j → ∞, and the lemma is proved.



Theorem 28.4 (Li) Let M m be a complete manifold satisfying conditions (Vμ ) and (M). Suppose E is a rank-n vector bundle over M. Let Sd (M, E) ⊂ (E) be a linear subspace of sections of E such that all u ∈ Sd (M, E) satisfy the properties: (a) |u| ≥ 0; and (b) |u|(x) ≤ O(r d (x)) as x → ∞. Then the dimension of Sd (M, E) is finite. Moreover, for all d ≥ 1, there exists a constant C > 0 depending only on μ such that dim Sd (M, E) ≤ n C CM d μ−1 . Proof Let K be a finite dimensional linear subspace of Sd (M, E) with k be any basis of K . Then for p ∈ M, ρ > 0, and any dim K = k and {u i }i=1

28

Polynomial growth harmonic functions

343

0 <  < 12 , we claim that they must satisfy the estimate k  i=1

B p (ρ)

|u i |2 ≤ n C CM  −(μ−1)

sup

u∈{A,U } B p ((1+)ρ)

|u|2 ,

(28.2)

where the supremum is taken over all u ∈ K of the form u = A, U  for some unit vector A = (a1 , . . . , ak ) ∈ Rk!with U = (u 1 , . . . , u k ).  2 To see this, we will estimate B p (ρ) |u i | by using an argument similar to Lemma 7.3. Observe that for any x ∈ B p (ρ), there exists a subspace K x ⊂ K which is of at most codimension n such that u(x) = 0 for all u ∈ K x . In particular, by an orthonormal change of basis, we may assume that u i ∈ K x k n for n + 1 ≤ i ≤ k and i=1 |u i |2 (x) = i=1 |u i |2 (x). Since |u i | ≥ 0, if we denote the distance from p to x by r (x), then the mean value inequality (M) implies that k 

|u i |2 (x) =

i=1

n 

|u i |2 (x)

i=1

≤ CM Vx−1 ((1 + )ρ − r (x))

n  i=1

≤

C M Vx−1 ((1 + )ρ

− r (x)) n

Bx ((1+)ρ−r (x))

|u i |2



sup

u∈{A,U } B p ((1+)ρ)

|u|2 . (28.3)

However, condition (Vμ ) and the fact that r (x) ≤ ρ imply that  C V Vx ((1 + )ρ − r (x)) ≥  ≥

(1 + )ρ − r (x) 2ρ (1 + )ρ − r (x) 2ρ

μ μ

Vx (2ρ) V p (ρ).

(28.4)

Therefore substituting this estimate into (28.3) and integrating over B p (ρ), we have k  i=1

B p (ρ)

|u i |2 ≤

nCV CM 2μ sup u2 V p (ρ) u∈{A,U } B p ((1+)ρ) × ((1 + ) − ρ −1 r (x))−μ d x. B p (ρ)

(28.5)

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Geometric Analysis

On the other hand, if we define f (r ) = ((1 + ) − ρ −1 r )−μ , then f  (r ) = μ ρ −1 ((1 + ) − ρ −1 r )−(μ+1) ≥ 0 and

B p (ρ)

((1 + ) − ρ −1 r (x))−μ d x =



ρ

A p (t) f (t) dt.

0

To estimate this, we integrate by parts and obtain ρ ρ ρ f (t) A p (t) dt = [ f (t) V p (t)]0 − f  (t) V p (t) dt. 0

(28.6)

0

Using f  (t) ≥ 0 and applying condition (Vμ ), we have ρ ρ f  (t) V p (t) dt ≥ ρ −μ V p (ρ) f  (t) t μ dt 0

0

≥ρ

−μ

 V p (ρ)

ρ [ f (t) t μ ]0

Substituting this into (28.6) yields ρ f (t) A p (t) dt = ρ −μ V p (ρ) μ 0

≤ ρ −1 V p (ρ) μ

ρ

0 ρ



ρ

−μ

f (t) t

μ−1

 dt .

0

f (t) t μ−1 dt

((1 + ) − tρ −1 )−μ dt

0

μ V p (ρ) ( −μ+1 − (1 + )−μ+1 ) μ−1 μ ≤ V p (ρ)  −(μ−1) . μ−1 =

The claim follows by combining this with (28.5). k To complete the proof of the theorem, let {u i }i=1 be an Aβρ -orthonormal basis of any finite dimensional subspace K ⊂ Sd (M, E). Clearly, it suffices to prove the estimate for k = dim K . Note that condition (Vμ ) implies that the volume growth of M is at most of order ρ μ , hence Lemma 28.3 implies that there is a ρ > 0 such that k  i=1

B p (ρ)

|u i |2 ≥ k β −(2d+μ+δ) .

28

Polynomial growth harmonic functions

345

On the other hand, by setting β = 1 + , (28.2) implies that k  |u i |2 ≤ n C CM  −(μ−1) because

!

i=1 2 B p ((1+)ρ) |u|

B p (ρ)

= 1 for all u ∈ {A, U }. For d ≥ 1, the estimate on

k follows by setting  = (2d)−1 and observing that (1 + (2d)−1 )−(2d+μ+δ) is bounded from below.  Note that extra care is used to obtain the order μ − 1 in Theorem 28.4 because this is sharp on Euclidean space (B.3). Corollary 28.5 (Colding–Minicozzi) Let M m be a complete Riemannian manifold with nonnegative Ricci curvature. There exists a constant C > 0 depending only on m such that the dimension of the Hd (M) is bounded by h d (M) ≤ C d m−1 for all d ≥ 1. Finite dimensionality of Hd (M) can be obtained by relaxing both the volume comparison condition (Vμ ) and the mean value inequality condition (M). However, in the general case, the order of dependency in d will not be sharp. Definition 28.6 A complete manifold M is said to satisfy a weak mean value inequality (WM) if there exist constants CWM > 0 and b > 1 such that for any nonnegative subharmonic function f defined on M it must satisfy f (y) dy. f (x) ≤ CWM Vx−1 (ρ) Bx (bρ)

for all x ∈ M and ρ > 0. Theorem 28.7 Let M be a complete manifold satisfying the weak mean value property (WM). Suppose that the volume growth of M satisfies V p (ρ) = O(ρ μ ) as ρ → ∞ for some point p ∈ M. Then Hd (M) is finite dimensional for all d ≥ 0 and dim Hd (M) ≤ CWM (2b + 1)(2d+μ) . Proof Let us set β = 2b + 1, where b is the constant in (WM). By Lemma 28.3, for any δ > 0, there exists ρ > 0 such that for u 1 , . . . , u k , an orthonormal basis of Hd (M) with respect to the inner product Aβρ (u, v), we have k  u i2 ≥ k β −(2d+μ+δ) . (28.7) i=1

B p (ρ)

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Geometric Analysis

k u i2 (x) is subharmonic, the maxiOn the other hand, since the function i=1 mum principle implies that there exists a point q ∈ ∂ B p (ρ) such that k 

u i2 (x) ≤

i=1

k 

u i2 (q)

(28.8)

i=1

for all x ∈ B p (ρ). Again, by an orthonormal change of basis, we may assume that u j (q) = 0 for 2 ≤ j ≤ k. Applying the weak mean value inequality (WM) to u 21 and noting that B p (ρ) ⊂ Bq (2ρ) ⊂ B p ((2b + 1)ρ), we get V p (ρ) u 21 (q) ≤ Vq (2ρ) u 21 (q) u 21 ≤ CWM ≤ CWM

Bq (2bρ)

B p ((2b+1)ρ)

u 21

= CWM . Thus integrating (28.8) over the ball B p (ρ) gives k  i=1

B p (ρ)

u i2 ≤ V p (ρ)

k 

u i2 (q)

i=1

≤ V p (ρ) u 21 (q) ≤ CWM .

(28.9)

From (28.7) and (28.9) we conclude that k β −(2d+μ+δ) ≤ C WM , hence k ≤ C WM β (2d+μ) as δ > 0 is arbitrary. This completes the proof.



Corollary 28.8 Let M be a complete manifold. Suppose there exist constants C1 , C2 > 0 and μ > 2 such that for all p ∈ M, ρ > 0, and for all f ∈ Hc1,2 (M) we have

(μ−2)/μ B p (ρ)

| f |2μ/(μ−2)

≤ C1 V p (ρ)−2/μ ρ 2 ×

B p (ρ)

−2/μ

|∇ f | + C2 V p (ρ) 2

B p (ρ)

f 2.

28

Polynomial growth harmonic functions

347

Then dim Hd (M) < C β d for all d ≥ 0, for some constants C > 0 and β > 1 depending only on C1 , C2 , μ, and b. Proof Applying the Moser iteration process given by Lemma 19.1 the Sobolev type inequality in the hypothesis implies the weak mean value inequality (WM). Also, by choosing f to be a cutoff function satisfying # 1 if x ∈ B p (1), f (x) = 0 if x ∈ M \ B p (2), and |∇ f | ≤ 1, and applying to the Sobolev inequality in the hypothesis we conclude that (μ−2)/μ

Vp

−2/μ

(1) ≤ C1 V p

(ρ)(C1 ρ 2 + C 2 )V p (2).

This implies that V p (ρ) = O(ρ μ ) as ρ → ∞. The corollary follows from Theorem 28.7.  In the case when M m has nonnegative sectional curvature, it was shown by d h i (M) has Li and Wang in [LW3] that if h d (M) = dim Hd (M), then i=1 an upper bound which is asymptotically sharp as d → ∞. In particular, they showed that if 0 ≤ α0 ≤ ωm is a constant given by lim inf ρ −m V p (ρ) = α0 , ρ→∞

then lim inf d −(m−1) h d ≤ d→∞

2α0 . (m − 1)! ωm

Moreover, lim inf d −(m−1) h d = d→∞

2 (m − 1)!

if and only if M = Rm . The proof of this will not be presented here, however, this motivates the following question.

348

Geometric Analysis

Question If M is a complete manifold with nonnegative sectional curvature, then is it true that h d (M) ≤ h d (Rm )     m+d −1 m+d −2 = + ? d d −1

29 L q harmonic functions

Recall that Yau’s theorem (Lemma 7.1) asserts that a complete manifold does not admit any nonconstant L q harmonic functions for q ∈ (1, ∞). In this chapter, we will discuss the validity of this Liouville property for L q harmonic functions when q ∈ (0, 1]. The condition to ensure the Liouville property for the case when q ∈ (0, 1) is quite different from the condition for the case q = 1. Both of these cases were first considered by Li and Schoen [LS], when they obtained the sharp curvature condition for q ∈ (0, 1). In the same paper, they also obtained an almost sharp condition for q = 1, while the sharp version was later proved by Li in [L4] using the heat equation method. Theorem 29.1 presented below is the argument from [LS], while Theorem 29.3 is from [L4]. Counterexamples for both theorems when the curvature conditions have been violated were given in [LS] indicating the sharpness of these conditions. Theorem 29.1 (Li–Schoen) Let M m be a complete manifold. There exists a constant δ(m) > 0 depending only on m such that if the Ricci curvature of M satisfies the lower bound Ri j (x) ≥ −δ(m) r −2 (x)

for r (x) → ∞,

where r (x) is the distant to a fixed point p ∈ M, then any nonnegative, L q subharmonic function must be identically zero for q ∈ (0, 1). Proof Let f be a nonnegative L q subharmonic function defined on M. According to the mean value inequality (14.23) of Corollary 14.8, for nonnegative subharmonic functions, we have the estimate  √  −1 q −m ¯ exp C2 ρ R Vx (ρ) f q, f (x) ≤ C 1 V (2ρ) ρ Bx (ρ)

349

350

Geometric Analysis

if Ri j ≥ −(m − 1)R on Bx (2ρ) with constants C1 , C2 > 0 depends only on m and q, and V¯ (2ρ) is the volume of the geodesic ball of radius 2ρ in the model space of constant −R curvature. Let us now apply the above mean value inequality to any point x sufficiently far from p with 5ρ ≤ r (x). The curvature assumption asserts that the quantity  √  V¯ (2ρ) ρ −m exp C2 ρ R is bounded, hence f q (x) ≤ C 3 Vx−1 (ρ),

(29.1)

where C3 also depends on the L q -norm of f. We will now estimate Vx (ρ) using the volume comparison theorem. Let γ be a minimal normal geodesic joining p to x with γ (0) = p and γ (T ) = x. Define the values {ti } with t0 = 0, t1 = 1 + β, and ti =  2 ij=0 β j − 1 − β i , for some fixed β>

2 21/m

−1

> 1.

Let tk ≤ T be the largest value not bigger than T. We let the points xi = γ (ti ) and they obviously satisfy the properties that r (xi , xi+1 ) = β i + β i+1 and r (xk , x) < β k +'β k+1 . Moreover, the ( closure of the set of geodesic balls  {Bxi (β i )} covers γ 0, 2 kj=1 β j − 1 and the balls have disjoint interiors. We now claim that  k βm V p (1). (29.2) Vxk (β k ) ≥ C 4 (β + 2)m − β m To see this, the volume comparison theorem implies that Vxi (β i ) ≥ Ai (Vxi (β i + 2β i−1 ) − Vxi (β i )) ≥ Ai Vxi−1 (β i−1 ), where ! βi Ai =

√

0

R(xi ,β i +2β i−1 )

! (β i +2β i−1 ) √ βi

R(xi

√

R(xi ,β ,β i +2β i−1 )

sinhm−1 (t) dt

i +2β i−1 )

sinhm−1 (t) dt

L q harmonic functions

29

351

with Ri j ≥ −(m − 1)R(xi , β i + 2β i−1 ) denotes the lower bound of the Ricci curvature on Bxi (β i + 2β i−1 ). Iterating this inequality, we conclude that Vxk (β k ) ≥ V p (1)

k -

Aj.

(29.3)

j=1

Since r (xi ) = 2

i

j=0 β

j

− 1 − β i , the curvature assumption implies that ⎛

R(xi , β i + 2β i−1 ) ≤ δ(m) ⎝2

i−2 

⎞−2 β j⎠

j=0

for i ≥ 2. Therefore √ / δ(m) βi i i i−1 β R(xi , β + 2β ) ≤ i−2 j 2 j=0 β √ β2 δ(m) ≤ i−2 − j 2 j=0 β √ δ(m) ≤ β(β − 1)(1 − β −i+1 )−1 , 2 which can be made arbitrarily small for a fixed β by choosing δ(m) to be sufficiently small. In particular, simply by approximating sinh(t) by t, Ai can be approximated by Ai ∼ =

(β i )m (β i + 2β i−1 )m − (β i )m βm . (β + 2)m − β m

Combining this with (29.3), we conclude (29.2). To estimate Vx (β k+1 ), we consider the following cases. First, let us assume that r (x, xk ) ≤ β k (β − 1). In this case, we see that Bxk (β k ) ⊂ Bx (β k+1 ),

352

Geometric Analysis

hence Vxk (β k ) ≤ Vx (β k+1 ). In particular, we conclude that  Vx (β k+1 ) ≥ C 4

βm (β + 2)m − β m

k V p (1).

(29.4)

On the other hand, if r (x, xk ) > β k (β − 1), then we have Bxk (β k ) ⊂ Bx (r (x, x k ) + β k )\ Bx (r (x, xk ) − β k ). Using the volume comparison theorem, we conclude that Vx (β k ) ≥ A (Vx (r (x, xk ) + β k ) − Vx (x(x, xk ) − β k ) ≥ A Vxk (β k ), where ! βk

A=

√

R(x,r (x,x k )+β k )

sinhm−1 (t) dt . √ ! (r (x,xk )+β k ) R(x,r (x,xk )+β k ) √ sinhm−1 (t) dt 0

(r (x,xk )−β k )

Since

/

(r (x, x k ) + β ) R(x, r (x, xk k

R(x,r (x,x k )+β k )

) + βk )

/

≤ (β + 2β ) R(x, r (x, xk ) + β k ) √ δ(m) β(β − 1) ≤ 2 k+1

k

can be made sufficiently small, we can approximate A by A∼

βm . (β + 2)m

Combining the value of A with (29.2), we again conclude that  k βm k+1 V p (1) Vx (β ) ≥ C 5 (β + 2)m − β m for some constant C5 > 0.

(29.5)

29

L q harmonic functions

353

Note that when x → ∞, k → ∞. Since the choice of β ensures that βm > 1, (β + 2)m − β m the right-hand side of (29.5) tends to infinity. Using the value R = β k+1 in (29.1), we conclude that f (x) → 0 as x → ∞. The maximum principle now asserts that f must be identically 0.  Corollary 29.2 Let M m be a complete manifold. There exists a constant δ(m) > 0, depending only on m, such that if the Ricci curvature of M satisfies the lower Ri j (x) ≥ −δ(m) r −2 (x)

for r (x) → ∞,

then M does not admit any nontrivial L q harmonic functions for q ∈ (0, 1). Proof Since the absolute value of a harmonic function is subharmonic, this corollary follows directly from Theorem 29.1.  Theorem 29.3 (Li) Let M m be a complete manifold. Suppose the Ricci curvature of M satisfies the lower Ri j (x) ≥ −C (1 + r (x))2 for some constant C > 0. Then any nonnegative, L 1 subharmonic function must be constant. Proof Let f be a nonnegative, L 1 subharmonic function defined on M. By solving the heat equation with f as initial datum, we obtain f (x, t) = H (x, y, t) f (y) dy M

with f (x, 0) = f (x). Recall that the heat semi-group is contractive in L 1 because of Theorem 12.4, hence H (x, y, t) dy ≤ 1. (29.6) M

We now claim that ∂ f (x, t) ≥ 0 ∂t

(29.7)

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Geometric Analysis

for all x ∈ M and for all t ∈ (0, ∞). To see this, we differentiate under the integral sign and obtain ∂ ∂ f (x, t) = H (x, y, t) f (y) dy ∂t M ∂t =  y H (x, y, t) f (y) dy. M

If we can justify integration by parts, namely, H (x, y, t) f (y) dy = H (x, y, t)  f (y) dy, M

(29.8)

M

then the subharmonicity of f implies (29.7). Indeed, (29.8) follows from Green’s identity       H (x, y, t) f (y) dy − H (x, y, t)  f (y) dy    B p (ρ)  B p (ρ)     ∂ ∂   H (x, y, t) H (x, y, t) f (y) − f (y) dy  =   ∂ B p (ρ) ∂r ∂r ∂ B p (ρ) |∇ H |(x, y, t)| f (y) dy + H (x, y, t) |∇ f |(y) dy = ∂ B p (ρ)

∂ B p (ρ)

and by showing that there exists a sequence {ρi } with ρi → ∞ such that the boundary terms on the right-hand side when setting ρ = ρi tend to 0 as ρi → ∞. Using the mean value inequality (14.23) of Corollary 14.8, we obtain the growth estimate  √  f, sup f ≤ C1 V¯ (4ρ) ρ −m exp C2 ρ R V p−1 (2ρ) B p (2ρ)

B p (ρ)

where −(m − 1)R is the lower bound of the Ricci curvature on B p (4ρ). However, the assumption on the curvature of M implies that  √  V¯ (4ρ) ≤ C6 exp 4(m − 1)ρ R   ≤ C 6 exp C7 ρ 2 for some constants C6 , C7 > 0. This implies that   sup f ≤ C8 V p−1 (2ρ) exp C 9 ρ 2 , B p (ρ)

where C8 depends also on the L 1 -norm of f .

(29.9)

L q harmonic functions

29

355

Let φ be a nonnegative cutoff function satisfying ⎧ 0 on B p (ρ − 1) ∪ (M \ B p (2ρ)), ⎪ ⎪ ⎪ ⎪ ⎨ r (x) − ρ + 1 on B p (ρ)\ B p (ρ − 1), φ(x) = 1 on B p (ρ)\ B p (ρ + 1), ⎪ ⎪ 2ρ + 1 − r (x) ⎪ ⎪ ⎩ on B p (2ρ + 1)\ B p (ρ + 1). ρ The subharmonicity of f implies that φ2 f  f 0≤ M





= −2

φ 2 |∇ f |2

φ f ∇φ, ∇ f  − M

1 ≤2 |∇φ|2 f 2 − 2 M

M



φ 2 |∇ f |2 . M

Hence using the definition of φ, we obtain the estimate 2 |∇ f | ≤ 4 f2 B p (ρ+1)\B p (ρ)

B p (2ρ+1)

for ρ ≥ 1. Combining the above inequality with (29.9) and applying Schwarz inequality, we obtain

1/2 B p (ρ+1)\B p (ρ)

|∇ f | ≤

B p (ρ+1)\B p (ρ)

|∇ f |2

1/2

V p (ρ + 1)

≤ C8 exp(C10 ρ 2 ).

(29.10)

To estimate H (x, y, t) we recall Theorem 13.4 which asserts that     −1/2 √ −1/2 √ H (x, y, t) ≤ C11 Vx t Vy t  / −r 2 (x, y) −2 (ρ + R)t + C12 × exp (4 + )t 

for all x, y ∈ B p (ρ). For a fixed x ∈ M and for sufficiently large ρ, we may assume that x ∈ B p (ρ/4). In this case, if y ∈ ∂ B p (ρ), then the curvature assumption implies that       √ −αρ 2 −1/2 √ −1/2 √ + C12 ρ t . t Vy t exp H (x, y, t) ≤ C11 Vx t (29.11)

356

Geometric Analysis

Using the volume comparison theorem, we have √    √ √ Vx t ≤ Vy r (x, y) + t − Vy r (x, y) − t √  V¯ r + √t 

√  t ≤ Vy V¯ t √    √  m t C13 t 2 exp C14 ρ ρ + t . ≤ Vy Applying this to (29.11), we conclude that   √   √ −αρ 2 t exp H (x, y, t) ≤ C15 Vx−1 + C16 ρ ρ + t . t

(29.12)

To estimate the gradient of H , we consider the integral φ 2 (y) |∇ H |2 (x, y, t) M



= −2

H (x, y, t)∇φ(y), φ(y)∇ H (x, y, t) M



φ 2 (y) H (x, y, t) H (x, y, t)

− M



|∇φ| (y) H (x, y, t) + 2

≤2

2

M

−

1 2

φ 2 (y) |∇ H |2 (x, y, t) M

φ 2 (y) H (x, y, t) H (x, y, t). M

This implies that |∇ H |2 B p (ρ+1)\B p (ρ)



≤

φ 2 |∇ H |2 M





≤4

|∇φ| H − 2 2

≤4

φ 2 H H

2

M

B p (ρ+1)\B p (ρ−1)

M



H2 + 2

H |H |

H +2 2

B p (2ρ+1)\B p (ρ−1)

1/2 



≤4

B p (2ρ+1)\B p (ρ−1)

B p (2ρ+1)\B p (ρ−1)

H

2

1/2 (H )

2

.

M

(29.13)

29

L q harmonic functions

357

Using the upper bound (29.12) for H and (29.6), we obtain the estimate H 2 (x, y, t) B p (2ρ+1)\B p (ρ−1)

≤ ≤

sup

y∈B p (2ρ+1)\B p (ρ−1)

C 15 Vx−1

H (x, y, t)

  √   √ −αρ 2 + C16 ρ ρ + t . t exp t

We now claim that there exists a constant C17 > 0, such that (H )2 (x, y, t) ≤ C17 t −2 H (x, x, t).

(29.14)

(29.15)

M

To see this, we first prove the inequality for any Dirichlet heat kernel H defined on a compact subdomain  of M. Using the fact that the heat kernel on M can be obtained by taking the limits of Dirichlet heat kernels on a compact exhaustion of M, (29.15) follows. Indeed, if H (x, y, t) is a Dirichlet heat kernel defined on a compact subdomain  ⊂ M, we can write H using the eigenfunction expansion H (x, y, t) =

∞ 

e−λi t ψi (x, ) ψi (y),

i=1

where {ψi } is the orthonormal basis of the space of L 2 functions with Dirichlet boundary value satisfying the equation ψi = −λi ψi . Differentiating with respect to the variable y, we have H (x, y, t) = −

∞ 

λi e−λi t ψi (x) ψi (y).

i=1

Therefore, using the fact that s 2 e−2s ≤ C 17 e−s for all 0 ≤ s < ∞, we conclude that ∞  (H )2 (x, y, t) dy ≤ C 17 t −2 e−λi t ψi2 (x) M

i=1

= C17 t

−2

H (x, x, t).

358

Geometric Analysis

Combining (29.13), (29.14), and (29.15), we obtain |∇ H |2 B p (ρ+1)\B p (ρ)



≤ C18

− 12

Vx

   √   √ 1 −αρ 2 t + t −1 H 2 (x, x, t) exp + C19 ρ ρ + t . 2t

Applying Schwarz inequality and the volume comparison theorem, we conclude that 1/2    −1/2 √ |∇ H | ≤ C20 Vx t + t −1 H 1/2 (x, x, t) B p (ρ+1)\B p (ρ)

 × exp

  √ −αρ 2 + C21 ρ ρ + t . 4t

(29.16)

Using (29.9), (29.10), (29.12), and (29.16), we conclude that for a fixed x ∈ M and for any sufficiently small fixed t > 0 |∇ H |(x, y, t) f (y) dy B p (ρ+1)\B p (ρ)



+

B p (ρ+1)\B p (ρ)

H (x, y, t) |∇ f |(y) dy → 0

as ρ → ∞. Hence, the mean value theorem implies that there is a sequence of {ρi } with ρi → ∞ such that |∇ H |(x, y, t) f (y) dy + H (x, y, t) |∇ f |(y) dy → 0, ∂ B p (ρi )

∂ B p (ρi )

and (29.8) is verified. In particular, we confirm the monotonicity of f (x, t) for t sufficiently small. Monotonicity of f (x, t) for all t follows from the semigroup property of the heat equation. Let us also observe that since ∂ H (x, y, t) dy = H (x, y, t) dy, ∂t M M and because (29.16) implies that H (x, y, t) dy ≤ B p (ρi )

∂ B p (ρi )

→0

|∇ H |(x, y, t) dy

L q harmonic functions

29

359

as ρi → ∞, we conclude that ∂ H (x, y, t) dy = 0, ∂t M hence H (x, y, t) dy = 1

(29.17)

M

for all x ∈ M and for all t. To finish the theorem, since (29.17) implies that f (x, t) d x = H (x, y, t) f (y) d y d x M



M

=

M

f (y) dy M

and by applying the monotonicity of f (x, t), we conclude that f (x, t) = f (x), hence  f (x) = 0. On the other hand, for any arbitrary constant a > 0, let us we define the function g(x) = min{ f (x), a}. It must be nonnegative satisfying g(x) ≤ f (x), |∇g|(x) ≤ |∇ f |(x), and g(x) ≤ 0. In particular, it will satisfy the same estimates, (29.9) and (29.10), as f . Hence we can show that ∂ H (x, y, t) g(y) dy = H (x, y, t) g(y) dy ∂t M M ≤ 0. Since g is also L 1 , the same argument as before implies that g(x) = 0, hence regularity theory asserts that g must be smooth. Since a is arbitrary, this is impossible unless f is identically constant. 

360

Geometric Analysis

Corollary 29.4 Let M m be a complete manifold. Suppose the Ricci curvature of M is bounded from below by Ri j (x) ≥ −C (1 + r (x))2 for some constant C > 0. Then M does not admit any nonconstant L 1 harmonic functions.

30 Mean value constant, Liouville property, and minimal submanifolds

In this chapter, we will assume that M satisfies a slightly stronger version of the mean value property (M) as in Definition 28.2. Definition 30.1 A manifold is said to satisfy an L 1 mean valued inequality (M1 ) if there exists a constant CM1 > 0 such that for any x ∈ M, ρ > 0, and any nonnegative subharmonic function f defined on M, it must satisfy −1 f (y) dy. f (x) ≤ CM1 Vx (ρ) Bx (ρ)

Note that by applying Schwarz’s inequality, (M1 ) implies (M), with 2 . It turns out that the value of C CM < CM M1 plays a significant role in both 1 the regularity theory and the Liouville property of a manifold. It was proved by Li and Wang [LW2] that when C M1 is sufficiently close to 1, the mean value inequality (M1 ) becomes more powerful. Theorem 30.2 (Li–Wang) Let M be a complete manifold satisfying the mean value inequality (M1 ). If CM1 < 2 and M has subexponential volume growth, then M has the Liouville property, i.e., M does not admit any nonconstant bounded harmonic functions. Proof

Let f be a harmonic function defined on M. Let us define s(r ) = sup f (x), B¯ p (r )

i(r ) = inf f (x), B¯ p (r )

and ω(r ) = s(r ) − i(r ). 361

362

Geometric Analysis

Assume that x, y ∈ B¯ p (r ) such that f (x) = s(r ) and f (y) = i(r ). In fact, the maximum principle asserts that x, y ∈ ∂ B p (r ). For any ρ > r, the mean value inequality applying to the functions f − i(ρ) and s(ρ) − f gives CM1 ( f − i(ρ)) ≥ CM1 ( f − i(ρ)) B p (ρ)

Bx (ρ−r )

≥ (s(r ) − i(ρ)) Vx (ρ − r ) and

C M1

(30.1)

B p (ρ)

(s(ρ) − f ) ≥ CM1

B y (ρ−r )

(s(ρ) − f )

≥ (s(ρ) − i(r )) Vy (ρ − r ).

(30.2)

By taking ρ > 2r , we have B p (ρ − 2r ) ⊂ Bx (ρ − r ) and B p (ρ − 2r ) ⊂ B y (ρ − r ), hence we have Vx (ρ − r ) ≥ V p (ρ − 2r ) and Vy (ρ − r ) ≥ V p (ρ − 2r ). Combining these inequalities with (30.1) and (30.2), and adding the two estimates, we conclude that C M1 ω(ρ) V p (ρ) ≥ (ω(r ) + ω(ρ)) V p (ρ − 2r ). This implies that

 C M1

 V p (ρ) − 1 ω(ρ) ≥ ω(r ). V p (ρ − 2r )

(30.3)

Suppose now that f is a nonconstant bounded harmonic function. Then by scaling we may assume that the total oscillation sup f − inf f = 1, hence ω(ρ)  1 as ρ → ∞. In particular, (30.3) becomes C M1

V p (ρ) − 1 ≥ ω(r ). V p (ρ − 2r )

(30.4)

30

Mean value constant, Liouville property, minimal submanifolds

363

For any r > 0, we claim that the sequence V p (2(i + 1)r )/V p (2ir ) must satisfy lim inf i→∞

V p (2(i + 1)r ) = 1. V p (2ir )

Indeed, if this is not the case, then there exists i 0 > 0 such that V p (2(i + 1)r ) ≥1+ V p (2ir ) for i ≥ i 0 . Iterating this inequality k times beginning from i = i 0 , we have V p (2(k + i 0 )r ) ≥ (1 + )k V p (2i 0 r ). Let k → ∞, this implies that M has exponential volume growth, and hence contradicts the assumption. Therefore there is a subsequence i j such that V p (2(i j + 1)r ) →1 V p (2i j r ) as j → ∞. Setting ρ j = 2(i j + 1)r , then V p (ρ j ) → 1. V p (ρ j − 2r ) Together with (30.4), this implies that (CM1 − 1) ≥ ω(r ) for all r > 0. However, this contradicts the assumptions that ω(r )  1 and  CM1 < 2, hence f must be identically constant. Theorem 30.3 (Li–Wang) Let M be a complete manifold satisfying the mean value inequality (M1 ) and the volume comparison condition Vμ . If CV CM1 < 2, then there exists a constant 0 < α < 1 depending only on CV CM1 and μ such that any harmonic function f defined on M satisfying | f (x)| = O(r α (x)), where r is the distance to some fixed point p ∈ M, must be identically constant. Proof Following the proof of Theorem 30.2, (30.1) and (30.2) are valid for any harmonic functions defined on M. On the other hand, using the volume growth condition, we have   ρ −r μ Vx (ρ − r ) ≥ Vx (ρ + r ) CV ρ +r ≥ V p (ρ)

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Geometric Analysis

and similarly  CV

ρ −r ρ +r

μ

Vy (ρ − r ) ≥ V p (ρ).

Substituting the above inequalities into (30.1) and (30.2), and taking their sum, we conclude that   ρ −r μ −1 . CM1 ω(ρ) ≥ C V (ω(r ) + ω(ρ)) ρ +r This implies that     ρ +r μ − 1 ω(ρ) ≥ ω(r ). C V C M1 ρ −r

(30.5)

Setting ρ = (1 + β)r, this becomes     2+β μ C V C M1 − 1 ω((1 + β)r ) ≥ ω(r ). β Iterating this inequality k times yields  k   2+β μ − 1 ω((1 + β)k r ) ≥ ω(r ). C V C M1 β

(30.6)

On the other hand, the assumption on the growth rate of f implies that ω((1 + β)k r ) ≤ C (1 + β)αk r α , and hence when combined with (30.6) gives  k   μ 2 ω(r ) . C V C M1 + 1 − 1 (1 + β)α ≥ β C rα

(30.7)

Obviously, by taking β sufficiently large depending on μ and using the assumption that C V CM1 < 2, we may assume that 

μ C V C M1 + 2 2 ≤ , +1 β 2CV CM1

which implies  μ   CV C M1 2 C V C M1 +1 −1 ≤ . β 2

30

Mean value constant, Liouville property, minimal submanifolds

365

Therefore there exists α sufficiently small depending on β and C V CM1 such that   μ  2 + 1 − 1 (1 + β)α < 1. CV CM1 β Under these choices, the left-hand side of (30.7) tends to 0 as k → ∞, hence ω(r ) = 0 and f is identically constant.  In general, the proof of a growth estimate for ρ → ∞ can also be modified to obtain a regularity estimate by considering ρ → 0. In this spirit, one obtains a regularity type result for harmonic functions. Of course, when M is smooth, this does not yield any new insight since all harmonic functions are smooth. However, the argument in the next theorem can be applied to singular manifolds where the issue of regularity is not so apparent at a singular point. Theorem 30.4 (Li–Wang) Let M be a singular manifold satisfying the mean value inequality M1 and the weak volume growth condition Vμ in a neighborhood B p (ρ) of a point p ∈ M. If CV CM1 < 2, then any bounded harmonic function defined on B p (ρ) must be C α at p for some 0 < α < 1 depending on CV CM1 and μ. Proof As in the proof of Theorem 30.3, by setting r = βρ for some β < 1 in (30.5), we have     1−β μ − 1 ω(ρ) ≥ ω(βρ). C V C M1 1+β Interating this k times, we obtain    k 1−β μ CV CM1 − 1 ω(ρ) ≥ ω(β k ρ). 1+β H¨older continuity follows if we can find α > 0 such that   k  1−β μ − 1 ≤ C(β k ρ)α . C V C M1 1+β This can be achieved by finding α such that  C V C M1

1−β 1+β

μ

− 1 ≤ βα.

366

Geometric Analysis

To see this, we choose β sufficiently small that   1 − β μ 2 + CV CM1 ≤ , 1+β 2CV CM1 which implies

 C V C M1

1−β 1+β

μ

−1≤

C V C M1 . 2

Therefore, there exists α sufficiently small that β α ≥ C V CM1 /2, and the theorem follows.  Lemma 30.5 Let M m be a minimal varifold in R N . Let r¯ (x, y) be the extrinsic distance function between two points x, y ∈ M obtained by the restriction of the Euclidean distance function to M, and B¯ x (ρ) = {y ∈ M | r¯ (x, y) < ρ} be the extrinsic ball centered at x of radius ρ. For any regular point x ∈ M and any nonnegative subharmonic function h defined on M, it must satisfy the mean value inequality m h(y) dy ωm ρ h(x) ≤ B¯ x (ρ)

for all x ∈ M and ρ ≥ 0. Proof For a point x ∈ M, it is known that the extrinsic distance function r¯x (y) = r¯ (x, y) satisfies |∇ r¯x | ≤ 1 and ¯r x =

m−1 . r¯x

If we define r¯ 2−m (y) − r 2−m G¯ x (y) = x m(m − 2) ωm with ωm = V¯ (1), then a direct computation shows that G¯ x = −

(m − 1)¯r x−m (m − 1)¯r x−m |∇ r¯x |2 + m ωm m ωm

≤ 0. Since x is a regular point, if we denote the Dirichlet Green’s function on the extrinsic ball B¯ x (r ) with pole at x by G x , then G¯ x ∼ G x when y → x.

30

Mean value constant, Liouville property, minimal submanifolds

367

Hence, together with the fact that G x (y) = 0 = G¯ x (y) when y ∈ ∂ B¯ x (r ), the maximum principle implies that G¯ x (y) ≥ G x (y). This also implies that ∂G x (y) ∂ G¯ x (y) ≥ ∂μ ∂ν =−

r¯x1−m (y) ∂ r¯x (y) m ωm ∂ν

≥−

r¯x1−m (y) m ωm

on ∂ B¯ x (r ) with ν being the outward pointing unit normal vector on ∂ B¯ x (r ). Using the definition of the Green’s function, if h is a nonnegative subharmonic function, we have −h(x) = G x (y) h(y) dy =

B¯ x (r )

B¯ x (r )

≥−

G x (y) h(y) +

r 1−m m ωm

∂ B¯ x (r )

∂ B¯ x (r )

∂G x (y) h(y) ∂ν

h(y).

Multiplying both sides by r m−1 and integrating over [0, ρ], we obtain the mean value inequality ρ h h= |∇ r¯x | ¯ ¯ Bx (ρ) Bx (r ) 0 ρ h ≥ 0

B¯ x (r )

≥ ωm ρ m h(x).



Theorem 30.6 (Li–Wang) Let M m be a minimal varifold in R N . Let r¯ (x, y) be the extrinsic distance function between two points x, y ∈ M. For a fixed point p ∈ M, suppose the density function θ p (ρ) =

V¯ p (ρ) ωm ρ m

368

Geometric Analysis

has an upper bound θ p (ρ) ≤ θ¯ < 2, where V¯ p (ρ) is the volume of the extrinsic ball of radius ρ with the center at p. Then there exists 0 < α < 1 depending only on θ¯ and m such that if f is a harmonic function satisfying the growth condition | f (x)| = O(¯r αp (x)), then f must be identically constant. Proof If f is a harmonic function, then applying a similar argument to that in Theorem 30.3 and Lemma 30.5, we obtain f − V¯ p (ρ) i(ρ) ≥ ω¯ m (ρ − r )m f (x) − ωm (ρ − r )m i(ρ) B¯ p (ρ)

and V¯ p (ρ) s(ρ) −

B¯ p (ρ)

f ≥ ωm (ρ − r )m s(ρ) − ωm (ρ − r )m f (y)

for any smooth points x, y ∈ ∂ B¯ p (r ). In particular, since smooth points are dense in B¯ p (r ), this implies that 



ρ θ p (ρ) ρ −r



m −1

ω p (ρ) ≥ ω p (r ).

The assumption that θ p (ρ) ≤ θ¯ < 2 and the argument of Theorem 30.3 yield the desired conclusion.  Combining the arguments of Theorem 30.4 and Theorem 30.6, we have the following regularity theorem. However, we should point out that it was proved by Simon [Si] that under the assumption of Theorem 30.7, a Poincar´e inequality is valid. In which case, the Moser iteration argument implies C α regularity also. Theorem 30.7 (Simon) Let M m be a stationary varifold in R N . Suppose the multiplicity at a point p ∈ M defined by θ p = lim θ p (ρ) ρ→0

is strictly less than 2. Then any harmonic function defined in a neighborhood of p is C α for some 0 < α < 1 depending only on θ p and m.

30

Mean value constant, Liouville property, minimal submanifolds

369

Corollary 30.8 Let M m be a minimal stationary varifold in R N . Suppose the volume growth of M is Euclidean, i.e., V¯ x (ρ) ≤ θ¯ ωm ρ m for some constant θ¯ ≥ ωm . Then there exists a constant C1 > 0 depending only on θ¯ and m such that h d (M) ≤ C d m−1 for all d ≥ 1. Proof Lemma 30.5 and the volume assumption assert that M satisfies the mean value inequality (M) for extrinsic balls. On the other hand, since ωm ρ m ≤ V¯x (ρ) ≤ θ¯ ωm ρ m , ¯ The it satisfies the weak volume comparison condition (Vm ) with CV = θ. theorem follows from the argument of Theorem 30.4. 

31 Massive sets

In this chapter, we will introduce the notion of d-massive sets. The notion of a 0-massive set was first introduced by Grigor’yan [G2] when he established the relationship between massive sets and harmonic functions. While it is not clear if such a relationship exists for d > 0, the notion of d-massive sets has important geometric and analytic implications. Definition 31.1 For any real number d ≥ 0, a d-massive set  is a subset of a manifold M that admits a nonnegative, subharmonic function f defined on  with the boundary condition f =0

on

∂,

and satisfying the growth property f (x) ≤ C r d (x) for all x ∈  and for some constant C > 0. The function f is called a potential function of . We also let m d (M) be the maximum number of disjoint d-massive sets admissible on M. We will now give a proof of the Grigor’yan theorem [G2]. Theorem 31.2 (Grigor’yan) Let M m be a complete Riemannian manifold. The maximum number of disjoint 0-massive sets admissible on M is given by the dimension of the space of bounded harmonic functions on M, i.e., m 0 (M) = h 0 (M). Proof Note that m 0 (M) ≥ 1 because the constant function 1 is a nonnegative bounded harmonic function on M. Let i with 1 ≤ i ≤ k be a set of disjoint 370

31

Massive sets

371

0-massive sets in M. By scaling, we may assume that there are nonnegative subharmonic functions f i on i such that f i = 0 on

∂i

and sup f i = 1. i

For a fixed 1 ≤ i ≤ k, a point p ∈ M, and ρ > 0, let us solve the Dirichlet problem u ρ = 0 with boundary value ⎧ ⎪ f ⎪ ⎨ i uρ = 1 − fα ⎪ ⎪ ⎩0

B p (ρ)

on

on

∂ B p (ρ) ∩ i ,

on

∂ B p (ρ) ∩ α , for α = i,

on

∂ B p (ρ)\(∪kj=1  j ).

Since both the functions f i and 1 − f α are bounded between 0 and 1, the maximum principle implies that 0 ≤ u ρ ≤ 1 on B p (ρ). Applying the maximum principle on each  j separately, we conclude that u ρ ≥ fi

on

B p (ρ) ∩ i

and u ρ ≤ 1 − fα

on

B p (ρ) ∩ α ,

for α = i.

The fact that u ρ is bounded implies that there exists a sequence ρ j → ∞ such that uρ j → ui , where u i is a harmonic function defined on M satisfying 0 ≤ u i ≤ 1. Moreover, ui ≥ fi

on

i

and ui ≤ 1 − fα

on

α .

Clearly, if {x j } is a sequence of points in M such that f i (x j ) → 1, then u i (x j ) → 1. Similarly, if {y j } is a sequence of points such that f α (y j ) → 1, then u i (y j ) → 0.

372

Geometric Analysis

k forms a linearly independent set. We claim that the set of functions {u i }i=1 Indeed, let

u=

k 

ai u i

i=1

be a linear combination such that u is identically 0. For each 1 ≤ i ≤ k, evaluating u on a sequence of points {x j } such that f i (x j ) → 1 we conclude that ai = 0. Hence {u i } are linearly independent and k ≤ h 0 (M). Since k is arbitrary, this implies that m 0 (M) ≤ h 0 (M). To prove the reverse inequality, we may assume that m 0 (M) is finite, otherwise the proposition is automatically true. Let Mˆ be the Stone–Cˇech compactification of M. Then every bounded continuous function on M can 0 ˆ Let {i }m (M) be a set of disjoint 0-massive be continuously extended to M. i=1 sets in M. For each i ∈ {1, . . . , m 0 (M)}, let us define the set Si =

7

{xˆ ∈ Mˆ | f (x) ˆ = sup f },

where the intersection is taken over all the potential functions f of i . The fact that f is subharmonic together with the maximum principle implies that Si ⊂ Mˆ \ M. We claim that Si = ∅. In fact, for each potential function f of i , the set {xˆ | f (x) ˆ = sup f } is a closed subset of Mˆ \ M. By the compactness ˆ of M \ M, we need only to show that for any finitely many potential functions f 1 , . . . , fl of i , l 7

{xˆ | f j (x) ˆ = sup f j } = ∅.

j=1

We will argue by induction on l. It is trivially true for one potential function. Let us assume that it is true for l potential functions that l 7 j=1

{xˆ | f j (x) ˆ = sup f j } = ∅.

31

Massive sets

373

After normalizing sup f j = 1, if we define the function f = f 1 + · · · + fl , then we have {xˆ | f (x) ˆ = sup f } =

l 7

{xˆ | f j (x) ˆ = sup f j }.

j=1

Note that both f and fl+1 are potential functions of i . If 7 ˆ = sup fl+1 } = ∅, {xˆ | f (x) ˆ = sup f } {xˆ | fl+1 (x) then for sufficiently small , the sets D1 = {x ∈ M | f (x) > sup f − } and D2 = {x ∈ M | fl+1 (x) > sup fl+1 − } are disjoint. Clearly both D1 and D2 are subsets of i with the properties that ∂ D1 ∩ ∂i = ∅ and ∂ D2 ∩ ∂i = ∅ because f = fl+1 = 0 on ∂i . Also, the functions g1 = f − sup f +  and g2 = fl+1 − sup fl+1 +  are potential functions of D1 and D2 , respectively. In particular, this implies that M has m 0 (M) + 1 disjoint massive sets given by {1 , . . . , i−1 , D1 , D2 , i+1 , . . . , k0 }, which is a contradiction. Therefore, l+1 7

{xˆ | f j (x) ˆ = sup f j } = {xˆ | f (x) ˆ = sup f } ∩ {xˆ | fl+1 (x) ˆ = sup fl+1 }

j=1

= ∅, and the claim that Si is nonempty follows. We now show that for each i there exists a potential function h i of i such that ˆ = sup h i } = Si . {xˆ | h i (x) The function h i will be called a minimal potential function of i . For an arbitrary open set U in Mˆ such that Si ⊂ U , note that 2 Mˆ \U ⊂ Mˆ \ Si = {xˆ ∈ Mˆ | f (x) ˆ < sup f },

374

Geometric Analysis

where the union is over all potential functions f of i . The compactness of Mˆ \U implies that there exist finitely many potential functions f 1 , . . . , fl of i such that Mˆ \U ⊂

l 2

{xˆ | f j (x) ˆ < sup f j }.

j=1

ˆ = Let us define g = f 1 + · · · + fl , which has the property that {xˆ | g(x) sup g} ⊂ U . One may assume by scaling g that 0 ≤ g ≤ 1 on M and sup g = 1. ˆ n = 1, 2, . . . , such that Un ⊂ Now choose a sequence of open sets Un ⊂ M, ∞ Un+1 and ∩n=1Un = Si . For each Un , there exists a potential function gn of i such that 0 ≤ gn ≤ 1, sup gn = 1 and {xˆ | gn (x) ˆ = sup gn } ⊂ Un . By defining hi =

∞ 

2−n gn ,

n=1

it is clear that h i is a minimal potential function of i satisfying ˆ = sup h i } = Si . {xˆ | h i (x) From now on, h i will denote a minimal potential function of i . For a bounded subharmonic function v on M, consider the set S = {xˆ | v(x) ˆ = sup v}. We claim that S must contain some Si . Moreover, for each j, either S ∩ S j = ∅ or S j ⊂ S. In fact, let us first argue that S ∩ Si = ∅ for some i. If this is not the case, then for  > 0 sufficiently small the sets  = {x ∈ M | v(x) > sup v − } and ˜ i = {x ∈ M | h i (x) > sup h i − }  ˜ i = ∅. Clearly  ˜ i ⊂ i , and each  ˜ i is a massive set with must satisfy  ∩  potential function h i − sup h i + . Also,  is a massive set with potential function v − sup v + . Therefore ˜ 1, . . . ,  ˜ k0 } {, 

31

Massive sets

375

are m 0 (M) + 1 disjoint massive sets of M, which is impossible, hence S ∩ Si = ∅ for some i. To see that Si ⊂ S, let us consider the function w = h i + v. Note that {xˆ | w(x) ˆ = sup w} = Si ∩ S ⊂ Si . Thus, for sufficiently small  > 0, the massive set W = {x | w(x) > sup w − } ⊂ i , has a potential function given by f = w − sup w + . In particular, by extending f to be zero outside W , f is a potential function of i with {xˆ | f (x) ˆ = sup f } = {xˆ | w(x) ˆ = sup w} = Si ∩ S. The minimality of Si implies that Si ⊂ Si ∩ S, hence Si ⊂ S. This argument also shows that for any j, either S ∩ S j = ∅ or S j ⊂ S. For each i , let h i be a minimal potential function of i with ˆ = sup h i } = Si . {xˆ | h i (x) After normalization, we may assume that sup h i = 1. Using the construction discussed in the first part of this proof, there exists m 0 (M) bounded harmonic functions { f i } with the properties that 0 ≤ f i ≤ 1, fi = 1

on

Si ,

and f i = 0 on

Sα

for α = i.

The claim that h 0 (M) ≤ m 0 (M) follows if we show that any bounded harmonic function f can be written as linear combinations of the set { f i }. Too see this, we first observe that the constant function is spanned by { f i }. Indeed, if we let g=

0 (M) m

i=1

then g≥0

fi ,

376

Geometric Analysis

with g = 1 on

Si

for 1 ≤ i ≤ m 0 (M).

In fact, sup g = 1 because otherwise the set {xˆ | g(x) ˆ = sup g} 8 would be disjoint from Si . In particular, there exists a sufficiently small constant  such that the set {x | g(x) ≥ sup g − } 8 is a 0-massive set disjoint from Si , hence contradicting the definition of m 0 (M). Let us now consider h to be any nonconstant bounded harmonic function. Let S0 (+) = {xˆ | h(x) ˆ = sup h} and ˆ = inf h}. S0 (−) = {xˆ | h(x) The above argument showed that there is at least one Si such that Si ⊂ S¯0 (+) and at least one Si such that Si ⊂ S¯0 (−). Moreover S j ∩ S¯0 (+) ∪ S¯0 (−) = ∅ if S j is not contained in S¯0 (+) ∪ S¯0 (−). Let I0 (+) and I0 (−) be those is such that Si ⊂ S¯0 (+) and Si ⊂ S¯0 (−), respectively. If we define ⎛ ⎞ ⎛ ⎞   f i ⎠ + +b0 ⎝ fi ⎠ g 0 = a0 ⎝ i∈I0 (+)

i∈I0 (−)

with a0 = sup h and b0 = inf h, then g0 = a0 on Si for all i ∈ I0 (+) and g0 = b0 on Si for all i ∈ I0 (+). In particular, the harmonic function h 1 = h − g0 satisfies the properties that h 1 = 0 on

2

Si

i∈(I0 (+)∪I0 (−))

and h1 = h

on

Sj

for j ∈ / (I0 (+) ∪ I0 (−)).

31

Massive sets

377

Applying the same process to h 1 taking combinations with the remaining f j for j ∈ / (I0 (+) ∪ I0 (−)), we obtain a harmonic function h 2 such that 2 h 2 = 0 on Si i∈(I1 (+)∪I1 (−))

and h2 = h1

on

Si

for j ∈ / (I1 (+) ∪ I1 (−)),

where I1 (+) and I1 (−) are those is such that Si ⊂ {xˆ | h 1 (x) ˆ = sup h 1 } and ˆ = inf h 1 }, respectively. In particular, Si ⊂ {xˆ | h 1 (x) 2 h 2 = 0 on Si i∈(I0 (+)∪I0 (−)∪I1 (+)∪I1 (−))

and h2 = h

on

Si

for j ∈ / (I0 (+) ∪ I0 (−) ∪ I1 (+) ∪ I1 (−)).

Since m 0 (M) < ∞, we can apply this inductive procedure finitely many times and end up with a function h k that vanishes on Si for all 1 ≤ i ≤ m 0 (M). This function h k must be identically 0. Indeed, if this is not the case, then either sup h k or inf h k is not 0. Let us assume that sup h k > 0. then for some  > 0 the set given by 8

ˆ = sup h k − } {xˆ | h k (x)

is disjoint from Si and produces another 0-massive set, which is a contradiction. This proves the inequality h 0 (M) ≤ m 0 (M), 

and hence the theorem.

As pointed out earlier, there is no direct relationship between h d (M) and m d (M) for d > 0. However, there are parallel theories concerning the two numbers as demonstrated by the following theorem that mirrors Theorem 28.4 when applied to polynomial growth harmonic functions. Theorem 31.3 (Li–Wang) Let M m be a complete manifold satisfying conditions (Vμ ) and (M). For all d ≥ 1, there exists a constant C > 0 depending only on μ such that m d (M) ≤ C CM d μ−1 . Proof The proof follows the proof of Theorem 28.4. Instead of using an orthonormal basis for a finite dimensional subspace of harmonic functions,

378

Geometric Analysis

we use a set of potential functions f i with normalized L 2 -norm. Since the functions have disjoint support, they are automatically perpendicular to each other. We also do not need to change bases and consider the space K x = {u | u(x) = 0} because for any x ∈ M there is at most one f i that does not vanish on x. The rest of the argument is exactly the same.  Similarly, we also have the following finiteness theorem that mirrors Theorem 28.7. Theorem 31.4 Let M be a complete manifold satisfying the weak mean value property (WM). Suppose that the volume growth of M satisfies V p (ρ) = O(ρ μ ) as ρ → ∞ for some point p ∈ M. Then m d (M) ≤ CWM (2b + 1)(2d+μ) . The next theorem gives a sharp estimate of m d on R2 . Theorem 31.5 On R2 , m d (R2 ) ≤ 2d for all d ≥ 0. Proof Let {1 , . . . , k } be disjoint d-massive sets and {u 1 , . . . , u k } be their corresponding potential functions. Note that since ∂u i |∇u i |2 ≤ ui , ∂r i ∩B(ρ) i ∩∂ B(ρ) Schwarz inequality implies that 1 1 |∇u i |2 ≤ 2λ12 (i ∩ ∂ B(ρ)) 2λ12 (i ∩ ∂ B(ρ)) i ∩B(ρ)

i ∩∂ B(ρ)

≤ λ1 (i ∩ ∂ B(ρ)) 

+ ≤

i ∩∂ B(ρ)

i ∩∂ B(ρ)

∂u i ∂r

i ∩∂ B(ρ)

|∇u i |2 ,

∂u i ∂r

u i2

2

¯ i |2 + |∇u

=

i ∩∂ B(ρ)

ui



i ∂ B(ρ)

∂u i ∂r

2

(31.1)

31

Massive sets

379

where λ1 (i ∩ ∂ B(ρ)) denotes the first Dirichlet eigenvalue on i ∩ ∂ B p (ρ). Using the fact that π2 , A(i ∩ ∂ B(ρ))2

λ1 (i ∩ ∂ B(ρ)) ≥ we conclude that 2

k 

k 

1/2

λ1 (i ∩ ∂ B(ρ)) ≥ 2π

i=1

i=1

1 . A(i ∩ ∂ B(ρ))

(31.2)

On the other hand, when combined with the inequality 2

k ≤

k  i=1

≤ 2πρ

A(i ∩ ∂ B(ρ))

k  i=1

k  i=1

1 A(i ∩ ∂ B(ρ))



1 , A(i ∩ ∂ B(ρ))

(31.1) and (31.2) imply that k  i=1

! !i ∩∂ B(ρ)

|∇u i |2

i ∩B(ρ) |∇u i

|2

≥

k2 . ρ

(31.3)

Observing that

∂ |∇u i | = ∂r i ∩∂ B(r )





2

|∇u i |

2

i ∩B(ρ)

,

(31.3) can be written as k

∂ k2 2 ln |∇u i | ≥ . ∂r r i ∩B(ρ) i=1

Integrating both sides from ρ0 to ρ yields k !

  2 ρ i ∩B(ρ) |∇u i | 2 ! ln ≥ k ln . 2 ρ0 i ∩B(ρ0 ) |∇u i | i=1

380

Geometric Analysis

The growth assumption on the u i s asserts that k |∇u i |2 ≤ (C ρ 2d−2 V (ρ))k i=1 i ∩B(ρ)

≤ C k ρ 2kd , hence we have 2kd ln ρ + C 1 ≥ k 2 ln ρ − k 2 ln ρ0 . Letting ρ → ∞ implies that 2d ≥ k, as was to be proven.



32 The structure of harmonic maps into a Cartan–Hadamard manifold

In this chapter, we will apply the notion of massive sets to study the structure of the image of a harmonic map whose target is a hyperbolic space. In fact, Li and Wang [LW1] developed this theory which applies when the target is a strongly negatively curved Cartan–Hadamard manifold, or when it is a two-dimensional visibility manifold. Throughout this chapter we shall assume that N is a Cartan–Hadamard manifold, namely, N is simply connected and has nonpositive sectional curvature. It is well known that N can be compactified by adding a sphere at infinity S∞ (N ). The resulting compact space N¯ = N ∪ S∞ (N ) is homeomorphic to a closed Euclidean ball. Two geodesic rays γ1 and γ2 in N are called equivalent if their Hausdorff distance is finite. Then the geometric boundary S∞ (N ) is simply given by the equivalence classes of geodesic rays in N . A sequence of points {xi } in N¯ converges to x ∈ N¯ if for some fixed point p ∈ N , the sequence of geodesic rays { pxi } converges to a geodesic ray γ ∈ x. In this case, we say γ is the geodesic segment px joining p to x. Recall that a subset C in N is strictly convex if any geodesic segment between any two points in C is also contained in C. For a subset K in N , the convex hull of K , denoted by C(K ), is defined to be the smallest strictly convex subset C in N containing K . The convex hull can also be obtained by taking the intersection of all convex sets C ⊂ N containing K . When N is a Cartan–Hadamard manifold, there is only one geodesic segment joining a pair of points in N . In this case, there is only one notion of convexity, and we will simply say a set is convex when it is a strictly convex set. For the purpose of this chapter, we will need a notion of convexity for N¯ . Since a geodesic line is a geodesic segment joining the two end points in S∞ (N ), it still makes sense to talk about geodesics joining two points in N¯ . However, it is not true in general that any two points in S∞ (N ) can always be joined by a geodesic segment given by a geodesic line, as indicated 381

382

Geometric Analysis

by two nonantipodal points in S∞ (Rn ). If every pair of points in S∞ (N ) can be joined by a geodesic line in N , then N is said to be a visibility manifold. This class of manifolds was extensively studied in [EO]. A typical example of a visibility manifold is a Cartan–Hadamard manifold with sectional curvature bounded from above by a negative constant −a < 0. To remedy the situation when N is not a visibility manifold, we define a generalized notion of geodesic segment joining two points at infinity. Definition 32.1 A geodesic segment γ joining a pair of points x and y in N¯ is the limiting set of a sequence of geodesic segments {γi } in N with end points {xi } and {yi } such that xi → x and yi → y. We will denote γ by x y. Observe that if x y ∩ S∞ (N ) = {x, y}, then x y must be a geodesic line in N and hence a geodesic segment in the traditional sense. For the case of two nonantipodal points in S∞ (R2 ), the shortest arc on S1 = S∞ (R2 ) joining the two points is the geodesic segment in the sense defined above. If the two points are antipodal in S∞ (R2 ), say the north pole and the south pole, then there are infinitely many geodesic segments joining them. Each vertical line is a geodesic segment in the genuine sense. Also, both arcs on S1 joining the two poles are geodesic segment joining them. Using this definition, for a pair of points in S∞ (N ), it is possible to have more than one geodesic segment joining them. The convexity we will define will be in the sense of strictly convex. Definition 32.2 A subset C of N¯ is a convex set if for every pair of points in C, any geodesic segment joining them is also in C. Definition 32.3 For a subset A in N¯ , we define its convex hull C(A) to be the smallest convex subset of N¯ containing A. In what follows, when we say that a subset is closed, we mean that it is closed in N¯ unless otherwise noted. In general, we denote the closure for a ¯ For a given sequence of closed subsets {Ai } decreasing subset A in N¯ by A. to A, it is natural to ask whether the convex hull of Ai in N¯ decreases to the convex hull of A. For this purpose, we introduce the following definition. Definition 32.4 A Cartan–Hadamard manifold N is said to satisfy the separation property if for every closed convex subset A in N¯ and every point p not in A, there exists a closed convex set C properly containing A and separating p from A, i.e., A ⊂ C, A ∩ S∞ (N ) is contained in the interior of C ∩ S∞ (N ) and p is not in C.

32

The structure of harmonic maps into a Cartan–Hadamard manifold 383

For a two-dimensional visibility manifold or a Cartan–Hadamard manifold with constant negative curvature, it is easy to check that the separation property holds. In fact, for a point p not in the closed convex set A, pick up a point q ∈ A such that r ( p, q) = r ( p, A). Then the convexity of A and the first variation formula imply that for z ∈ A, ∠(zq, qp) ≥ π/2. Let x be the midpoint of the geodesic segment between p and q, and C = {y ∈ N : ∠(yx, x p) ≥ π/2}. Then C is closed, convex as ∂C is evidently totally geodesic and C properly separates p from A. Lemma 32.5 A Cartan–Hadamard manifold N satisfies the separation property if and only if for every closed subset A and monotone decreasing sequence 9∞ Ai = A, of closed subsets { Ai } in N¯ such that i=1 ∞ C(Ai ) = C(A). ∩i=1

Proof Suppose that N satisfies the separation property. Let { Ai } be a 9∞ decreasing sequence of closed subsets in N with i=1 Ai = A. Obviously, C(A) ⊂

∞ 7

C(Ai )

i=1

from the definition of convex hull. Assume the contrary that ∞ 7

C(Ai ) = C(A).

i=1

9∞ C(Ai ) but not in C(A). The separation Then there exists a point p ∈ i=1 property asserts that there is a closed convex subset C properly separating p from C(A). Let C  = {x ∈ N : r (x, C) ≤ } be the -neighborhood of C. For sufficiently small  > 0, C¯  also properly separates p from C(A). Since A ∩ S∞ (N ) is contained in the interior of C ∩ S∞ (N ) and Ai is decreasing to A, we conclude that for i sufficiently large, Ai ⊂ C¯  . Thus, C(Ai ) ⊂ C¯  and p ∈ C¯  , which is a contradiction. Conversely, to show that N satisfies the separation property, let A be a closed convex subset and p be a point not in A. We identify N¯ with the closed unit ball of the Euclidean space endowed with the canonical metric. Let Ai be the

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Geometric Analysis

tubular neighborhood of A of size 1/i. It is then clear that Ai is a decreasing ∞ A = A. Hence by the assumption, sequence of closed subsets with ∩i=1 i ∞ ∩i=1 C(Ai ) = C(A) = A.

The fact that p ∈ / A implies that p ∈ / C(Ai ) for sufficiently large i. By choosing C = C(Ai ), it is clear that C properly separates p from A.  According to our definition of a convex hull, it is possible that C(K ) ∩ S∞ (N ) is a much bigger set than K ∩ S∞ (N ). In fact, if we consider K to be the y-axis in R2 , then K ∩ S∞ (R2 ) consists of the two poles in S1 . However, C(K ) = R2 because every line given by x = constant is a geodesic joining the two poles of S1 . Hence, C(K ) ∩ S∞ (R2 ) = S1 . On the other hand, if we assume in addition that N satisfies the following separation property at infinity, then C(K ) ∩ S∞ (N ) = K ∩ S∞ (N ). Definition 32.6 Let N be a Cartan–Hadamard manifold. N is said to satisfy the separation property at infinity if for any closed subset A of S∞ (N ) and any point p ∈ S∞ (N )\ A, there exists a closed convex subset C in N¯ such that A is contained in the interior of C ∩ S∞ (N ) and p not in C. It is easy to check that a two-dimensional visibility manifold alway satisfies the separation property at infinity. On the other hand, upon improving a result of Anderson [An], Borb´ely [Bo] has shown that the Cartan–Hadamard manifold N has the separation property at infinity provided that its sectional curvature satisfies −Ceαr (x) ≤ K N (x) ≤ −1 for some constant C > 0 and 0 ≤ α < 1/3, where r (x) is the distance from point x to a fixed point o ∈ N . The interested reader should refer to [Bo] for a detailed proof. The following simple lemma gives a condition equivalent to the separation property at infinity. Lemma 32.7 Let N be a Cartan–Hadamard manifold. Then for every closed set K in N¯ , C(K ) ∩ S∞ (N ) = K ∩ S∞ (N ) if and only if N satisfies the separation property at infinity. Proof Assume that N satisfies the separation property at infinity. For a given closed subset K , let A = K ∩ S∞ (N ). If A = S∞ (N ), then there is nothing to prove. So we may assume this is not the case. The closeness of K implies that A is closed. Given p ∈ S∞ (N )\ A, there is a closed convex subset C such that

32

The structure of harmonic maps into a Cartan–Hadamard manifold 385

A is contained in the interior of C ∩ S∞ (N ) and p is not in C. In particular, we conclude that sup r (x, C) = R < ∞. x∈K

Let us consider the R-neighborhood, C R = {x ∈ N : r (x, C) ≤ R}, of C. Then C¯ R is a closed convex subset and K ⊂ C¯ R . Moreover, C ∩ S∞ (N ) = C¯ R ∩ S∞ (N ). / C(K ) ∩ S∞ (N ). Therefore, C(K ) ⊂ C¯ R and p is not in C¯ R . In particular, p ∈ This shows that C(K ) ∩ S∞ (N ) = A. Conversely, to show that N satisfies the separation property at infinity, let A be a closed subset of S∞ (N ) and point p ∈ S∞ (N )\ A. Then there exists a closed subset K ⊂ S∞ (N ) such that A is in the interior of K and p ∈ / K. / C and A Let C = C(K ) and by the assumption C ∩ S∞ (N ) = K . Thus, p ∈ is contained in the interior of C ∩ S∞ (N ). Thus, N satisfies the separation property at infinity and the lemma is proved.  Theorem 32.8 (Li–Wang) Let M be a complete Riemannian manifold such that the dimension of the space of bounded harmonic functions H0 (M) is h 0 (M). Let u : M → N be a harmonic map from M into a Cartan–Hadamard manifold, N n . Let A = u(M) ∩ S∞ (N ), where S∞ (N ) = Sn−1 is the geometk ric boundary of N . Then there exists a set of points {yi }i=1 ⊂ u(M) ∩ N with 0 k ≤ h (M) such that  7  2 k , C Aj u(M) ⊂ {yi }i=1 j

where {A j } is a monotonically decreasing sequence of closed subsets of N¯ 9 properly containing A and j A j = A. In addition, if we assume that N has the separation property, then   2 k u(M) ⊂ C A {yi }i=1 . Proof

Let us pick a point y0 ∈ u(M). If 7

 u(M) ⊂ C A j ∪ {y0 } , j

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Geometric Analysis

then we are done. Hence we may assume that there exists an A j properly containing A such that the set

 u(M)\C A j ∪ {y0 } = ∅. Moreover, one can easily check that u(M)\C(A j ∪ {y0 }) is bounded in N . Since u is a harmonic map and the distance function r (y, C(A j ∪ {y0 })) to the set C(A j ∪ {y0 }) is convex, the composition function

 f (x) = r (u(x), C A j ∪ {y0 } ) is a bounded nonconstant subharmonic function on M. Thus, f attains its maximum at every point of some supremum set S1 . In particular, for xˆ1 ∈ S1 ˆ a subnet of {u(xα )} converges to and a net {x α } in M converging to xˆ1 in M, y1 ∈ N . Again, if

 u(M) ⊂ ∩ j C A j ∪ {y0 ∪ y1 } , then the theorem is true, otherwise by choosing a larger j if necessary, the function 

 g(x) = r u(x), C A j ∪ {y0 ∪ y1 } is a bounded nonconstant subharmonic function on M. If g achieves its maxiˆ = sup g for xˆ ∈ S1 . In particular, this implies mum on S1 , then g(x) sup g = g(xˆ1 ) 

 = r y1 , C A j ∪ {y0 ∪ y1 } = 0, which is impossible. Therefore we may assume g achieves its maximum on S2 . For a net {x α } in M converging to a point xˆ2 in S2 , there exists a subnet of {u(xα )} that converges to y2 ∈ N . Suppose that we have chosen l points y1 , . . . , yl described in the above procedure. If 7

 u(M) ⊂ C A j ∪ {yi }li=1 , j

then we are done, otherwise by choosing a larger j if necessary, we define the function    2 l {yi }i=1 , h(x) = r u(x), C A j

32

The structure of harmonic maps into a Cartan–Hadamard manifold 387

which is a bounded nonconstant subharmonic function on M. We claim that 8 h cannot achieve its maximum on li=1 Si . Indeed, if it does, then h must achieve its maximum at every point on Si for some 1 ≤ i ≤ l. Thus using an argument similar to that used before 

 h(xˆi ) = r yi , C A j ∪ {yi }li=1 = 0, which is a contradiction, hence h achieves its maximum on some Si with i > l. We may assume that i = l + 1. Let us pick a point xˆl+1 ∈ Sl+1 and a net {x α } converging to xˆl+1 . Suppose yl+1 is an accumulation point of the net {u(xα )}. It is clear that this process must stop after at most m 0 (M) steps since there are only m 0 (M) massive sets. In particular, there exist k points {y1 , . . . , yk } with k ≤ m 0 (M) such that  7  2 k u(M) ⊂ . C Aj {yi }i=1 j

Moreover, yi ∈ u(M), and the proof of the first statement is completed. The second statement follows from Lemma 32.5  Corollary 32.9 If h 0 (M) = 1 and N is a Cartan–Hadamard manifold, then every bounded harmonic map M → N must be constant. Theorem 32.10 Let M be a complete manifold and we denote the space spanned by all positive harmonic functions on M by H+ (M). Suppose M has the property that h 0 (M) = dim H+ (M) < ∞. Assume that u : M → N is a harmonic map from M into a Cartan–Hadamard manifold N which either is a two-dimensional visibility manifold or has strongly negative sectional curvature, and that A = u(M) ∩ S∞ (N ) consists of at most one point. Then the set A is necessarily empty, and there exists a set k ⊂ u(M) ∩ N with k ≤ h 0 (M) such that of k points {yi }i=1

 k u(M) ⊂ C {yi }i=1 . In particular, if M does not admit any nonconstant positive harmonic functions, then every such harmonic map must be a constant map. Proof

Theorem 32.8 implies that

 k u(M) ⊂ C A ∪ {yi }i=1

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Geometric Analysis

k in N with k ≤ h 0 (M). If A contains exactly for some set of k points {yi }i=1 one point a, let γ be a geodesic line on (−∞, +∞) such that its restriction to (0, +∞) represents a. For each yi , there exists a unique point γ (ti ) such that r (yi , γ ) = r (yi , γ (ti )). Choose a point p = γ (t0 ) with t0 < ti for i = 1, 2, . . . , k. Let δ be the geodesic ray given by the restriction of γ onto (t0 , +∞), and denote the Busemann function associated to δ by β. Recall that if δ is parametrized by arc-length, then

β(y) = lim (t − r (y, δ(t))). t→∞

We claim that there exists a constant c such that r (y, p) ≤ β(y) + c 

k . In fact, by the convexity of the function r (y, γ ) and for y ∈ C {a} ∪ {yi }i=1 the choice of p, one easily checks that r (y, δ) = r (y, γ ) ≤ max r (yi , γ ) 1≤i≤k

=c

 k . Therefore, if we let y¯ ∈ δ be the point such that for any y ∈ C {a} ∪ {yi }i=1 r (y, δ) = r (y, y¯ ), then r (y, p) ≤ r (y, δ) + r ( y¯ , p) ≤ c + β( y¯ ) ≤ 2c + β(y). This justifies the claim that r (u(x), p) ≤ β(u(x)) + c for all x ∈ M. Since u is a harmonic map and N is a Cartan–Hadamard manifold, the function r (u(x), p) is subharmonic and the function β(u(x)) + c is superharmonic. The sub–super solution method yields a harmonic function f (x) on M such that r (u(x), p) ≤ f (x) ≤ β(u(x)) + c. Therefore f is an unbounded positive harmonic function on M, contradicting to our assumption that there is no such function. In conclusion, A must be empty and

32

The structure of harmonic maps into a Cartan–Hadamard manifold 389 k C(u(M)) = C({yi }i=1 ).

This proves our first statement. The second part of the theorem follows from  the first part by taking h 0 (M) = 1. Notice that the horoball of a visibility manifold intersects the geometric boundary at exactly one point (see [BGS]). Thus, we obtain the following Liouville type theorem which partially generalizes the results in [Sh] and [T]. Corollary 32.11 Suppose M satisfies dim H+ (M) = 1. Assume that N is either a two-dimensional visibility manifold or has strongly negative sectional curvature. Then every harmonic map from M into a horoball of N must be constant. Recall that a manifold is parabolic if it does not admit a positive Green’s function. Since a parabolic manifold has no massive subsets and every positive harmonic function must be constant, we can apply Theorem 32.8 to this case and obtain the following corollary. Corollary 32.12 Let u be a harmonic map from a parabolic manifold M into N n . Assume that N is either a two-dimensional visibility manifold or has strongly negative sectional curvature. Then u(M) ⊂ C(A), where A = u(M) ∩ S∞ (N ). Proof

In this case, we have h 0 (M) = 1, hence Theorem 32.8 implies that u(M) ⊂ C(A ∪ {y})

for some y ∈ u(M) ∩ N . Let us assume the contrary that u(M) is not contained in C(A). In particular, the parabolicity of M implies that the function r (u(x), C(A)) is unbounded. Lemma 32.5 then asserts that u(M)\C(W ) is nonempty for some open set W ⊂ S∞ (N ) which properly contains A. Let us consider the function f (x) = r (u(x), C(W )), which is a nonconstant, nonnegative, bounded subharmonic function on M. However, the parabolicity assumption on M implies that such a function does not exist. This completes our proof.  Theorem 32.13 Let M be a complete manifold such that the maximum number of disjoint d-massive sets of M is m d (M). Suppose u : M → N is a harmonic map from M into N , and N satisfies the separation property at infinity. Assume

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Geometric Analysis

that there exists a point o ∈ N such that r (u(x), o) = O(r d (x)) as x → ∞. Then 

k A = u(M) ∩ S∞ (N ) = {ai }i=1

with k  ≤ m d (M) − m 0 (M). If, in addition, N is either a two-dimensional visibility manifold or has strongly negative sectional curvature, then there exist k points {y j }kj=1 ⊂ u(M) ∩ N with k  + k ≤ m d (M) such that    u(M) ⊂ C {ai }ki=1 ∪ {y j }kj=1 . Proof Since a 0-massive set is always d-massive, we have m 0 (M) ≤ m d (M). Theorem 32.8 implies that there exist k points {y j }kj=1 ⊂ u(M) ∩ N with k ≤ m 0 (M), such that  7  u(M) ⊂ C A ∪ {y j }kj=1 . >0  If A contains at least k  points, then there exist disjoint open sets {Ui }ki=1 in N¯ such that Ui ∩ A = ∅ for i = 1, 2, . . . , k  . Since N is assumed to satisfy the separation property at infinity, Lemma 32.7 yields that u(M) is not a subset of   . In particular, the function C ( N¯ \Ui ) ∪ {y j }k

j=1

   f i (x) = r u(x), C ( N¯ \Ui ) ∪ {y j }kj=1 is not identically zero on u −1 (Ui ) and sup f i = ∞. Clearly, f i = 0 on the boundary of u −1 (Ui ) and f i (x) = O(r d (x)). This implies that each set u −1 (Ui ) is d-massive but not massive. In particular, since they are disjoint, k  ≤ m d (M) − m 0 (M). Thus A has at most m d (M) − m 0 (M) points, and the first conclusion follows. If, in addition, N either is a two-dimensional visibility manifold or has strongly negative sectional curvature, then Theorem 32.8 yields that    u(M) ⊂ C {ai }ki=1 ∪ {y j }kj=1 , and the estimate k  + k ≤ m d (M) follows from the argument. This completes our proof. 

32

The structure of harmonic maps into a Cartan–Hadamard manifold 391

Combining Theorem 32.13 with the estimates on m d (M) given by Theorem 31.3 and Theorem 31.4, we have the following corollary. Corollary 32.14 Let M be a complete manifold satisfying condition (M) and its volume growth V p (R) = O(R μ ) for some point p ∈ M. Suppose N is a Cartan–Hadamard manifold satisfying either of the following conditions: (1) it has strongly negative sectional curvature; (2) it is a two-dimensional visibility manifold. Let u : M → N be a harmonic map and suppose that there exists a point o ∈ N such that r (u(x), o) = O(r d (x)) as x → ∞. Then there exist sets of k = u(M) ∩ S∞ (N ) and k points {y j }kj=1 ⊂ u(M) ∩ N with k  points {ai }i=1  (2d+μ) k + k ≤ λ3 such that   k ∪ {y j }kj=1 . u(M) ⊂ C {ai }i=1 If M is further assumed to have property (Vμ ), then we have k  + k ≤ Cd μ−1 . We would like to remark that though Lemma 32.5 states that separation property is necessary and sufficient to conclude ∩∞ j=1 C(A j ) = C(A), after careful examination of the proof of Theorem 32.8, we only need to use the 9 fact that ∞ j=1 (A j ) is a bounded distance from C(A). With this in mind, using a theorem of Anderson and Borb´ely, Theorem 32.8 and hence all consequential theorems of this chapter are valid when N is a strongly negatively curved manifold. A complete treatment can be found in [LW1].

Appendix A Computation of warped product metrics

Let M m = R × N m−1 be the product manifold endowed with the warped product metric 2 ds M = dt 2 + f 2 (t) ds N2 ,

where ds N2 is a given metric on N . Our purpose is to compute the curvature on M with respect to this warped product metric. Let {ω˜ 2 , . . . , ω˜ m } be an orthonormal coframe on N with respect to ds N2 . m If we define ω1 = dt and ωα = f (t) ω˜ α for 2 ≤ α ≤ m, then the set {ωi }i=1 2 . The first structural forms an orthonormal coframe of M with respect to ds M equations assert that dωi = ωi j ∧ ω j , where ωi j are the connection 1-forms with the property that ωi j = −ω ji . On the other hand, direct exterior differentiation yields dω1 = 0 and dωα = f  ω1 ∧ ω˜ α + f ω˜ αβ ∧ ω˜ β = −(log f ) ωα ∧ ω1 + ω˜ αβ ∧β , where ω˜ αβ are the connection 1-forms on N and f  is the derivative of f with respect to t. Hence we conclude that the connection 1-forms on M are 392

Appendix A Computation of warped product metrics

393

given by ω1α = −ωα1 = (log f ) ωα

(A.1)

and ωαβ = ω˜ αβ .

(A.2)

The second structural equations also assert that dωi j − ωik ∧ ωk j = 12 Ri jkl ωl ∧ ωk , where Ri jkl is the curvature tensor on M. Exterior differentiation of (A.1) yields

 dω1α = (log f ) ω1 ∧ ωα + (log f ) −(log f ) ωα ∧ ω1 + ω˜ αβ ∧ ωβ . Hence substituting (A.1) and (A.2) into the above identity yields   dω1α − ω1β ∧ ωβα = (log f ) + ((log f ) )2 ω1 ∧ ωα . Also, exterior differentiation of (A.2) gives dωαβ = d ω˜ αβ , and dωαβ − ωα1 ∧ ω1β + ωαγ ∧ ωγβ = d ω˜ αβ − ω˜ αγ ∧ ω˜ γβ + ((log f ) )2 ωα ∧ ωβ ˜ αβγ τ ω˜ τ ∧ ω˜ γ + ((log f ) )2 ωα ∧ ωβ = 12 R ˜ αβγ τ f −2 ωτ ∧ ωγ + ((log f ) )2 ωα ∧ ωβ , = 12 R ˜ αβγ τ is the curvature tensor on N . In particular, the sectional curvature where R of the 2-plane section spanned by e1 and eα is given by   K (e1 , eα ) = − (log f ) + ((log f ) )2 , and the sectional curvature of the 2-plane section spanned by eα and eβ is given by K (eα , eβ ) = f −2 K˜ (eα , eβ ) − ((log f ) )2 ,

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where K˜ is the sectional curvature of N . Moreover the curvature tensor is given by ⎧ if j = α, k = 1, ⎨ (log f ) + ((log f ) )2 R1α jk = −(log f ) − ((log f ) )2 if j = 1, k = α, ⎩ 0 otherwise and

#

Rαβi j =

˜ αβγ τ + ((log f ) )2 (δατ δβγ − δαγ δβτ ) f −2 R 0

if i = γ , j = τ, otherwise.

The Ricci curvature is then given by  R1α jα R1 j = α

  = −(m − 1) (log f ) + ((log f ) )2 δ1 j and Rαβ =



(A.3)

Rαγβγ + Rα1β1

γ

  ˜ αβ − (log f ) + (m − 1)((log f ) )2 δαβ , = f −2 R ˜ αβ is the Ricci tensor on N . where R

(A.4)

Appendix B Polynomial growth harmonic functions on Euclidean space

In this appendix, we will determine all polynomial growth harmonic functions in Rm . We will also compute their dimensions. Recall that we denote the space of polynomial growth harmonic functions of order at most d on a complete manifold M by Hd (M) and its dimension is denoted by h d (M). Corollary 6.6 asserts that if M has nonnegative Ricci curvature, then h d (M) = 1 for all d < 1. Moreover, if M = Rm , with rectangular coordinates given by {x1 , . . . , x m }, and if f ∈ Hd (Rm ), then the function ∂ f /∂ xi is also harmonic in Hd−1 (Rm ) by Corollary 6.6. Hence if d < 2, then ∂ f /∂ xi must be constant functions for all xi . This implies that f must be a linear function spanned by the coordinate functions {x i } and the constant functions. Therefore we conclude that h d (Rm ) = m + 1

for

1 ≤ d < 2.

We now claim that any f ∈ Hd (Rm ) must be a harmonic polynomial of degree at most d. To see this, we argue by induction and assume that this is true for some d ≥ 2. To prove that this is also valid for d + 1, we consider f ∈ Hd+1 (Rm ). The above observation asserts that ∂ f /∂ xi ∈ Hd (Rm ) for all 1 ≤ i ≤ m. The induction hypothesis implies that each ∂ f /∂ xi = pi , where pi a harmonic polynomial of degree at most d. On the other hand, f (x1 , 0, . . . , 0) =

x1

p1 (t1 , 0, . . . , 0) dt1 + f (0, . . . , 0)

0

395

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Geometric Analysis

implies that f (x 1 , 0, . . . , 0) is a polynomial of at most degree d + 1 in x 1 . By using the formula f (x1 , . . . , x k , 0, . . . , 0) xk pk (x1 , . . . , xk−1 , tk , 0 . . . , 0) dtk + f (x1 , . . . , xk−1 , 0, . . . , 0) = 0

we can argue inductively that f (x 1 , . . . xm ) is a polynomial of degree at most d and the claim is proved. To determine h d (Rm ) for each d ∈ Z+ , we let x = x1 and y = (x2 , . . . , xm ). For any f ∈ Hd (Rm ), we can write f (x, y) =

d 

ad−i (y) x i ,

i=0

where a j are polynomials in the variables y = (x2 , . . . x m ) of degree at most j. Since f is harmonic, by separation of variables we have 0 = f = =

∂2 f + y f ∂x2 d  i=2

=

d−2 

ad−i i(i − 1) x i−2 +

d 

 y ad−i x i

i=0

(ad−2−i (i + 2)(i + 1) +  y ad−i ) x i +  y a1 x d−1 +  y a0 x d .

i=0

Note that since a0 and a1 are of degree 0 and 1, respectively, the last two terms vanish. Hence we conclude that −ad−2−i (i + 2)(i + 1) =  y ad−i for all 0 ≤ i ≤ d − 2. This gives an inductive formula for all the coefficients once ad and ad−1 are fixed and arbitrary polynomials in y of degree at most d and d − 1, respectively. Let P d (Rm−1 ) be the space of all polynomial of degree at most d defined on Rm−1 , and pd (Rm−1 ) = dim P d (Rm−1 ), Clearly, h d (Rm ) = p d (Rm−1 ) + pd−1 (Rm−1 ).

(B.1)

Appendix B Polynomial growth harmonic functions on Euclidean space 397 We now claim that  pd (Rm ) =

m+d d

 .

Again, to see this we write any polynomial of degree at most d as d 

f (x, y) =

ad−i (y) x i .

i=0

Obviously, each a j ∈ P j (Rm ) and hence p d (Rm ) =

d 

p i (Rm−1 ).

i=0

When m = 1, we see that  p (R ) = d + 1 = d

1

 d +1 . d

Using induction on m we have p d (Rm ) =

 d   m +i −1 . i i=0

It remains to prove that 

m+d d

 =

 d   m +i −1 . i i=0

One can see this by induction on d. Obviously, the identity is true for d = 1 since   m+1 =m+1 1 and 

m 0

 =1

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by convention. By the induction hypothesis, we have      d   m +i −1 m+d −1 m+d −1 = + i d −1 d i=0

=

(m + d − 1)! (m + d − 1)! + m! (d − 1)! (m − 1)! d!

(m + d − 1)! (d + m) m! d!   m+d = , d

=

and the identity is proved. Applying this to (B.1), we conclude that     m+d −1 m+d −2 + . h d (Rm ) = d d −1

(B.2)

Note that by Sterling’s formula h d (Rm ) ∼

2 d m−1 (m − 1)!

(B.3)

as d → ∞. We also note that d  i=0

   d  d   m +i −1 m +i −2 + h (R ) = i i −1 i

m

i=0

i=0

= h (R d

∼ as d → ∞.

m+1

2 m d m!

) (B.4)

References

[A] [An] [B] [BC] [BDG] [BGS] [Bm] [Bo] [Br] [Bu] [C]

[CCM] [Cg1] [Cg2] [Cg3] [CG1] [CG2] [CgY]

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402 [LT4] [LT5] [LT6] [LW1] [LW2] [LW3] [LW4] [LW5] [LW6] [LW7] [LW8] [LY1] [LY2] [Lz] [M] [MS] [NT] [O] [R] [Ro] [S] [SC] [Sh] [Si] [St] [STW] [SY1]

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Index

(VD), 203 CM1 , 361 CM , 341 CP ,φ (B p (ρ)), 207 CP (), 203 CV , 341 CSD , 98, 217 CVD,φ (B p (ρ)), 207 CVD (), 203 CWM , 345 I Dα (M), 86 I Nα (M), 86 L 1 mean valued inequality (M1 ), 361 L α subharmonic function, 68 L q harmonic functions, 281, 349 N  (M), 242 N 0 (M), 242 Ni (M), 242 Ni0 (M), 242 S Dα (M), 87 S Nα (M), 87 H (M), 73 Hd (M), 326 H+ (M), 387 K (M), 262 K(M), 264 K0 (M), 262 Lq (E), 281 λ1 (M), 91 μ1 (M), 91 Spec(), 267 d-massive set, 370 kth Neumann eigenfunction, 116 kth Dirichlet eigenfunction, 113 kth Dirichlet eigenvalue, 113 kth Neumann eigenvalue, 116 k  (M), 262

404

k(M), 264 k 0 (M), 262 m d (M), 370 pth Betti number, 29 adjoint operator, 26 area, boundary of geodesic ball, 14 barrier function, 242 Betti number, 28 Bochner Laplacian, 26 Bochner technique, 19 Bochner–Weitzenb¨ock formulas, 19 Buseman function, 35 C S N , 115 Cartan’s first structural equations, 19 Cartan’s second structural equations, 19 Cartan–Hadamard manifold, 381 coarea formula, 87 Cohn–Vossen–Hartman formula, 300 commutation formula, 21 comparison theorem, heat equation, 142 compatible with metric, 1 conformally equivalent, 311 connection 1-forms, 19 constant mean curvature, 82 contractive on L 1 , 170 contractive on L ∞ , 169 contractive on L p , 171 convex, 53, 131 convex hull C(A), 382 convex set, 382 covariant derivatives, 22 covariant differentiation, p-forms, 27 curvature operator, 29 curvature tensor, 2, 19 cut-locus, 10

Index de-Rham decomposition, 29 diameter, 14 Dirichlet α-isoperimetric constant, 86 Dirichlet α-Sobolev constant, 87 Dirichlet eigenfunction, 37 Dirichlet eigenvalue, 37 Dirichlet heat kernel, 108 Dirichlet Laplacian, 36 distance function, 32 dual 1-form, 31 dual coframe, 19 Duhamel principle, 106 eigenfunctions, 97 eigenvalue comparison theorem, 36 eigenvalues, 97 end, 241 ends, 36 essential spectrum μe (M), 267 exterior p-form, 21 finite index, 321 first nonzero eigenvalue, 37 first variation of area, 3 fundamental solution, 96 Gauss lemma, 10 geodesic, 7 geodesically star-shaped, 157 geometric boundary, 381 gradient estimate, 57 greatest lower bound for the L 2 -spectrum, 63 Green’s function, 189 half-space model, 67 harmonic p-form, 29 harmonic 1-form, 28 harmonic function, polynomial growth, 395 harmonic functions, 57 harmonic functions, linear growth, 76 Harnack type inequality, 62 heat equation, 96 heat kernel, 96 Hessian, 20 Hodge decomposition, 29 homology group, 84 horoball, 389 hyperbolic m-space, 67 induced metric, 77 infimum of the spectrum, 63 infimum of the spectrum of , 267 infinite volume, 17 infinitesimal generator, 30 injectivity radius, 110 isoperimetric, 86 isoperimetric inequalities, 86

405

kernel function, 96 Killing vector field, 30 Laplacian, 20 Laplacian, p-forms, 23 line, 34 Liouville property, 349 Marcinkiewicz interpolation theorem, 117 maximum principle method, 57 mean curvature vector, 2 mean value inequality, 68 mean value inequality (M), 341 mean value inequality at infinity, 327 measured Neumann Poincar´e inequality, 204, 207 measured Neumann Sobolev inequality, 208 metric, 1 minimal barrier function, 257 minimal Green’s function, 248 minimal heat kernel, 128 minimal hypersurfaces, 306 Myers’ theorem, 14 Nash–Mosers Harnack inequality, 216 Neumann α-isoperimetric constant, 86 Neumann α-Sobolev constant, 87 Neumann boundary condition, 48 Neumann heat equation, 105 Neumann heat kernel, 105 Neumann Poincar´e inequality, 203 nonparabolic, 241 normal variations, 8 normalized Sobolev inequality, 217 orthonormal frame, 19 oscillation, 195 parabolic, 241 parabolic equations, 122 parabolic Schr¨odinger equation, 57 parabolicity, 247 parallel, 27 Poincar´e inequality, 71 polynomial growth harmonic functions, 326 positive solutions, 134 potential function, 63, 370 pseudo-differential operator, 118 quasi-isometric, 239 quasi-isometric invariant, 247 ray, 34 Rayleigh principle, 38 Rayleigh quotient, 299 Ricci curvature, 2

406 Ricci identity, 21 Riemannian connection, 1 Riemannian manifold, 1 scaled local λ1 bound, 203 second covariant derivative, 20 second fundamental form, 2 second variational formula, 4 second variational formula, length, 6 sectional curvature, 2 semi-group property, 112 separation property, 382 separation property at infinity, 384 Sobolev constants, 86 Sobolev inequality, 98 Sobolev type inequalities, 86 space form, 12 spherical harmonic, 65 stability inequality, 8, 306 stable, 8 stable minimal hypersurface, 306 star operator, 23 strictly convex, 381

Index subadditive operator, 117 subharmonic functions, 68 third covariant derivative, 21 torsion free, 1 unit (m − 1)-sphere, 16 universal covering, 29 variational vector field, 4 visibility manifold, 382 Vitali covering, 212 volume doubling property, measure, 207 volume comparison condition (Vμ ), 341 volume comparison theorem, 10 volume doubling property, 203 volume entropy, 268 volume growth, 16 volume, geodesic ball, 14 warped product, 285, 286, 392 weak ( p, q) type, 117 weak mean value inequality (WM), 345