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 +
b
ai1 ... jb ... ka ... i p−1 ω jb ib ∧ ωka ia ,
b>a
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
where σ (i 1 , . . . , i p+1 ) denotes a permutation of the set (i1 , . . . , i p+1 ) and sgn(σ ) is the sign of σ. Recall that if ω is a p-form then δω = (−1)m( p+1)+1 ∗ d ∗ ω. The linear operator ∗ : p (M) → m− p (M) is determined by ∗(ωi p ∧ . . . ∧ ωi1 ) = sgn(σ (I, I c )) ωim ∧ . . . ∧ ωi p+1 , where σ (I, I c ) denotes a permutation by sending (i p , . . . i 1 , i m , . . . , i p+1 ) → (1, . . . , m). Let us now define β = ∗ω = ai1 ... i p sgn(σ (I, I c )) ωim ∧ . . . ∧ ωi p+1 .
24
Geometric Analysis
By setting bk1 ... km− p = sgn(σ (I, I c )) ai1 ... i p with K = (k1 , . . . , km− p ) = (i p+1 , . . . , i m ) = I c , we can write β = bK ωK and dβ = b K , j ω j ∧ ω K = (dbk1 ... km− p + bk1 ... jθ ... km− p ω jθ kθ ) ∧ ω K for 1 ≤ θ ≤ m − p. On the other hand, we also have dβ = sgn(σ (I, I c )) dai1 ... i p ∧ ωim ∧ . . . ∧ ωi p+1 + bk1 ... jθ... km− p ω jθ kθ ∧ ω K = sgn(σ (I, I c )) ai1 ... i p , j ω j ∧ ωim ∧ . . . ∧ ωi p+1 − sgn(σ (I, I c )) ai1 ... jα ... i p ω jα iα ∧ ωim ∧ . . . ∧ ωi p+1 + bk1 ... jθ ... km p ω jθ kθ ∧ ωim ∧ . . . ∧ ωi p+1 = sgn(σ (I, I c )) ai1 ... i α ... i p ,i α ωiα ∧ ωim ∧ . . . ∧ ωi p+1 − sgn(σ (I, K )) ai1 ... jα ... i p ω jα iα ∧ ωim ∧ . . . ∧ ωi p+1 + bk1 ... jθ ... km− p ω jθ kθ ∧ ωim ∧ . . . ∧ ωi p+1 . However, if jθ = i α for some 1 ≤ α ≤ p, this means that jθ ∈ I c = K . In this case, we have either bk1 ... jθ ... km− p = 0 when
jθ = kθ ,
or ω jθ kθ = 0 when
jθ = kθ .
If jθ = i α for some α, then we have bk1 ... jθ ... km− p = sgn(σ (i p , . . . , kθ , . . . , i1 , km− p , . . . , jθ , . . . , k1 )) ai1 ... αth slot
kθ ... i p αth slot
.
3 Bochner–Weitzenb¨ock formulas
25
In particular, bk1 ... jθ ... km− p ω jθ kθ = −sgn(σ (i p , . . . , i 1 , i m , . . . , im− p )) ai1 ...
kθ ... i p αth slot
ωiα kθ .
Using the skew symmetry ωiα kθ = − ωkθ iα , we conclude that dβ = sgn(σ (I, I c )) ai1 ... i α ... i p ,i α ωiα ∧ ωim ∧ . . . ∧ ωi p+1 , hence ∗dβ = sgn(σ (I, I c )) sgn(σ (i α , i m , . . . , i p+1 , i p , . . . , iˆα , . . . , i 1 )) × ai1 ... iα ... i p ,iα ωi p ∧ . . . ∧ ωˆ iα ∧ . . . ∧ ωi1 = (−1)m−α+ p(m− p) ai1 ... i p ,iα ωi p ∧ . . . ∧ ωˆ iα ∧ . . . ∧ ωi1 , and δω =
(−1)α+ p
2 +1
ai1 ... iα ... i p ,i α ωi p ∧ . . . ∧ ωˆ iα ∧ . . . ∧ ωi1 .
1≤α≤ p
Computing directly gives −ω = δ(ai1 ... i p , j ω j ∧ ωi p ∧ . . . ∧ ωi1 ) + d[(−1)α+ p = (−1)α+( p+1)
2 +1
2 +1
ai1 ... i p ,i α ωi p ∧ . . . ∧ ωˆ iα ∧ . . . ∧ ωi1 ]
ai1 ... i p , jiα ω j ∧ ωi p ∧ . . . ∧ ωˆ iα ∧ . . . ∧ ωi1
+ (−1)( p+1)+( p+1) + (−1)α+ p
2 +1
2 +1
ai1 ... i p , j j ωi p ∧ . . . ∧ ωi1
ai1 ... i p ,iα j ω j ∧ ωi p ∧ . . . ∧ ωˆ iα ∧ . . . ∧ ωi1 .
However, (3.3) implies that ai1 ... i p ,iα j = ai1 ... i p , jiα + Rkβ iβ iα j ai1 ... kβ ... i p , hence ω = (−1)α+ p Rkβ iβ iα j ai1 ... kβ ... i p ω j ∧ . . . ∧ ωˆ iα ∧ . . . ∧ ωi1 2
+ ai1 ... i p , j j ωi p ∧ . . . ∧ ωi1 = −Rkβ iβ jα iα ai1 ... kβ ... i p ωi p ∧ . . . ∧ ω jα ∧ . . . ∧ ωi1 + a I, j j ω I . If we write E(ω) = Rkβ iβ jα iα ai1 ... kβ ... i p ωi p ∧ . . . ∧ ω jα ∧ . . . ∧ ωi1
26
Geometric Analysis
and we define the Bochner Laplacian by ∇ ∗ ∇ω = a I, j j ω I , then ω = ∇ ∗ ∇ω − E(ω). Remark 3.1 If ω = f is a smooth function, then f =
m
f ii .
i=1
Remark 3.2 If ω = ai ωi is a 1-form, then E(ω) = Rki ji ak ω j = R jk ak ω j . Lemma 3.3 If M is compact, the operator δ is the adjoint operator to d, i.e., dω, η = ω, δη M
for all ω ∈ Proof
p−1 (M)
and η ∈
M
p (M).
Note that by the definition of ∗, we have
∗ ∗ (ωi p ∧ . . . ∧ ωi1 ) = ∗(sgn(σ (I, I c )) ωim ∧ . . . ∧ ωi p+1 = sgn(σ (I c , I )) sgn(σ (I, I c )) ωi p ∧ . . . ∧ ωi1 . On the other hand, it is clear that sgn(σ (I, I c )) = (−1)(m− p) p sgn(σ (I c , I )), which yields the formula that ∗ ∗ = (−1)(m− p) p on p-forms. We also observe that
ωi p ∧ . . . ∧ ωi1 ∧ ∗ ωi p ∧ . . . ∧ ωi1 = sgn(σ (I, I c )) ωi p ∧ . . . ∧ ωi1 ∧ ωim ∧ . . . ∧ ωi p+1 = ω1 ∧ . . . ∧ ωm . Also by linearity, if ω = a I ω I and θ = b I ω I , then ω ∧ ∗θ = a I b I ω1 ∧ . . . ∧ ωm = ω, θ d V.
3 Bochner–Weitzenb¨ock formulas
27
Let us now consider ω ∈ p−1 (M) and η ∈ p (M), then dω, η = dω ∧ ∗η = d (ω ∧ ∗η) + (−1) p ω ∧ d ∗ η = d (ω ∧ ∗η) + (−1) p (−1)(m− p+1)( p−1) ω ∧ ∗ ∗ d ∗ η = d (ω ∧ ∗η) + ω ∧ ∗δη = d (ω ∧ ∗η) + ω, δη. Integrating both sides over M the lemma follows from Stokes’ theorem. Lemma 3.4 Let ω =
I
a I ω I be a p-form on M. Then
|ω|2 = 2ω, ω + 2|∇ω|2 + 2E(ω), ω. Proof
The norm of ω is given by |ω|2 =
a 2I .
I
Let us choose an orthonormal coframe in a neighborhood of x ∈ M by parallel translation of an orthonormal frame {e1 , . . . , em } at the point x. Hence at x, we have ωi j = 0. Moreover, ∇ei e j = 0 along the geodesic tangent to ei which implies that ∇ei ∇ei e j = 0 at x. Direct differention yields |ω|2 = a 2I
jj
= 2a I (a I ) j j + 2((a I ) j )2 . Note that, in general, (a I ) j = a I, j and (a I ) j j = a I, j j , since the terms on the left denote differentiations of the function a I , while the terms on the right denote covariant differentiations of the p-form. However, by the choice of our frame, (a I ) j = da I (e j ) = a I, j − ai1 ... jα ... i p ω jα iα (e j ) = a I, j
28
Geometric Analysis
at the point x. Similarly, we also have a I, j j = dai1 ... i p , j (e j ) = d[da I (e j ) + ai1 ... jα ... i p ω jα iα (e j )](e j ) = (a I ) j j + ai1 ... jα ... i p e j (ω jα iα (e j )) at x. On the other hand, e j (ω jα iα (e j )) = e j ∇e j e jα , eiα = ∇e j ∇e j e jα , eiα + ∇e j e jα , ∇e j eiα =0 at x. Therefore, |ω|2 = 2∇ ∗ ∇ω, ω + 2|∇ω|2 = 2ω, ω + 2|∇ω|2 + 2E(ω), ω at the point x. Now we observe that both sides of the equation are globally defined, hence this formula is valid on M. Theorem 3.5 (Bochner [B]) Let M be a compact m-dimensional Riemannian manifold without boundary. Suppose M has nonnegative Ricci curvature, Ri j ≥ 0, then any harmonic 1-form ω must be parallel and R(ω, ω) ≡ 0. In particular, this implies that the first Betti number b1 (M) of M must be at most m. If in addition there exists a point x ∈ M such that the Ricci curvature is positive, then b1 = 0. Proof Let ω be a harmonic 1-form. By the Bochner formula and Remark 3.2, we have |ω|2 = 2|∇ω|2 + 2E(ω), ω = 2|∇ω|2 + 2R jk a j ωk , ai ωi = 2|∇ω|2 + 2R ji a j ai = 2|∇ω|2 + 2R(ω, ω),
(3.4)
3 Bochner–Weitzenb¨ock formulas
29
which is nonnegative. Hence, by the maximum principle and the fact that M is compact, |ω|2 must be identically constant. Moreover, (3.4) implies that |∇ω|2 = 0 and R(ω, ω) = 0 on M. Since the dimension of parallel 1-forms is at most m, we conclude that the dimension of harmonic 1-forms is at most m. The Betti number bound now follows from the Hodge decomposition theorem. If we further assume that R(x) > 0 for some point x ∈ M, then the fact that R(ω, ω) ≡ 0 implies that ω(x) = 0. On the other hand, since |ω|2 is constant, this shows that ω ≡ 0. Hence b1 (M) = 0. We would like to remark that if ω is a parallel 1-form on M, then by the de-Rham decomposition theorem, the universal covering M˜ of M must be isometrically a product of R × N , where R is given by ω. Hence, if M is a compact manifold with nonnegative Ricci curvature, then its universal covering M˜ must be a product of Rk × N for some manifold N with nonnegative Ricci curvature and k = b1 (M). Definition 3.6 The curvature operator of a Riemannian manifold is a linear map S : 2 (M) → 2 (M) given by S(ωi ∧ ω j ) = R jikl ωl ∧ ωk . Theorem 3.7 (Gallot–Meyer [GM]) Let M be a compact manifold. If the curvature operator is nonnegative on M, then any harmonic p-form must be
parallel. Hence the pth Betti number of M is at most mp . Moreover, if there exists a point x ∈ M such that S(x) > 0, then b p (M) = 0. Proof
Similar to the case of 1-forms, we have the identity |ω|2 = 2|∇ω|2 + 2E(ω), ω,
where E(ω), ω = Rkβ iβ jα iα ai1 ... kβ ... i p ai1 ... jα ... i p . Let us now define the 2-form ω¯ = ai1 ... jα ... i p ω jα ∧ ωiα . By the definition of S, we have S(ω) ¯ = Riα jα kl ai1 ... jα ... i p ωl ∧ ωk .
30
Geometric Analysis
Hence, S(ω), ¯ ω ¯ = Riα jα kl ai1 ... jα ... i p ωl ∧ ωk , ai1 ... jβ ... i p ω jβ ∧ ωiβ = Riα jα iβ jβ ai1 ... jα ... i p ai1 ... jβ ... i p = E(ω), ω. The theorem now follows from the argument of Theorem 3.5.
Definition 3.8 A vector field X = ai ei is a Killing vector field if ai, j + a j,i = 0 for all 1 ≤ i, j ≤ m. Lemma 3.9 The infinitesimal generator of a one-parameter family of isometries of M is a Killing vector field on M. Proof Let φt : M → M be a one-parameter family of isometries parametrized by t ∈ (−, ). If {e1 , . . . , em } is an orthonormal frame at a point x ∈ M given by the normal coordinates centered at x, then the fact that φt are isometries implies that dφt (ei ), dφt (e j ) = ei , e j = δi j . Differentiating with respect to t and evaluating at t = 0, we have 0=
d dφt (ei ), dφt (e j )|t=0 dt
= ∇T dφt (ei ), dφt (e j ) + ∇T dφt (e j ), dφt (ei ),
(3.5)
where T is the infinitesimal vector field given by φt . However, since one can view {∂/∂t, e1 , . . . , em } as tangent vectors given by a coordinate system of (−, ) × M, we have the property that [T, dφt (ei )] = 0 for all 1 ≤ i ≤ m. Hence we can rewrite (3.5) as 0 = ∇ei T, e j + ∇e j T, ei which is exactly the condition that T is a Killing vector field.
Theorem 3.10 Let M be a compact manifold with nonpositive Ricci curvature. Then any Killing vector field on M must be parallel. Moreover, if there exists a point x ∈ M such that the Ricci curvature satisfies R(x) < 0, then there are
3 Bochner–Weitzenb¨ock formulas
31
no nontrivial Killing vector fields. In particular, this implies that M does not have a one-parameter family of isometries. Proof Let X = ai ei be a Killing vector field. Its dual 1-form is given by ω = ai ωi . The commutation formula yields |ω|2 = 2ai,2 j + 2ai, j j ai = 2|∇ω|2 − 2a j,i j ai = 2|∇ω|2 − 2R(X, X ). We now apply the same argument as in the proof of Theorem 3.5.
4 Laplacian comparison theorem
Let M be a complete m-dimensional manifold. Suppose p is a fixed point in M and let us consider the distance function r p (x) = r ( p, x) to p. When there is no ambiguity, the subscript will be deleted and we will simply write r (x). The distance function in general is not smooth due to the presence of cut-points. However, it can be seen that it is a Lipschitz function with Lipschitz constant 1. In particular, we have |∇r |2 = 1 almost everywhere on M. Though r might not be a C 2 function, one can still estimate its Laplacian in the sense of distribution. Theorem 4.1 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 . Then the Laplacian of the distance function satisfies √ √ (m − 1) K cot( K r ) r (x) ≤ (m − 1)r −1 √ √ ⎩ (m − 1) −K coth( −K r ) ⎧ ⎨
for for for
K > 0, K = 0, K <0
in the sense of distribution. Proof For a smooth function f , let x ∈ M be a point such that ∇ f (x) = 0. Then locally the level surface N of f through the point x is a smooth 32
4 Laplacian comparison theorem
33
hypersurface. Let {e1 , . . . , em−1 } be an orthonormal frame tangent to N , and let em be the unit normal vector to N . The Laplacian of f at x is defined to be f (x) =
m−1
(ei ei − ∇ei ei ) f (x) + (em em − ∇em em ) f (x)
i=1
=
m−1
(ei ei − (∇ei ei )t ) f (x)
i=1
−
m−1
(∇ei ei )n f (x) + (em em − ∇em em ) f (x)
i=1
− → = N f (x) + H f (x) + (em em − ∇em em ) f (x) − → = H f (x) + (em em − ∇em em ) f (x), where N denotes the Laplacian of N with respect to the induced metric and − → H denotes the mean curvature vector of N . If we take f = r, then for a point x which is not in the cut-locus of p, we have N = ∂ B p (r ). Moreover, the unit normal vector em = ∂/∂r . Hence, we have r (x) = H (x), where H (x) is the mean curvature of ∂ B p (r ) with respect to ∂/∂r . Therefore according to Corollary 2.2 the theorem is true for those points that are not in the cut-locus of p. Using the same notation as in Theorem 2.1, let us denote √ √ ⎧ (m − 1) K cot( K r ) for 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.
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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)
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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 .
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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)
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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
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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
.
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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 )
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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
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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)
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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
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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
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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)
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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
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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
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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),
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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)
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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).
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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)
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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
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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
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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)
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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
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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)
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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
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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¯
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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)
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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
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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,
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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
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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,
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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
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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
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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 .
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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)
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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).
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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) → ∞
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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)
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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
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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)
208
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 ] ,
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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 .
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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 ] ,
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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
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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)
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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
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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 ] ,
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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 .
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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 understan