Yamabe-type Equations on Complete, Noncompact Manifolds

Progress in Mathematics Volume 302 Series Editors Hyman Bass Joseph Oesterlé Yuri Tschinkel Alan Weinstein Paolo Mas...

Author: Paolo Mastrolia; Marco Rigoli; Alberto G Setti

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Progress in Mathematics Volume 302

Series Editors Hyman Bass Joseph Oesterlé Yuri Tschinkel Alan Weinstein

Paolo Mastrolia Marco Rigoli Alberto G. Setti

Yamabe-type Equations on Complete, Noncompact Manifolds

Paolo Mastrolia Dipartimento di Matematica Università degli Studi die Milano Milano, Italy

Marco Rigoli Dipartimento di Matematica Università degli Studi die Milano Milano, Italy

Alberto G. Setti Dipartimento di Fisica e Matematica Università dell’Insubria Como, Italy

ISBN 978-3-0348-0375-5 ISBN 978-3-0348-0376-2 (eBook) DOI 10.1007/978-3-0348-0376-2 Springer Basel Heidelberg New York Dordrecht London Library of Congress Control Number: 2012944419 Mathematics Subject Classification (2010): Primary: 53C21, 58-02; Secondary: 35J60, 35B45, 58J50 © Springer Basel 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper

Springer Basel AG is part of Springer Science+Business Media (www.birkhauser-science.com)

Contents Introduction 1 Some Riemannian Geometry 1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Moving frames and the ﬁrst structure equations . 1.1.2 Covariant derivative of tensor ﬁelds . . . . . . . . 1.1.3 Meaning of the ﬁrst structure equations . . . . . 1.1.4 Curvature: the second structure equations . . . . 1.1.5 Einstein manifolds and Schur’s Theorem . . . . . 1.2 Comparison theorems . . . . . . . . . . . . . . . . . . . 1.2.1 Ricci identities . . . . . . . . . . . . . . . . . . . 1.2.2 Cut locus and regularity of the distance function 1.2.3 The Laplacian comparison theorem . . . . . . . . 1.2.4 The Bishop-Gromov comparison theorem . . . . 1.2.5 The Hessian comparison theorem . . . . . . . . . 1.3 Some formulas for immersed submanifolds . . . . . . . . 2

3

1

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

7 7 8 10 12 14 16 18 18 21 22 28 31 32

Pointwise conformal metrics 2.1 The Yamabe equation . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 The derivation of the Yamabe equation . . . . . . . . . 2.1.2 The Kazdan-Warner obstruction . . . . . . . . . . . . . 2.1.3 The Weyl and Cotton tensors . . . . . . . . . . . . . . . 2.2 Some applications in the compact case . . . . . . . . . . . . . . 2.2.1 A rigidity result of Obata . . . . . . . . . . . . . . . . . 2.2.2 A result by M. F. Bidaut-V´eron and L. V´eron . . . . . . 2.2.3 A version of Theorem 2.12 on manifolds with boundary 2.2.4 A rigidity result of Escobar . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

37 37 37 40 43 49 49 57 62 67

General nonexistence results 3.1 Some spectral considerations . . . . . . . . . . . . . . . . . . . . . 3.1.1 The main nonexistence result . . . . . . . . . . . . . . . . . 3.2 The endpoint case K = −1 and the Poisson equation . . . . . . . .

73 74 78 93

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v

vi

Contents 3.3

A reﬁned version of Theorem 3.2 . . . . . . . . . . . . . . . . . . .

4 A priori estimates 4.1 Estimates from below . . . . . . . . . . . . . 4.2 Estimates from above . . . . . . . . . . . . . 4.3 Sharpness of the previous results . . . . . . . 4.4 Some further estimates . . . . . . . . . . . . . 4.5 Nonexistence results for the Yamabe problem 5

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Uniqueness 5.1 A sharp integral condition . . . . . . . . . . . . . . . . . . . . . 5.2 A remark on the asymptotic behaviour of solutions: examples Rm and Hm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Uniqueness via the weak maximum principle . . . . . . . . . . 5.3.1 A useful form of the weak maximum principle . . . . . . 5.3.2 A comparison result . . . . . . . . . . . . . . . . . . . . 5.3.3 Uniqueness of ground states . . . . . . . . . . . . . . . . 5.4 Some geometric applications and further uniqueness . . . . . . 5.4.1 Conformal diﬀeomorphisms . . . . . . . . . . . . . . . . 5.4.2 Uniqueness for the Yamabe problem . . . . . . . . . . . 5.4.3 An L∞ a priori estimate . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . in . . . . . . . . . . . . . . . . . .

98 105 105 111 115 117 121 127 127 130 132 133 140 143 146 146 148 149

6

Existence 6.1 A general procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Another comparison result . . . . . . . . . . . . . . . . . . . 6.1.2 More basic spectral theory and a result of Li, Tam and Yang 6.1.3 Two useful lemmas . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Existence of a maximal solution . . . . . . . . . . . . . . . . 6.2 Subsolutions and existence . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Existence with λL 1 (M ) < 0 . . . . . . . . . . . . . . . . . . . 6.2.2 λL (M ) < 0: some suﬃcient conditions . . . . . . . . . . . . 1 6.2.3 A more general case . . . . . . . . . . . . . . . . . . . . . . 6.3 Global sub- and supersolutions . . . . . . . . . . . . . . . . . . . . 6.4 The case of the Yamabe problem . . . . . . . . . . . . . . . . . . . 6.5 Appendix: the Monotone Iteration Scheme . . . . . . . . . . . . . .

157 158 158 158 162 165 166 166 170 177 180 188 191

7

Some special cases 7.1 A nonexistence result . . . . . . . . . . . . . . . . 7.1.1 A Rellich-Pohozaev formula . . . . . . . . 7.1.2 A nonexistence result for hyperbolic space 7.1.3 An integral obstruction . . . . . . . . . . 7.2 Special symmetries and existence . . . . . . . . . 7.3 The case of Euclidean space and further results . 7.3.1 A linear comparison result . . . . . . . . .

197 197 207 211 219 221 227 227

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Contents

vii

7.3.2 7.3.3

Back to Corollary 5.8 . . . . . . . . . . . . . . . . . . . . . 229 The Euclidean space . . . . . . . . . . . . . . . . . . . . . . 231

Bibliography

239

List of Symbols

247

Index

253

Introduction This book originates from a graduate course given at the University of Milan in 2007. Our goal is twofold: ﬁrst, to present a self-contained introduction to the geometric and analytic aspects of the Yamabe problem on a complete noncompact Riemannian manifold, treating existence, nonexistence, uniqueness and a priori estimates of the solutions. Secondly, we intend to describe in a way accessible to the nonspecialist a range of methods and techniques that can be successfully applied to more general nonlinear equations which arise in applications. The classical Yamabe problem concerns the possibility of pointwise conformally deforming a metric of scalar curvature S(x) on the manifold M to a new metric with prescribed scalar curvature K(x). In the case where K is constant it is a natural higher dimensional generalization of the Poincar´e–K¨ obe Uniformization Theorem for Riemann surfaces and can be seen as a way to select a privileged metric on the manifold. If , is the original metric of the Riemannian manifold M and we denote with , = ϕ2 , , ϕ > 0, a conformally deformed metric, then the two scalar curvatures S(x) and S(x) are related by the equation ϕ2 S(x) = S(x) − 2(m − 1)

Δϕ |∇ϕ|2 − (m − 1)(m − 4) 2 ϕ ϕ

(see equation (2.7) in Chapter 2), where Laplacian, gradient, and norm are those of the metric , . In the case where the dimension m of the manifold is greater than or equal to three, it is useful to set 2

ϕ = u m−2 so that the above equation takes the form m+2 m−1 m−2 Su = Su − 4 Δu. m−2

Thus the Yamabe problem amounts to ﬁnding a positive solution u of the familiar Yamabe equation m+2 cm Δu − Su + Ku m−2 = 0, (1)

1

2

Introduction

where cm = 4 m−1 m−2 and K = S, the prescribed scalar curvature of the conformally deformed metric. If M is compact and K is constant, after an initial attempted solution by H. Yamabe [Yam60], the problem was solved thanks to eﬀorts of N. Trudinger [Tru68], T. Aubin [Aub76] and R. Schoen [Sch84] (see the nice survey paper by J.M. Lee and T.H. Parker, [LP87], for a complete and self-contained treatment). The solution was obtained using variational methods, and one of the m+2 main analytic diﬃculties stems from the fact that m−2 is the critical exponent for 2m

the Sobolev embedding W 1,2 → L m−2 . A natural generalization of the classical Yamabe problem is the case where K is nonconstant and/or M is noncompact. In this direction we mention the pioneering work of J. L. Kazdan and F. W. Warner, [KW74a], [KW74b], [KW75a], [KW75b]. It should also be mentioned that even the classical Yamabe problem of deforming the metric to one of constant scalar curvature in the noncompact setting is in general not solvable, as ﬁrst shown by Z. R. Jin, [Jin88]. The Yamabe problem for noncompact manifolds with variable prescribed curvature is the subject of the present monograph. Indeed, we describe methods which allows us to consider the more general Yamabe-type equations (resp. inequalities) of the form (2) Δu + a(x)u − b(x)uσ = 0 (resp. ≥ 0) where σ > 1, and we study nonexistence, a priori estimates, uniqueness and existence. Equations of the form (2) and still more general diﬀerential inequalities of the form uΔu + a(x)u2 − b(x)uσ+1 ≥ −A|∇u|2 (3) arise in complex analysis (e.g. in the study of the structure of complete K¨ahler manifolds, [LY90], [Li90] and [LR96], in the Schwarz Lemma for the ratio of volume elements of K¨ahler manifolds of the same dimension, [Gri76], in the study of pluriharmonic functions on a K¨ ahler manifold, [PRS08]), in the study of harmonic maps with bounded dilation ([EL78] and [PRS08] Chapter 8), in the classiﬁcation of locally conformally ﬂat manifolds ([PRS07]), in the study of Yang-Mills ﬁelds, and in population dynamics, to quote only a few examples. Existence and nonexistence of positive solutions of (2) clearly depend on the geometry of the underlying manifold, typically encoded by curvature or volume growth of geodesic balls, on properties of the coeﬃcients (typically the relative signs of the coeﬃcients a(x) and b(x)) and their asymptotic behavior and on the mutual interplay of the two. This interplay can be taken into account in terms of the relative asymptotic behavior of the coeﬃcients versus the geometry at inﬁnity of the manifold or, at a deeper level, in terms of spectral properties of Schr¨ odinger operators naturally associated to the equation. From the geometrical interpretation of the equation, it is natural to expect it will be easier to have existence when a and b are “close” enough, for instance they have the same sign, while it will be more diﬃcult to have existence (and therefore

Introduction

3

it will be easier to prove nonexistence) when a and b are farther apart, for instance when they have opposite sign. This expectation is conﬁrmed by both the existence and the nonexistence results that we will describe. The geometry of the manifold also plays a natural role in the uniqueness results as well in the a priori estimates on the solutions. The latter have a particular geometric interest since they are responsible for the completeness/noncompleteness of the deformed metric. As mentioned above, we use a variety of techniques adapted to the geometric situation at hand in which the lack of compactness and of symmetry and homogeneity prevents the use of more standard tools typical of compact situations or of the Euclidean setting. In particular, for existence we will essentially use the method of sub-super solutions, [Ama76], [Sat73]. Nonexistence will be obtained using Liouville-type results which in turn are obtained using either integral formulas or a method based on the coupling of the supposed solution of the Yamabe-type inequality with that of an appropriate Schr¨odinger-type inequality associated to it, in a manner reminiscent of the classical generalized maximum principle. Uniqueness will be obtained using variants of the weak maximum principle (see, e.g., [PRS05b]) and of clever integration by parts arguments. Finally, a priori estimates will be typically obtained using an elaboration of the old idea of the proof of the Schwarz’s Lemma by L. H. Ahlfors, [Ahl38], which is at the heart of the maximum principle. The book is divided into seven chapters. In the ﬁrst chapter we give a quick review of Riemannian geometry using the method of moving frames. While we assume basic knowledge of Riemannian geometry, several computations will be carried out in full detail in order to acquaint the reader with notation and formalism. We concentrate on derivation of the symmetry properties of the curvature tensors together with a number of other identities that will be repeatedly used in the sequel. In particular, we will describe the commutation rules for covariant derivatives up to fourth order. Then we describe comparison results for the Laplacian of the Riemannian distance function, and for the volume of geodesic balls in terms of lower bounds for the Ricci curvature. We point out that our treatment, which follows that of [PRS05b], does not use Jacobi ﬁelds. In Chapter 2 we ﬁrst derive equations for the change of curvature tensors under a conformal change of the metric and introduce the Yamabe equation. As a side product of our computations we obtain decomposition of the Riemann curvature tensor in its irreducible components and we exhibit the conformal invariance of the Weyl tensor. Then, we brieﬂy consider the case where M is compact to illustrate the interplay between geometry and analysis, with a few illuminating examples such as the Kazdan-Warner obstruction, a result of Obata on Einstein manifolds and a far-reaching “generalization” due to V´eron-V´eron, through which we prove further results of Escobar. Along the way we give a detailed proof, which inspires to Petersen’s treatise [Pet06a], of a famous rigidity result of Obata. The goal is also to give some geometrical feeling on the subject that will enable us to

4

Introduction

proceed with the noncompact case: the case of the rest of our investigation. The core of the monograph begins with Chapter 3, devoted to nonexistence results. As mentioned above, since our methods apply to more general situations which have a wide range of applications, we consider in fact diﬀerential inequalities of the form (2) and (3). We describe several nonexistence results; in most of them we assume that u satisﬁes suitable integrability conditions, that b(x) is nonnegative and that there exists a positive solution ϕ to the diﬀerential inequality Δϕ + Ha(x)ϕ ≤ −K

|∇ϕ|2 ϕ

with H, K parameters satisfying H > 0, K > −1. Note that in the special case where K = 0 the latter condition amounts to the fact that the bottom of the spectrum of the operator −Δ − Ha(x) is nonnegative. Since −Δ is a nonnegative operator, the condition is trivially satisﬁed if a(x) ≤ 0 on M and may be interpreted as a measure of smallness in a spectral sense of the positive part of a(x). This agrees with the heuristic intuition on the eﬀect of the relative signs of a(x) and b(x) on the existence of solutions. The existence of the positive function ϕ enters the proof in two diﬀerent ways. In Theorem 3.2 one uses the functions ϕ to obtain an integral inequality involving u and its gradient from which one concludes that u is constant, and therefore necessarily identically zero. In a second group of results, the function ϕ is combined with the solution u to give rise to a diﬀusion-type diﬀerential inequality for which we prove a Liouville theorem. This yields the desired triviality. We also show that when σ is greater than or equal to the critical exponent (m + 2)/(m − 2), then, by performing an appropriate change of the metric and of the solution, the nonexistence results can be improved to allow even some controlled negativity of the coeﬃcient b(x). Chapter 4 is devoted to establishing a priori upper and lower estimates for the asymptotic behavior of solutions of the diﬀerential inequalities Δu + a(x)u − b(x)uσ ≥ 0, resp. Δu + a(x)u − b(x)uσ ≤ 0, under assumptions on a(x) and b(x) related to an assumed radial lower bound for the Ricci curvature. As brieﬂy mentioned above, the results are obtained by applying Alhfors’s old idea, namely, one considers an auxiliary function deﬁned in terms of the solution u which by construction attains an extremum, and applies the usual maximum principle. Clearly, the heart of the method consists in ﬁnding the best auxiliary function for the problem at hand. We exhibit examples showing that our estimates are essentially sharp. Some further estimates, which cannot be obtained with the previous method, are provided by direct comparison with the aid of the maximum principle (see section 4.3). The chapter ends with some nonexistence results for the Yamabe problem, which complement those described in Chapter 3. In Chapter 5 we discuss some uniqueness results for positive solutions of Yamabe-type equations (2). The ﬁrst, the very general Theorem 5.1, states that

Introduction

5

if the coeﬃcient b(x) is nonnegative and not identically zero, then two solutions whose diﬀerence is L2 -integrable are necessarily the same. Although very general, it is sharp, and, remarkably, the assumption on the L2 -integrability cannot be replaced by an Lp condition with p > 2. The result is obtained by means of a clever elementary integral inequality. The second result, Theorem 5.2, follows by a comparison argument which relies on a version of the weak maximum principle (see Theorem 5.3) which is interesting in its own. While in most of the results available in the literature, uniqueness is obtained by requiring that solutions have a rather precisely determined asymptotic behavior, our result applies to solutions whose behavior at inﬁnity is speciﬁed in a much less stringent manner, see (5.13); moreover, the conclusion is reached assuming only conditions on the volume growth of the manifold. Counterexamples show the sharpness of each result. The chapter ends with a geometric application to the group of conformal diﬀeomorphisms of a complete manifold and to the uniqueness of solutions of the geometric Yamabe problem. Chapter 6 deals with existence results for Yamabe-type equations (2) on the complete, noncompact, Riemannian manifold M . The main tool is the monotone iteration scheme in various forms, and we give a rather detailed description of it in the appendix at the end of the chapter. The application of the scheme in this context goes back to W. M. Ni, [Ni82], in the Euclidean setting and to P. Aviles and R. C. McOwen, [AM85] and [AM88], for noncompact manifolds. After having introduced some preliminary material on spectral theory, and a useful comparison result, the main body of the chapter is then devoted to the construction of (global and local) super- and subsolutions for the problem. In general terms, supersolutions are obtained under assumptions on the sign of b(x) and of the ﬁrst eigenvalue of L = Δ + a(x) on appropriate domains. Because of the combination of signs of the coeﬃcients, subsolutions are harder to ﬁnd. We give a number of suﬃcient conditions which ensure that such subsolutions do exist: among them the spectral condition λL 1 (M ) < 0, for which we provide a new suﬃcient condition contained in Theorem 6.11. Furthermore, we mention Theorem 6.15, in which existence is guaranteed under a very week growth condition on b(x), and also Theorem 6.16, where a further weakining on the condition on the sign of b(x) is balanced by the necessity of imposing a constant negative lower bound on the Ricci curvature. We explicitly note that the assumptions of our existence theorems match those of the nonexistence results in the previous chapters. In the last chapter, Chapter 7, we consider some particular cases where the symmetry of the geometry allows one to use special techniques and to obtain stronger results. Typically this happens in Euclidean and Hyperbolic spaces, and more generally in the case of models (in the sense of R. Greene and H. Wu, [GW79]), or manifolds with special symmetry. The speciﬁc feature of models which make the analysis more precise is that the Laplacian of the distance function from the origin is given explicitly, as opposed to the case of a general manifold where only upper and lower bounds may be obtained under suitable curvature assumptions, by means of the the Laplacian and Hessian comparison theorems,

6

Introduction

and where the possible presence of the cut locus raises additional diﬃculties. We describe reﬁned techniques adapted to the situation at hand and obtain results that, as a by-product, show the degree of sharpness of the general theory and methods we have developed dealing with generic complete Riemannian manifolds. It seems worth remarking that, in the speciﬁc case of Hyperbolic space, we provide a nonexistence result with the aid of a Rellich-Pohozaev type formula (see Theorem 7.7) and, even more, in Proposition 7.9 we introduce an integral obstruction to the existence of a conformal deformation which is of a diﬀerent nature with respect to the Kazdan-Warner condition. Many of the results presented in this monograph have been obtained over the years by the authors jointly with many collaborators. To all of them we wish to extend our thanks and appreciation. In particular we are indebted to S. Pigola and M. Rimoldi who provided us with the proof of Theorem 2.10 in Chapter 2.

Chapter 1

Some Riemannian Geometry In this chapter we give a quick review of Riemannian geometry using the moving frame formalism. While we assume basic knowledge of Riemannian geometry, several computations will be carried out in full detail in order to acquaint the reader with notation and formalism. After having introduced frame and coframes, we will describe connection and curvature in terms of the connection and curvature forms. Symmetry properties of the curvature tensors will be described in detail and we will derive a number of identities that will be repeatedly used in the sequel. In particular, we will obtain the commutation rules for covariant derivatives up to fourth order. Along the way, we will introduce Einstein manifolds and prove Schur’s lemma. Then we introduce basic results for the Riemannian distance function from a ﬁxed reference point o ∈ M , and discuss brieﬂy the cut locus and some of its properties. We will then describe comparison results for the Laplacian of the Riemannian distance function and for the volume of geodesic balls in terms of lower bounds for the Ricci curvature. We point out that our treatment, which follows that of [PRS05b], does not use Jacobi ﬁelds. The chapter ends with a brief section on the geometry of immersed submanifolds to ﬁx notation and terminology used in the second part of Chapter 2.

1.1

Preliminaries

Let (M, , ) be a Riemannian manifold of dimension m with metric , . The aim of this section is to ﬁx notation and to describe the essential facts of the geometry of ´ Cartan’s formalism. The usefulness of this approach will become (M, , ) using E. apparent in the sequel, and it will prove to be particularly eﬀective in Chapter 2 in the derivation of the formulae which express the change of the Riemannian, Ricci and scalar curvature under a conformal change of the metric.

P. Mastrolia et al., Yamabe-type Equations on Complete, Noncompact Manifolds, Progress in Mathematics 302, DOI 10.1007/978-3-0348-0376-2_1, © Springer Basel 2012

7

8

1.1.1

Chapter 1. Some Riemannian Geometry

Moving frames and the ﬁrst structure equations

Let p ∈ M and (U, ϕ) a local chart such that U p, with coordinate functions x1 , . . . , xm , m = dim (M ) . If q is a generic point in U we have, at q, , = , ij dxi ⊗ dxj ,

(1.1)

where dxi denotes the diﬀerential of the function xi and , ij are the (local) ∂ ∂ components of the metric, deﬁned by , ij = ∂x i , ∂xj . In relation (1.1), and throughout the book, we adopt the Einstein summation convention for repeated indices. Applying in q the Gram-Schmidt orthonormalization process we can ﬁnd linear combinations of the 1-form dxk , which we will call θi , such that , = δij θi ⊗ θj ,

(1.2)

where δij is the Kronecker symbol. Since, as q varies in U , the previous process gives rise to coeﬃcients that are C ∞ -functions of q, the set of 1-forms θi , i = 1, . . . , m, deﬁne an orthonormal system on U for the metric , , i.e., a (local) orthonormal (o.n.) coframe. We will sometimes write , =

m

(θi )2

i=1

instead of (1.2). We also deﬁne the (local) dual orthonormal frame {ei }, i = 1, . . . , m, as the set of m (local) vector ﬁelds satisfying, on the open set U , θj (ei ) = δij

(1.3)

(where δij is just a suggestive way of writing the Kronecker symbol, reﬂecting the position of the indexes in the pairing of θi and ej ). We have the following Proposition 1.1. Let θi be a local o.n. coframe on M , deﬁned on an open set U ; then there exist unique 1-forms i θj , i, j = 1, . . . , m on U such that, ∀ i, j = 1, . . . , m, dθi = −θji ∧ θj ,

(1.4)

θij

(1.5)

θji +

= 0.

The forms θji are called the Levi-Civita connection forms associated to the o.n. i coframe θ . Remark. Equations (1.4) are classically known as the ﬁrst structure equations.

1.1. Preliminaries

9

Proof. Let us assume the existence of the forms θji satisfying (1.5) and (1.4) and determine their expression. Of course θji = aijk θk for some aijk ∈ C ∞ (U ) (where C ∞ (U ) denotes the set of smooth functions deﬁned on the open set U ), and (1.5) is equivalent to aijk + ajik = 0.

(1.6)

The 2-forms dθi can be written, for some (unique) coeﬃcients bijk ∈ C ∞ (U ), as dθi =

1 i j b θ ∧ θk , 2 jk

bijk + bikj = 0.

Since (1.4) must hold we have: 1 i j 1 b θ ∧ θk = −aijk θk ∧ θj = aijk θj ∧ θk = (aijk − aikj )θj ∧ θk . 2 jk 2 It follows that bijk = aijk − aikj .

(1.7)

Cyclic permutations of the indices i, j, k and use of (1.6) and (1.7) yield bkij = akij − akji = −aikj + ajki ;

(1.8)

bjki

(1.9)

=

ajki

−

ajik

=

ajki

+

aijk .

Adding (1.7) and (1.9) and subtracting (1.8) we get aijk =

1 i (b − bkij + bjki ). 2 jk

(1.10)

The previous relation determines the expression of the forms θji and also proves uniqueness. Now deﬁne 1 θji = (bijk − bkij + bjki )θk , (1.11) 2 where the bijk ’s satisfy bijk + bikj = 0. It is clear that aijk =

1 1 j bik − bkji + bikj = − bijk − bkij + bjki = −aijk , 2 2

thus (1.6) is met, and then the θji deﬁned in (1.11) satisfy (1.5); it is also immediate to verify that they satisfy (1.4).

10

Chapter 1. Some Riemannian Geometry

1.1.2

Covariant derivative of tensor ﬁelds

The Levi-Civita connection forms are the starting point to deﬁne a covariant derivative of tensor ﬁelds in the following way. Following standard notation we denote with Tp M the tangent space at p ∈ M and with Tp∗ M the cotangent space at p. We recall that a tensor ﬁeld T of type (r, s) is a law that assigns to all points p ∈ M a multilinear map Tp : Tp∗ M × · · · × Tp∗ M × Tp M × · · · × Tp M → R s times

r times

with the usual diﬀerentiability request with respect to the variable p (see e.g. [Lee03]). The set of tensor ﬁelds of type (r, s) will be denoted by Trs (M ). Let us begin considering the case of a vector ﬁeld X on M , i.e., a tensor of type (1, 0). We denote with X(M ) the set of all (smooth) vector ﬁelds on M . Let θi a local o.n. coframe and {ei } the dual frame. Deﬁnition 1.2. The covariant derivative of the vector ﬁeld X, ∇X, is the tensor ﬁeld of type (1, 1) ∇X deﬁned in the following way: if X = X i ei , ∇X = (dX i ) ⊗ ei + X i ∇ei , having deﬁned ∇ei = θij ⊗ ej . Setting Xki θk = dX i + X j θji , ∇X can be then written as ∇X = (dX i + X j θji ) ⊗ ei = Xki θk ⊗ ei , and Xki is said to be the covariant derivative of the coeﬃcient X i . If Y ∈ X(M ) we deﬁne the covariant derivative of X in the direction of Y as the vector ﬁeld ∇Y X = ∇X(Y ), which in components reads as ∇Y X = Xki θk (Y )ei = Xki Y k ei . Deﬁnition 1.3. The divergence of the vector ﬁeld X ∈ X(M ) is the trace of ∇X, that is, (1.12) div X = tr (∇X) = ∇ei X, ei = Xii . Analogously, for a 1-form ω (i.e., a tensor of type (0, 1)), we have:

1.1. Preliminaries

11

Deﬁnition 1.4. The covariant derivative of the 1-form ω, ∇ω, is the tensor ﬁeld of type (0, 2) deﬁned in the following way: if ω = ωi θi , ∇ω = (dωi ) ⊗ θi + ωi ∇θi , with ∇θi = −θji ⊗ θj . Note that, setting ωik θk = dωi − ωj θij , it follows that ∇ω = ωik θk ⊗ θi . If Y ∈ X(M ) we deﬁne the covariant derivative of ω in the direction of Y as the 1-form ∇Y ω = ∇ω(Y ), which in components reads as ∇Y ω = ωik θk (Y )θi = ωik Y k θi . For a 0-form f ∈ C ∞ (M ) we set ∇f = df

(exterior diﬀerential of f ).

We point out that this notation may give rise to some ambiguity; indeed, in the literature (and also in this book) ∇f often denotes the gradient of f , i.e., the vector ﬁeld dual to the 1-form df : in this case, using standard notation (see for instance [Lee97]), we can write ∇f = (df ) , where is the sharp map from the ∗ cotangent bundle T M to the tangent bundle T M deﬁned by (df ) , Y = ∇f, Y = df (Y ) = Y (f ), for all Y ∈ X(M ). Note that, in components, setting df = fj θj for some smooth i coeﬃcients fj , we have (∇f ) = δ ij (df )j = δ ij fj = fi (that is, in an orthonormal frame, diﬀerential and of a function have the same coeﬃcients with respect gradient to the (dual) basis θi and {ei }). Finally, ∇ can be extended in a natural way to a generic tensor T , in order to deﬁne a connection on each tensor bundle Trs (M ): this extension of ∇ satisﬁes the Leibniz rule and some other nice properties, like the commutativity with the trace on any pair of indices (see again [Lee97]). Although the covariant derivative has been introduced by means of ilocally deﬁned objects, it is possible to show (using the transformation laws of θ and i θj as the coframe changes) that the new tensor ﬁeld thus obtained is globally deﬁned. Remark. One can verify that the previous deﬁnition matches the “canonical” one usually given in terms of the Koszul formalism (see for example [Lee97], [Pet06a]). Indeed, as we will see before long, the operator ∇ coincides precisely with the Levi-Civita connection associated to the metric , of M .

12

1.1.3

Chapter 1. Some Riemannian Geometry

Meaning of the ﬁrst structure equations

We now want to discuss the geometric meaning of condition (1.5), namely θji + θij = 0. To this purpose let us compute the covariant derivative of the metric tensor , . Using the Leibniz rule, and recalling that for everytangent vector Xp = X(p) ∈ Tp M (with p in the domain of the o.n. coframe θi ) it holds that ∇Xp θi = −θji (Xp )θj , we have ∇Xp , = ∇Xp (δij θi θj ) = δij (∇Xp θi ⊗ θj + θi ⊗ ∇Xp θj ) = δij (−θki (Xp )θk ⊗ θj − θkj (Xp )θi ⊗ θk ) = −θki (Xp )θk ⊗ θi − θki (Xp )θi ⊗ θk = −(θki + θik )(Xp )θk ⊗ θi , and therefore ∇ , = 0 if and only if θji + θij = 0, i.e., condition (1.5) is equivalent to the parallelism of the metric (in other words: ∀ X, Y, Z ∈ X(M ), X Y, Z = ∇X Y, Z + Y, ∇X Z). On the other hand, the ﬁrst structure equations (1.4) tell us that the metric is torsion-free. Indeed, let X and Y be two vector ﬁelds on M and [X, Y ] their Lie bracket, deﬁned by [X, Y ](f ) = X(Y (f )) − Y (X(f )),

∀f ∈ C ∞ (M ).

(1.13)

We claim that condition [X, Y ] = ∇X Y − ∇Y X

∀ X, Y ∈ X(M )

(1.14)

is equivalent to the validity of (1.4). Note that the left-hand side of (1.14) is independent of the choice of a metric on M . Since the torsion of a generic connection ∇ on M is the (0, 2) tensor ﬁeld Tor(X, Y ) = ∇X Y − ∇Y X − [X, Y ], this justiﬁes the expression “torsion-free” used above. To prove the equivalence, recall that the exterior diﬀerential of a 1-form ω is intrinsically deﬁned by dω(X, Y ) = X(ω(Y )) − Y (ω(X)) − ω([X, Y ]); moreover, as a consequence of the deﬁnition of covariant derivative, (∇X ω)(Y ) = X(ω(Y )) − ω(∇X Y ), so that

X θi (Y ) − θi (∇X Y ) = (∇X θi )(Y ) = −θji (X)θj (Y ),

(1.15)

1.1. Preliminaries

13

that is,

X θi (Y ) + θji (X)θj (Y ) = θi (∇X Y ). Then we compute dθi + θji ∧ θj (X, Y ), that is dθi (X, Y ) + θji ∧ θj (X, Y ) = X(θi (Y )) − Y (θi (X)) − θi ([X, Y ]) + θji (X)θj (Y ) − θji (Y )θj (X) = X(θi (Y )) + θji (X)θj (Y )) − Y (θi (X)) − θji (Y )θj (X)) − θi ([X, Y ]) = θi (∇X Y − ∇Y X − [X, Y ]),

and the claim follows. Remark. By the fundamental theorem of Riemannian geometry (see for instance [Lee97] or [Pet06b]), we deduce that the connection ∇ coincides, as we said previously, with the Levi-Civita connection of the metric , . We now deﬁne the Lie derivative of Y in the direction of X to be LX Y = [X, Y ], so that condition (1.14) can be written in the form LX Y = ∇X Y − ∇Y X.

(1.16)

LX f = X(f )

(1.17)

(LX ω)(Y ) = LX (ω(Y )) − ω(LX Y ),

(1.18)

Setting also ∞

for f ∈ C (M ), and

if ω is a 1-form, we can extend LX to a generic tensor ﬁeld requiring R-linearity and the validity of the Leibniz rule (see also [Lee03], [Pet06a]). Using (1.15), we compute the Lie derivative of the metric in the direction of X, LX , (note that this latter has to be a covariant tensor of order 2, that is, a (0, 2)-tensor): (LX , )(Y, Z) = ((LX θi ) ⊗ θi + θi ⊗ (LX θi ))(Y, Z) = θi (Z)(LX θi )(Y ) + θi (Y )(LX θi )(Z) = θi (Z) LX (θi (Y )) − θi (LX Y ) + θi (Y ) LX (θi (Z)) − θi (LX Z) = θi (Z)X(θi (Y )) − θi (Z)θi (∇X Y − ∇Y X) + θi (Y )X(θi (Z)) − θi (Y )θi (∇X Z − ∇Z X) = θi (Z)(∇X θi )(Y ) + θi (Y )(∇X θi )(Z) + θi (Z)θi (∇Y X) + θi (Y )θi (∇Z X) = (∇X θi ⊗ θi + θi ⊗ ∇X θi )(Y, Z) + ∇Y X, Z + Y, ∇Z X = (∇X , )(Y, Z) + ∇Y X, Z + Y, ∇Z X = ∇Y X, Z + Y, ∇Z X ,

14

Chapter 1. Some Riemannian Geometry

where in the last equality we have used the fact that the metric is parallel with respect to the Levi-Civita connection. Thus, we have proved the useful identity (LX , )(Y, Z) = ∇Y X, Z + Y, ∇Z X

(1.19)

for all X, Y, Z ∈ X(M ), which will be repeatedly used in the sequel. Note that equation (1.19) in components reads as (1.20) (LX , )ij = ∇ei X, ej + ei , ∇ej X = Xij + Xji . It can be proved that the Lie derivative of Y in the direction of X has the following geometric meaning (see e.g. [Lee03]): (ϕ−t )∗ Yϕt (p) − Yp d (LX Y )p = , (ϕ−t )∗ Yϕt (p) = lim t→0 dt t=0 t where ϕt is the local ﬂow generated by X and (ϕt )∗ is the push-forward. The analogous applies to LX h, with h a generic tensor ﬁeld (see also Chapter 2 for the special case of LX , ).

1.1.4

Curvature: the second structure equations

We now consider the second structure equations. Let θi be a local o.n. frame and i the corresponding Levi-Civita connection forms. The curvature forms Θij θj associated to the coframe are deﬁned through the second structure equations, dθji = −θki ∧ θjk + Θij .

(1.21)

Obviously the Θij are 2-forms. Since, according to (1.5), θji + θij = 0, it follows immediately that the Θij ’s satisfy the same antisymmetry condition Θij + Θji = 0.

(1.22)

Using the basis {θi ∧ θj }, 1 ≤ i < j ≤ m, of the space of skew-symmetric 2-forms 2 (U ) on the open set U , we may write Θij =

1 i k i Rjkt θk ∧ θt Rjkt θ ∧ θt = 2

(1.23)

k
i for some coeﬃcients Rjkt ∈ C ∞ (U ) satisfying i i Rjkt + Rjtk = 0,

(1.24)

j i + Rikt = 0. Rjkt

(1.25)

while (1.22) implies that

1.1. Preliminaries

15

From (1.24) and (1.25) we thus deduce the symmetries j i i Rjkt = −Rjtk = −Rikt .

(1.26)

Diﬀerentiating the ﬁrst structure equation, using the second and the properties of the exterior diﬀerential we have 0 = d(dθi ) = −d(θji ∧ θj ) = −dθji ∧ θj + θji ∧ dθj = θki ∧ θjk ∧ θj − Θij ∧ θj − θji ∧ θkj ∧ θk = θj ∧ Θij , that is θj ∧ Θij = 0.

(1.27)

These identities go under the name of the ﬁrst Bianchi identities. Using (1.23) we get i i i i θj ∧ θk ∧ θt = (Rjkt + Rktj + Rtjk )θj ∧ θk ∧ θt , 0 = Rjkt 1≤j
and then we deduce the ﬁrst Bianchi identities in the classical form i i i Rjkt + Rktj + Rtjk = 0.

(1.28)

It is possible also to show that another consequence of (1.26) and (1.28) is the last, important symmetry i k Rjkt = Rtij . (1.29) i One can verify that the coeﬃcients Rjkt gives rise to a (global) (1, 3)-tensor, called the Riemann curvature tensor, i θ k ⊗ θ t ⊗ θ j ⊗ ei . Riem = Rjkt

We warn the reader that there are a number of diﬀerent conventions for the Riemann curvature tensor (see the discussion in [Lee97]). We will often use the curvature tensor of type (0, 4) given by R = Rijkt θi ⊗ θj ⊗ θk ⊗ θt , i with Rijkt = Rjkt . Note that we have performed the classical operation of lowering indices using the metric tensor, that is, in our chosen orthonormal frame, s i = Rjkt Rijkt = δis Rjkt

(compare with the discussion about gradient and diﬀerential in section 1.1.2). We complete the description of the symmetries of the curvature tensor noting that, from the ﬁrst Bianchi identities, we can obtain the second Bianchi identities

16

Chapter 1. Some Riemannian Geometry

involving the covariant derivatives of the components of the curvature tensor, namely: Rijkt,l + Rijtl,k + Rijlk,t = 0. (1.30) Tracing the curvature tensor on its ﬁrst and third indices (or, equivalently, on its second and fourth) we obtain the Ricci tensor Ric = R(ei , ·, ei , ·) = R(·, ei , ·, ei , ) = Rijik θj ⊗ θk = Rjk θj ⊗ θk , where {ei } is the dual basis of θj . Note that, according to (1.29), Ric is a symmetric (0, 2)-tensor; in other words, ∀ X, Y ∈ X(M ) Ric(X, Y ) = Ric(Y, X),

(1.31)

Rij = Rji .

(1.32)

and in components The scalar curvature S is deﬁned as the trace of Ric, i.e., S = Ric(ei , ei ) = Rijij = Rii . The sectional curvature of the 2-plane π ⊂ Tp M spanned by the vectors u and v is deﬁned to be R(u, v, u, v) Kp (π) = 2 ∈ R. u, u v, v − u, v It is not diﬃcult to verify that the right-hand side of the above formula is in fact independent of the chosen basis of π. Clearly, if u v are an orthonormal basis for π, then Kp (π) = R(u, v, u, v). We note that a common notation for the sectional curvature of the plane π spanned by u and v is Kp (π) = Sect(u ∧ v).

1.1.5

Einstein manifolds and Schur’s Theorem

Deﬁnition 1.5. The manifold (M, , ), dim(M ) = m ≥ 2, is said to be Einstein if Ric = λ , ,

(1.33)

for some λ ∈ R. Using the moving frame formalism we now show that, if the dimension of the manifold is greater than or equal to 3 and equation (1.33) holds for some λ ∈ C ∞ (M ), where C ∞ (M ) is the set of smooth functions deﬁned on M , then

1.1. Preliminaries

17

λ is automatically constant. Indeed, ﬁrst note that, tracing equation (1.33), we immediately obtain S λ= . (1.34) m We now trace the second Bianchi identities (1.30) with respect to the indices i and l to get Rijkt,i + Rijti,k + Rijik,t = 0. Since covariant derivative commutes with contractions, the previous relation yields Rijkt,i = Rjt,k − Rjk,t ,

(1.35)

whence, contracting again this time with respect to j and k, we get Rikkt,i = Rkt,k − Rkk,t that is, 2Rkt, k = St .

(1.36)

Now because of (1.33) and (1.34) we have Rkt =

S δkt , m

and using again the fact that the metric tensor is parallel we deduce that Rkt,l =

1 Sl δkt . m

Now tracing with respect to k and l, we get Rkt,k = substituting in (1.36), we obtain

1 St ; m

(1.37)

2 − 1 St = 0, m

and we conclude that if m ≥ 3 and M is connected, then the scalar curvature, and therefore λ, are constant. We have thus proved the well-known result of Schur: Theorem 1.6. Let (M, , ) be a connected Riemannian manifold of dimension m ≥ 3. If Ric = λ , for some λ ∈ C ∞ (M ), then M is Einstein.

18

Chapter 1. Some Riemannian Geometry

Note that, if m ≥ 3, then (M, , ) is Einstein if and only if the traceless Ricci tensor S T = Ric − , (1.38) m is identically null. Observe also that, in our o.n. coframe, Tij = Rij −

S δij . m

(1.39)

Using (1.37) (sometimes called Schur’s identities) we shall obtain a remarkable formula, an inﬁnitesimal version of the Kazdan-Warner obstruction, that will be discussed in detail below (see Section 2.1.2).

1.2 1.2.1

Comparison theorems Ricci identities

We now want to recall that the curvature tensor can be interpreted as an obstruction to the validity of Schwarz’s theorem for mixed derivatives of third order and higher. Since this will be useful in the sequel, we consider the case of a function u : M → R which we assume to be at least C 3 (M ). If du = ui θi

(1.40)

for some smooth coeﬃcients ui , the Hessian of u is deﬁned as the 2-covariant tensor Hess(u) = ∇du of components uij given by uij θj = dui − uk θik ,

(1.41)

Hess(u) = uij θj ⊗ θi .

(1.42)

that is, The Laplacian of u is the trace of the Hessian, that is, Δu = tr (Hess(u)) = uii .

(1.43)

One can verify that the operator Δ deﬁned in (1.43) is the Laplace-Beltrami operator associated to the metric , (see for instance [Pet06b]). Since second derivatives commute we expect Hess(u) to be a symmetric tensor. This can be veriﬁed as follows: we diﬀerentiate equation (1.40) and use the structure equations to get 0 = dui ∧ θi + ui dθi = (uij θj + uk θik ) ∧ θi − ui θki ∧ θk = uij θj ∧ θi 1 = (uij − uji )θj ∧ θi , 2

1.2. Comparison theorems thus 0=

since θ ∧ θ j

i

19

(uij − uji )θj ∧ θi ;

1≤j
(1 ≤ j < i ≤ m) is a basis for the 2-forms we deduce uij = uji ,

(1.44)

as expected. The third derivatives of u are deﬁned by the rule uijk θk = duij − ukj θik − uik θjk .

(1.45)

Note that according to (1.44), the right-hand side is symmetric with respect to i and j and therefore uijk = ujik . (1.46) We now diﬀerentiate (1.41) and use the structure equations to get duik ∧ θk − uij θkj ∧ θk = −dut ∧ θit + uk θtk ∧ θit − uk Θki 1 k = −(utk θk + uk θtk ) ∧ θit + uk θtk ∧ θit − uk Rijt θj ∧ θt , 2 thus

1 t (duik − utk θit − uit θkt ) ∧ θk = − ut Rijk θj ∧ θk , 2

and, by (1.45), 1 t θj ∧ θk . uikj θj ∧ θk = − ut Rijk 2 Skew-symmetrizing we get 1 1 t (uikj − uijk )θj ∧ θk = − ut Rijk θj ∧ θk , 2 2 thus t = uikj + ut Rtijk . uijk = uikj + ut Rijk

(1.47)

Identities like (1.47) are generally called Ricci identities. As a ﬁrst application of (1.47) we derive the well-known Bochner-Weitzenb¨ ock 2 formula. We denote with |∇u| the squared norm of the vector ﬁeld ∇u and with 2 |Hess(u)| the squared norm of Hess(u) with respect to the natural ﬁber metric induced by , on the tensor bundle T02 (M ) (see e.g. [Lee97], [Pet06a]). Note 2 2 that, in components, |∇u| = ui ui and |Hess(u)| = uij uij . Lemma 1.7. Let u ∈ C 3 (M ); then 1 2 2 Δ|∇u| = |Hess(u)| + Ric (∇u, ∇u) + ∇Δu, ∇u . 2

(1.48)

20

Chapter 1. Some Riemannian Geometry

m 2 Proof. We set v = k=1 (uk ) = uk uk . To compute Δv we need to trace the second covariant derivative of v. Using the Leibniz rule we have vi = 2uk uki and diﬀerentiating once more vit = 2ukt uki + 2uk ukit . Tracing with respect to i and t we obtain 1 2 2 Δ|∇u| = ukt ukt + uk uktt = |Hess(u)| + uk uktt ; 2

(1.49)

however, from (1.47) we deduce uktt = uttk + us Rstkt = uttk + us Rsk , from which uk uktt = ∇Δu, ∇u + Ric (∇u, ∇u). Putting together (1.49) and (1.50) we obtain the desired identity.

(1.50)

We end this section with another useful commutation relation, not so easily available in the literature. First we deﬁne the “(1, 1) versions” of the Hessian (of a suﬃciently smooth function u on M ) and of the Ricci tensor, i.e., the two tensor ﬁeld of type (1, 1), respectively denoted by hess(u) and ric, such that, for all X, Y ∈ X(M ), hess(u)(X), Y = Hess(u)(X, Y ) (1.51) and ric (X), Y = Ric (X, Y ).

(1.52)

Note that, with the notation of section 1.1.2, we can write

hess(u)(X) = (Hess(u)(X, ·)) and

ric (X) = (Ric (X, ·)) . We are now ready to prove the following Lemma 1.8. Let u ∈ C ∞ (M ). Then utkkt − ukktt =

1 ∇S, ∇u + tr (hess(u) ◦ ric). 2

(1.53)

1.2. Comparison theorems

21

Proof. We start from the commutation relations (1.47). By taking covariant derivative we deduce (1.54) uijkt − uikjt = ust Rsijk + us Rsijk, t . Diﬀerentiating both sides of (1.45), using the structure equations and (1.45) itself, we arrive at 1 uijkl θl ∧ θk = − (utj Rtilk + uit Rtjlk )θl ∧ θk , 2 from which, interchanging k and l and adding, we deduce uijkl − uijlk = utj Rtikl + uit Rtjkl . Now, (1.53) follows immediately from (1.54), (1.55), (1.36) and tracing.

1.2.2

(1.55)

Cut locus and regularity of the distance function

We recall a few facts on the cut locus and the Riemannian distance function that will be repeatedly used in the sequel, referring to Chavel’s book [Cha06] for proofs and further details. Let o be a point in the complete manifolds (M, , ), and let γ be a geodesic issuing from o. It is known that γ is locally minimizing. A point q in the image of γ is said to be a cut point for o along γ if γ minimizes the distance from o to q, but ceases to be minimizing after beyond q. The set of cut points of o along geodesic issuing emanating from o is the cut locus of o, and is denoted by cut(o). It turns out that cut(o) is a closed set of measure zero with respect to the Riemannian measure, and that the set Do = M \ cut(o) is an open starshaped domain, which is in fact the maximal domain of the normal geodesic coordinates centered at o. At the tangent space level, we say that v is in the tangent cut locus of o, Cut(o), if the geodesic γv with initial velocity v minimizes distances for t ∈ [0, 1] and does not minimize distances for t > 1. Thus cut(o) is the image of Cut(o) under the exponential map expo , and the set Eo = {tv ∈ To M : v ∈ Cut(o), 0 ≤ t < 1}, is the maximal starshaped domain with respect to 0 on which expo is a diﬀeomorphism, and Do = expo (Eo ). Moreover if r(x) denotes the Riemannian distance function from o, namely, r(x) = d(x, o) = | exp−1 (x)|, then r(x) is smooth on Do \ {o}. Let BR (o) denote the geodesic ball centered at o with radius R, and let ∂BR (o) be its its boundary. Following R.L. Bishop, [Bis77] we say that q is an ordinary cut point for o if there are two or more minimizing geodesics joining o and q. Cut points which are not ordinary are said to be singular . Bishop proves that ordinary cut points are dense in cut(o) ([Bis77], Main Theorem). Since it is easy to verify that the distance function r(x) is not C 1 at ordinary cut points (see [Bis77], Proposition), we deduce that if r(x) is smooth on the punctured ball BR (o) \ {o}, then BR (o) ∩ cut(o) = ∅.

22

1.2.3

Chapter 1. Some Riemannian Geometry

The Laplacian comparison theorem

Now we show how (1.47) is the starting point to derive the classical Laplacian comparison theorem without using Jacobi ﬁelds. Fix a reference point o in (M, , ), and let γ be a minimizing geodesic parameterized by arclength issuing from o; we adopt the standard notation γ˙ to denote the tangent vector of γ. Note that, since γ is a geodesic, we have ∇γ˙ γ˙ = 0. We deﬁne a unit vector ﬁeld Y ⊥ γ˙ along γ by parallel translation (see e.g. [Lee97]); note that γ(t) is an integral curve of ∇r, that is, γ(t) ˙ = (∇r)(γ(t)). To perform calculations we let θi be a local orthonormal coframe and {ei } its dual frame. Then dr = ri θi

and

Y = Y j ej .

By Gauss’ lemma (see for instance [dC92]) ri ri ≡ 1 and since γ is a geodesic we obtain ri rij = 0, j = 1, . . . m. (1.56) Therefore Y j rij ri = 0. Diﬀerentiating the latter equation and using the fact that Y is parallel yields (ri Y j rijk + rij rik Y j )θk = 0; hence, if γ˙ = γ˙ k ek , since rijk = rjik , rist Y i γ˙ s Y t = −rij rik Y j Y k .

(1.57)

Now in formula (1.47) we take u(x) = r(x) to deduce rijk γ˙ k Y j Y i − rist Y i γ˙ s Y t = −Rijkt Y i γ˙ j Y k γ˙ t .

(1.58)

Thus, inserting (1.57) into (1.58), we get rijk γ˙ k Y j Y i + rij rik Y j Y k = −Rijkt Y i γ˙ j Y k γ˙ t . Recalling the deﬁnition (1.51) of hess we let hess(r)(Y ) = ∇Y ∇r, so that Hess(r)(Y, X) = hess(r)(Y ), X . Having set we deﬁne

hess2 (r)(Y ) = hess(r)(hess(r)(Y )), Hess2 (r)(Y, X) = hess2 (r)(Y ), X .

(1.59)

1.2. Comparison theorems

23

Then, since ∇γ˙ Y = 0, (1.59) can be reinterpreted in the form d ˙ (Hess(r)(γ)(Y, Y )) + Hess2 (r)(γ)(Y, Y ) = − Sectγ (Y ∧ γ). dt

(1.60)

Note that (1.56) rewrites as hess(r)(∇r) ≡ 0.

(1.61)

We sum (1.60) over an orthonormal basis {Y i } (i = 2, . . . , m) of γ˙ ⊥ , and use (1.61) to get d 2 (Δr)(γ) + |Hess(r)| (γ) = − Ric(∇r, ∇r)(γ). (1.62) dt Thus, using Newton’s inequality 2

|Hess(r)| ≥ we obtain

(Δr)2 , m−1

d (Δr ◦ γ)2 (Δr ◦ γ) + ≤ − Ric(∇r ◦ γ, ∇r ◦ γ). dt m−1

(1.63)

We recall that, in the literature, Ric(∇r, ∇r) is called radial Ricci curvature. It follows that, if we assume Ric(∇r, ∇r) ≥ −(m − 1)G(r)

(1.64)

for some function G ∈ C 0 ([0, +∞), then (Δr ◦ γ)2 d (Δr ◦ γ) + ≤ (m − 1)G(t). (1.65) dt m−1 √ √ Now we recall that Δr = ( g(r, u))−1 ∂ g/∂r (see for instance [Cha06]), where √ g is the square root of the determinant of the metric in polar geodesic coordinates √ (r, u) centered at o. Also, g = detA(r, u) where A(r, u) is the matrix solution of the diﬀerential equation in u⊥ ⊂ To M , A (r, u) + R(r, u)A(r, u) = 0, satisfying the initial conditions A(0, u) = 0, A (0, u) = Id, and R(r, u) is the composition of the curvature operator at expo (ru) with parallel translation along the geodesic γu (t) = expo (tu) (see again [Cha06], p. 114). Thus A(r, u) = r Id +O(r2 )

and

A (r, u) = Id +O(r)

and we conclude that Δr = log(detA) = tr(A A−1 ) =

m−1 + O(r) r

(1.66)

24

Chapter 1. Some Riemannian Geometry

(see [Pet06a], page 136, for a diﬀerent derivation). Hence, having set ϕ(t) = Δr ◦ γ, using (1.65) and (1.66) and again the fact that γ is parameterized by arclength we deduce that, under assumption (1.64), ⎧ ϕ(t)2 ⎪ ⎪ ⎨ ϕ (t) + ≤ (m − 1)G(t), m−1 (1.67) ⎪ m−1 ⎪ ⎩ ϕ(t) = + o(1) as t → 0+ . t Of course, in order to make sense from the analytical point of view, (1.67) has to be interpreted with the image of γ inside of the domain Do of the normal geodesic coordinates centered at o, or, in other words, outside the cut locus of o. To analyze (1.67) we now need two simple calculus lemmas. Lemma 1.9. Let G ∈ C 0 ([0, +∞)) and let ϕ, ψ ∈ C 2 ((0, +∞)) ∩ C 1 ([0, +∞)) be solutions of the problems: ϕ − Gϕ ≤ 0, ψ − Gψ ≥ 0, i) (1.68) ii) ϕ(0) = 0; ψ(0) = 0, ψ (0) > 0. If ϕ(r) > 0 for r ∈ (0, T ) and ψ (0) ≥ ϕ (0), then ψ(r) > 0 in (0, T ) and ϕ ψ ≤ , ψ ≥ ϕ on (0, T ). ϕ ψ

(1.69)

Proof. Since ψ (0) > 0, ψ > 0 in a neighborhood of 0. We observe in passing that if G is assumed to be nonnegative, then, integrating (1.68) ii), we have r G(s)ψ(s) ds, ψ (r) ≥ ψ (0) + 0

so that ψ is positive in the interval where ψ ≥ 0, and we conclude that, in fact, ψ > 0 on (0, +∞). In the general case, where no assumption is made on the sign of G, we let β = sup {t : ψ > 0 in (0, t)}; τ = min {β, T }. The function ψ ϕ − ψϕ ∈ C 0 ([0, +∞)) vanishes in r = 0, and it satisﬁes (ψ ϕ − ψϕ ) = ψ ϕ − ψϕ ≥ 0 in (0, τ ). Thus, ψ ϕ − ψϕ ≥ 0 on [0, τ ), and, dividing through by ϕψ, we deduce that ϕ ψ ≥ in (0, τ ). ψ ϕ

1.2. Comparison theorems

25

Integrating between ε and r, 0 < ε < r < τ , yields ϕ(ε) ψ(r) ψ(ε)

ϕ(r) ≤ and, since lim+

ε→0

ϕ(ε) ϕ (0) = ≤ 1, ψ(ε) ψ (0)

we conclude that in fact ϕ(r) ≤ ψ(r) in [0, τ ). Since ϕ > 0 in (0, T ) by assumption, this in turn forces τ = T , for, otherwise, τ = β < T and we would have ϕ(β) > 0, while, by continuity, ψ(β) = 0, a contradiction. Lemma 1.10. Let G ∈ C 0 ([0, +∞)) and let gi ∈ C 1 ((0, Ti )), i = 1, 2 be solutions of the Riccati diﬀerential inequalities i) g1 +

g12 − αG ≤ 0; α

ii) g2 +

g22 − αG ≥ 0 α

(1.70)

satisfying the condition gi (t) =

α + O(1) as t → 0+ t

(1.71)

for some α > 0. Then T1 ≤ T2 and g1 (t) ≤ g2 (t) in (0, T1 ). Proof. Since gi = α−1 gi satisfy the conditions in the statement with α = 1, without loss of generality we assume α = 1. Observe that the functions gi (s) − 1s are bounded and integrable in a neighborhood of s = 0, thus we deﬁne ϕi ∈ C 2 ((0, Ti )) ∩ C 1 ([0, Ti )) on [0, Ti ), by setting ϕi (t) = te

t 0

(gi (s)− 1s ) ds

.

Then ϕi (0) = 0, ϕi > 0 on (0, Ti ) and straightforward computations show that ϕi (t) = gi (t)ϕi (t), and

ϕi (0) = 1

ϕ1 ≤ Gϕ1

on (0, T1 );

ϕ2

on (0, T2 ).

≥ Gϕ2

An application of Lemma 1.9 shows that T1 ≤ T2 and g1 = (0, T1 ), as required.

ϕ1 ϕ1

≤

ϕ2 ϕ2

= g2 on

We are now ready to prove the next Laplacian comparison theorem, which is a simpliﬁed (but suﬃcient for our purposes) version of that appearing in [PRS08]:

26

Chapter 1. Some Riemannian Geometry

Theorem 1.11. Let (M, , ) be a complete manifold of dimension m ≥ 2. Having ﬁxed a reference point o ∈ M , let r(x) = distM (x, o). Assume that the radial Ricci curvature Ric(∇r, ∇r) of M satisﬁes Ric(∇r, ∇r) ≥ −(m − 1)G(r)

(1.72)

for some nonnegative function G ∈ C 0 ([0, +∞)). Let h ∈ C 2 ([0, +∞)) be a solution of the problem h − Gh ≥ 0, (1.73) h(0) = 0, h (0) = 1. Then the inequality Δr(x) ≤ (m − 1)

h (r(x)) h(r(x))

(1.74)

holds pointwise on M \({o} ∪ cut(o)) and weakly on all of M . Proof. Fix any x ∈ M \({o} ∪ cut(o)) and let γ : [0, l] → M be a minimizing geodesic from o to x parameterized by arclength. We then arrive to (1.67), where the diﬀerential inequality is in (0, l]. Since g = (m − 1) hh satisﬁes g (t) +

g(t)2 ≥ (m − 1)G(t) on (0, +∞) m−1

(1.75)

and (1.71) with α = m − 1, an application of Lemma 1.10 to (1.67) and (1.75) gives h (t) in (0, l]. ϕ(t) ≤ (m − 1) h(t) Thus, in particular, since γ(l) = x and r(x) = l, Δr(x) ≤ (m − 1)

h (r(x)) , h(r(x))

showing the validity of (1.74) pointwise within the cut locus. It remains to show the validity of (1.74) weakly in all of M , which is guaranteed by the following lemma. Lemma 1.12. Set Do = M \ cut(o) and suppose that Δr ≤ α(r) pointwise on Do \{o},

(1.76)

for some α ∈ C 0 ((0, +∞)). Let v ∈ C 2 (R) be nonnegative and set u(x) = v(r(x)) on M . Suppose either i) v ≤ 0

or

ii) v ≥ 0.

(1.77)

Then we respectively have i) Δu ≥ v (r) + α(r)v (r); weakly on M .

ii) Δu ≤ v (r) + α(r)v (r)

(1.78)

1.2. Comparison theorems

27

Proof. Let Eo be the maximal star-shaped domain in To M on which expo is a diﬀeomorphism onto its image Do , so that we have cut(o) = ∂(expo (Eo )). Since Eo is a star-shaped domain, we can exhaust Eo by a family {Eon } of relatively compact, star-shaped domains with smooth boundary. We set Don = expo (Eon ) so that n Do ⊂ Don+1 and Don = Do . n

The fact that each Eon is star-shaped implies ∂r = ∇r, νn > 0 on ∂Don , ∂νn

(1.79)

where νn denotes the outward unit normal to ∂Don . Now we assume the validity of (1.77) i). Since r ∈ C ∞ (Don \{o}), computing we get Δu ≥ v + α(r)v

pointwise on Don \{o}.

(1.80)

Let 0 ≤ ϕ ∈ C0∞ (M ), where C0∞ (M ) denotes the set of smooth functions with compact support on M . We claim that, ∀ n, uΔϕ ≥ (v + α(r)v )ϕ + εn , Don

Don

where εn → 0 as n → +∞. Since M = Do ∪ cut(o) and cut(o) has measure 0, inequality (1.78) i) will follow by letting n → +∞. To prove the claim we ﬁx δ > 0 small and we apply the second Green formula (see e.g. [GT01]) on Don \Bδ (o) to obtain ∂u ∂ϕ ϕ , (1.81) uΔϕ = ϕΔu − −u ∂νn ∂νn Don \Bδ (o) Don \Bδ (o) ∂Don \∂Bδ (o) where νn is the outward unit normal to ∂Don \∂Bδ (o). We note that, according to (1.77) i) and (1.79), ∂r ∂u = v (r) ≤ 0 on ∂Don . ∂νn ∂νn Using this, (1.79) and (1.81) we obtain uΔϕ ≥ (v + α(r)v )ϕ + εn + Iδ , Don

Don

with

∂ϕ , ∂νn [uΔϕ − (v + α(r)v )ϕ] − εn =

Iδ =

u

∂Don

Bδ (o)

! ∂Bδ (o)

" ∂ϕ ∂u u −ϕ , ∂r ∂r

28

Chapter 1. Some Riemannian Geometry

∂· means ∇r, ∇·. Clearly, Iδ → 0 as δ ↓ 0+ ; on the other where the notation ∂r ∞ hand, since ϕ ∈ C0 (M ) and cut(o) has measure 0, using the divergence and Lebesgue theorems we see that, as n → +∞, div(u∇ϕ) → div(u∇ϕ) = div(u∇ϕ) = 0. εn = Don

Do

M

This proves the claim and the validity of (1.78) i). The case (1.77) ii) and (1.78) ii) can be dealt with in a similar way. Remark. We note that, for the above proofs to work, it is not necessary that (1.72) holds on the entire M : instead, for instance, if (1.72) is valid on BR (o), then (1.74) holds on BR (o)\({o} ∪ cut(o)) and weakly on BR (o).

1.2.4

The Bishop-Gromov comparison theorem

We now show how to get from the previous results a (somewhat generalized) version of what is known in the literature as the Bishop-Gromov comparison theorem (see also [PRS08]). We denote with vol BR (o) and vol ∂BR (o) the volume of the geodesic ball BR (o) and of its boundary ∂BR (o), respectively. Theorem 1.13. Let (M, , ) be a complete, m-dimensional Riemannian manifold satisfying Ric (∇r, ∇r) ≥ −(m − 1)G(r) on M (1.82) for some G ∈ C 0 ([0, +∞)), G ≥ 0, where r(x) = dist(x, o). Let h ∈ C 2 ([0, +∞)) be the nonnegative solution of the problem h − G(t)h = 0, (1.83) h(0) = 0, h (0) = 1. Then, for almost every R > 0, the function R →

vol ∂BR (o) h(R)m−1

(1.84)

is nonincreasing, and vol ∂BR (o) ≤ ωm h(R)m−1 ,

(1.85)

where ωm is the volume of the unit sphere in Rm . Moreover, vol BR (o) R → # R h(t)m−1 dt 0 is a nonincreasing function on (0, +∞).

(1.86)

1.2. Comparison theorems

29

Since it will be used in the proof of Theorem 1.13, and also in the next chapters, we ﬁrst recall the useful co-area formula. We denote with W 1,1 (M ) the Sobolev space consisting of functions in L1 (M ) with (weak) gradient also in L1 (M ). We also denote with Aft the t-level set (t ∈ R) of a function f on M , i.e., Aft = {x ∈ M |f (x) = t}. Following Schoen and Yau (see [SY94], p.89) we can state the following Proposition 1.14. Let M be a compact Riemannian manifold with boundary and f ∈ W 1,1 (M ). For any nonnegative measurable function g on M the following formula holds: % +∞ $ g dt. (1.87) g= −∞ M Aft |∇f | For a proof see the classical [Fed69]. Note that, in particular, if f (x) = r(x) = distM x, o, equation (1.87) becomes

D

$

%

g= M

g dt, 0

(1.88)

∂Bt (o)

where D = supM r(x). Proof of Theorem 1.13. In case o is a pole of M (see [GW79] and also the proof of Theorem 6.8 in Chapter 6) one integrates the divergence of the radial vector ﬁeld X = h(r(x))−m+1 ∇r on concentric balls BR (o), and uses the divergence and Laplacian comparison theorems. However, in general, objects are nonsmooth and inequalities are intended in the sense of distributions. Therefore, we have to take some extra care. The Laplacian comparison theorem asserts that Δr(x) ≤ (m − 1)

h (r(x)) h(r(x))

(1.89)

pointwise on the open, star-shaped, full measured set M \ cut(o) and weakly on all of M . Thus, ∀ 0 ≤ ϕ ∈ Lip0 (M ), −

∇r, ∇ϕ ≤ (m − 1)

h (r(x)) ϕ. h(r(x))

(1.90)

∀ ε > 0, consider the radial cut-oﬀ function ϕε (x) = ρε (r(x))h(r(x))−m+1 ,

(1.91)

30

Chapter 1. Some Riemannian Geometry

where ρε is the piecewise linear function ⎧ ⎪ 0, if ⎪ ⎪ ⎪ t−r ⎪ ⎪ if ⎨ ε , ρε (t) = 1, if ⎪ ⎪ R−t ⎪ if ⎪ ε , ⎪ ⎪ ⎩0, if Note that

t ∈ [0, r) t ∈ [r, r + ε) t ∈ [r + ε, R − ε) t ∈ [R − ε, R) t ∈ [R, +∞).

(1.92)

' χr, r+ε h (r(x)) χR−ε, R + − (m − 1) ρε h(r(x))−m+1 ∇r, − ε ε h(r(x))

&

∇ϕε =

for almost all x ∈ M , where χs, t is the characteristic function of the annulus Bt (o)\Bs (o). Therefore, using ϕε into (1.90) and simplifying, we get 1 1 −m+1 h(r(x)) ≤ h(r(x))−m+1 . ε BR (o)\BR−ε (o) ε Br+ε (o)\Br (o) Using the co-area formula (1.87)) we deduce that 1 r+ε 1 R −m+1 vol(∂Bt (o)) h(t) ≤ vol(∂Bt (o)) h(t)−m+1 ε R−ε ε r and, letting ε ↓ 0,

vol(∂BR (o)) vol(∂Br (o)) ≤ h(R)m−1 h(r)m−1

(1.93)

for almost all 0 < r < R. Letting r → 0 and recalling that h(r) ∼ r and vol(∂Br ) ∼ ωm rm−1 as r → 0 (which can be deduced, for instance, integrating equation (1.66) on a geodesic ball and using the divergence theorem and Gauss’ lemma), we conclude that, for almost any R > 0, vol ∂BR (o) ≤ ωm h(R)m−1 . To prove the second statement we note that, as observed in [CGT82], for general real valued functions f (t) ≥ 0, g(t) > 0, if t →

f (t) g(t)

is decreasing, then t →

t

0t 0

f g

is

decreasing. Indeed, since f /g is decreasing, if 0 < r < R, r R r R r R r R f R f (r) r f f g= g g≥ g g≥ g g = g f g r g(r) 0 g 0 r 0 r 0 r 0 r whence r f 0

R 0

r

g= 0

r

f 0

r

g+ 0

R

g≥

f r

r 0

r

f 0

r

g+ 0

R

g r

r

f=

R

g 0

f. 0

1.2. Comparison theorems

31

In particular, applying this observation to (1.93) and using the co-area formula (1.87) we deduce that vol Br (o) r → # r h(t)m−1 dt 0

is decreasing, concluding the proof.

We conclude with the following analytical result whose proof can be found in [PRS08]: Proposition 1.15. Assume h is a solution of h − H 2 (1 + r2 )δ/2 h = 0, h(0) = 0, h (0) = 1 where H > 0 and δ ≥ −2. Set H, ( H = 1 1 + 4H 2 ), 2 (1 +

if δ > −2 if δ = −2.

Then,

h (r) ≤ H rδ/2 (1 + o(1)) as r → +∞. h Moreover, there exists a constant C > 0 such that for r > 1,

⎧ 2H 1+δ/2 ⎪ exp , if δ ≥ 0 (1 + r) ⎪ 2+δ ⎨

2H −δ/4 1+δ/2 h(r) ≤ C r , if − 2 < δ < 0 exp 2+δ r ⎪ ⎪ ⎩ H r , if δ = −2.

1.2.5

The Hessian comparison theorem

For the sake of completeness we recall here the following Hessian comparison theorem; however, since it will only be used a very few times in the sequel, we only give its statement (the interested reader can ﬁnd the proof in [PRS08]). Recall that the radial sectional curvature Krad of a manifold is the sectional curvature of a 2-plane containing ∇r. Theorem 1.16. Let (M, , ) be a complete manifold of dimension m. Having ﬁxed a reference point o ∈ M , let r(x) = distM (x, o), and let Do = M \ cut(o) be the domain of the normal geodesic coordinates centered at o. Given a smooth even function G on R, let h be the solution of the Cauchy problem h − Gh = 0, (1.94) h(0) = 0, h (0) = 1,

32

Chapter 1. Some Riemannian Geometry

and let I = [0, r0 ) ⊆ [0, +∞) be the maximal interval where h is positive. If the radial sectional curvature of M satisﬁes Krad ≥ −G(r(x)) then Hess(r) ≤

on Br0 (o),

h (r(x)) { , − dr ⊗ dr} h(r(x))

(1.95)

(1.96)

on Do \ {o} ∪ Br0 (o) in the sense of quadratic forms. On the other hand, if Krad ≤ −G(r(x)) then Hess(r) ≥

on Br0 (o),

h (r(x)) { , − dr ⊗ dr}. h(r(x))

(1.97)

(1.98)

Remark. By taking traces in Theorem 1.16 we immediately obtain the corresponding estimates for Δr. However, as we have seen in Theorem 1.11, the estimate from above for the Laplacian of the distance function holds under the weaker assumption that the radial Ricci curvature (and not the full radial sectional curvature) is bounded from below by −(m − 1)G(r(x)). Furthermore the estimate in this latter case can be extended, in weak form, to the entire manifold. This is not the case for the above estimates on Hess(r).

1.3

Some formulas for immersed submanifolds

Let (N, , ) and M be respectively a Riemannian manifold and a manifold of dimensions n and m, with m ≤ n. Let f : M → N be an immersion and let f ∗ , be the metric induced on M by f , where f ∗ denotes the pull-back (note that f ∗ , is indeed a metric since f is an immersion). If g is a given metric on M and f : M → N is an immersion we will say that f is an isometric immersion if g = f ∗ , . To simplify notation we use the symbol , on M to denote the induced metric; more generally, from now on we shall omit the pull-back notation, being clear from the context where forms or tensors are considered. We ﬁx the following indexes convention: 1 ≤ i, j, k, . . . ≤ m,

m + 1 ≤ α, β, γ, . . . ≤ n,

1 ≤ a, b, c, . . . ≤ n.

Deﬁnition 1.17. Given the isometric immersion f : M → (N, , ), a Darboux coframe along f is a local o.n. coframe θi on N such that θα = 0

on M.

(1.99)

In particular, for a Darboux coframe along f we have , =

m i 2 θ i=1

on M.

(1.100)

1.3. Some formulas for immersed submanifolds

33

The dual {ea } of a Darboux coframe is called a Darboux frame along f and condition (1.99) is equivalent to say that the vectors {ei } (locally) span (the image of) T M in T N , while {eα } are orthogonal to T M (and span in fact T M ⊥ , the normal bundle that will be deﬁned afterward). The existence of Darboux coframes along f can, of course, be proved analytically but the above geometric meaning is “evidence” of their existence. We let {θba } be the Levi-Civita connection forms relative to {θa } on N . Then θji on M are the Levi-Civita connection forms of the induced metric. Indeed, obviously, θji + θij = 0

on M.

(1.101)

Furthermore, from the ﬁrst structure equations on N and by (1.99), dθi = −θji ∧ θj .

(1.102)

We now recall the following elementary but useful Lemma 1.18 (Cartan’s lemma). Let U ⊂ M be an open set of the Riemannian assume that a set of manifold (M, , ). Let {θi } be a local basis of T ∗ U , and 1-forms {ωJi } on U , where J is any set of indices, satisﬁes i ωJi ∧ θi = 0. Then, there exist smooth functions biJ,k on U such that ωJi = biJ,k θk

and

biJ,k = bkJ,i ,

that is, the matrix B = (biJ,k )ik is an n × n symmetric matrix of 1-forms. Proof. We can write ωJi as ωJi = biJ,k θk for some smooth functions biJ,k on U . Then from i ωJi ∧ θi = 0 we deduce biJ,k θk ∧ θi = (biJ,k − bkJ,i )θk ∧ θi , 0= i,k

i

which easily implies the thesis.

To obtain further information we diﬀerentiate equations (1.99), use (1.102) and (1.99) again to obtain 0 = dθα = −θiα ∧ θi − θβα ∧ θβ = −θiα ∧ θi . Hence, by Cartan’s lemma 1.18, there exist (locally deﬁned) smooth functions hα ij such that j (1.103) θiα = hα ij θ , with α hα ij = hji .

(1.104)

hα ij

are the coeﬃcients of the second fundamental tensor II (a tensor along f ) The of the immersion, which in the present setting is deﬁned by i j II = hα ij θ ⊗ θ ⊗ eα .

(1.105)

34

Chapter 1. Some Riemannian Geometry

One can verify that II is globally deﬁned and symmetric. The mean curvature vector ﬁeld is 1 1 eα . H = II(ei , ei ) = hα m m ii If ν is a globally deﬁned unit normal vector ﬁeld, we deﬁne the mean curvature in the direction of ν as hν = H, ν . If m + 1 = n and both the hypersurface M and N are orientable, we can choose Darboux frames along f preserving orientations, that is such that θ1 ∧ · · · ∧ θm+1 and θ1 ∧ · · · ∧ θm give the correct orientations respectively of N and M . In this case, the vector ﬁeld em+1 dual to θm+1 on N is a global normal vector ﬁeld on M . The mean curvature in the direction of em+1 is called the mean curvature of the immersed hypersurface. Moreover, we have the following deﬁnitions: • if II ≡ 0 on M , then the immersion is said to be totally geodesic; • if II − H , ≡ 0 on M , then the immersion is said to be totally umbilical, and an umbilical point p is a point of M where IIp − Hp , p = 0; • if H ≡ 0 on M , then the immersion is said to be minimal (this last deﬁnition comes from the variational principle of the area functional). On M we can consider the second structure equations dθji = −θki ∧ θjk + Ωij

(1.106)

1M i k Rjkl θ ∧ θl . 2

(1.107)

with Ωij the curvature forms Ωij =

Our aim is to relate the curvature of M with that of N . We let Θab =

1N a c Rbcd θ ∧ θd 2

be the curvature forms on N . Pulling back the second structure equations of N and using (1.107), (1.103) we obtain dθji = −θki ∧ θjk − θαi ∧ θjα + Θij α k l = −θki ∧ θjk + hα ik hjl θ ∧ θ +

1N i k Rjkl θ ∧ θl . 2

Therefore, skew-symmetrizing in k and l, Ωij = dθji + θki ∧ θjk =

k 1 α α N i α l h h − hα il hjk + Rjkl θ ∧ θ . 2 ik jl

1.3. Some formulas for immersed submanifolds

35

From the above we thus obtain the Gauss equations M

i i α α α Rjkl = N Rjkl + hα ik hjl − hil hjk .

(1.108)

We now diﬀerentiate (1.103) and use (1.102) and the structure equations to obtain a classical set of commutation relations: j = −θjα ∧ θij − θγα ∧ θiγ + Θα 0 = dθiα − d hα ij θ i j α j k α j γ − dhα ij ∧ θ + hij θk ∧ θ + hij θγ ∧ θ j γ α k k α j α j k = hα jk θi ∧ θ − hik θγ ∧ θ − dhij ∧ θ + hij θk ∧ θ 1 α + N Rijk θj ∧ θk . 2

Setting β α k α α k α k hα ijk θ = dhij − hkj θi − hik θj + hij θβ

(1.109)

the above rewrites as k hα ijk θ

1 α + N Rijk θk 2

∧ θj = 0.

α , with Hence, by Cartan’s lemma 1.18 there exist lijk α α lijk = likj ,

such that k hα ijk θ +

1N α k α Rijk θ = lijk . 2

However, 1N α k α Rikj θ = likj ; 2 thus, subtracting and recalling the symmetries of the curvature tensor, k hα ikj θ +

N α α hα ijk − hikj = − Rijk .

(1.110)

These commutation rules are known as Codazzi equations. Note that the coeﬃcients hα ijk deﬁned in (1.109) are the coeﬃcients of the covariant derivative of II. Although for the sequel we will not use the Van der Waerden-Bortolotti covariant derivation, we brieﬂy describe it in a few words in the above formalism. Given the immersion f : M → (N, , ), on M we have a well-deﬁned bundle, the normal bundle T M ⊥ that pointwise is the orthogonal complement of Tp M in Tp N . Given a Darboux coframe along f we locally deﬁne a covariant derivative by setting Deα = θαβ ⊗ eβ .

36

Chapter 1. Some Riemannian Geometry

)

* θβα are called the connection forms (one veriﬁes that this deﬁnition is meaningful globally). We let the curvature forms Φα β be deﬁned via the second structure equations as follows: (1.111) dθβα = −θγα ∧ θβγ + Φα β, and setting Φα β =

1⊥ α i Rβij θ ∧ θj ; 2

α are called the components of the normal curvature tensor. Comparing the ⊥ Rβij (1.111) with the pull-back of the second structure equations of N , that is,

dθβα = −θγα ∧ θβγ − θiα ∧ θβi + Θα β we deduce β α α Φα β = θi ∧ θi + Θ β .

A simple computation as those presented above shows that ⊥

β N α α α β Rβij = hα ki hkj − hkj hki + Rβij .

These equations are often called the Ricci equations.

(1.112)

Chapter 2

Pointwise conformal metrics At the beginning of this chapter we introduce the basic formalism and the derivation of the geometric Yamabe equation. Then, we concentrate on the case where M is compact to illustrate the interplay between geometry and analysis, with a few illuminating examples such as the Kazdan-Warner obstruction, a result of Obata on Einstein manifolds, the far-reaching “generalization” of Bidaut-V´eron and V´eron and a result of Escobar. Along the way we give a detailed proof, which inspires to P. Petersen’s treatise [Pet06a], of a famous rigidity result of Obata. In this way, we hope to provide some geometrical feeling on the subject of this monograph that will enable us to proceed with the noncompact case: the case of the rest of our investigation.

2.1 2.1.1

The Yamabe equation The derivation of the Yamabe equation

Let (M, , ) be a Riemannian manifold and consider a pointwise conformal deformation of the metric , , that is, a new metric on M of the form , = ϕ2 , ,

(2.1)

the curvature tensor with ϕ a strictly positive smooth function. Denoting with R of , , we want to determine the relationship between R and R. Let θi , i = 1, . . . , m = dim M , be a local orthonormal coframe on (M, , ) with corresponding Levi-Civita connection forms θji . Then, in the new metric , , θi = ϕθi , i = 1, . . . , m, (2.2)

is a local orthonormal coframe on M, , . To determine the corresponding connection forms we could use the general theory developed in Chapter 1, but it

P. Mastrolia et al., Yamabe-type Equations on Complete, Noncompact Manifolds, Progress in Mathematics 302, DOI 10.1007/978-3-0348-0376-2_2, © Springer Basel 2012

37

38

Chapter 2. Pointwise conformal metrics

is easy to deduce that, if dϕ = ϕt θt , the 1-forms ϕj i ϕi j θ − θ θji = θji + ϕ ϕ

(2.3)

are skew-symmetric and satisfy the ﬁrst structure equation. Thus, they are the desired connection forms relative to the coframe deﬁned in (2.2). In order to determine the curvature forms, we use the structure equations and the expression for the components of the Hessian (see equation (1.41)) to compute ij = dθji + θki ∧ θjk Θ ϕj ϕj i ϕi ϕi ∧ θi + ∧ θj − dθj + θki ∧ θjk dθ − d = dθji + d ϕ ϕ ϕ ϕ = −θki ∧ θjk + Θij + (−ϕ−2 ϕk ϕj θk + ϕ−1 dϕj ) ∧ θi − ϕ−1 ϕj θki ∧ θk − (−ϕ−2 ϕk ϕi θk + ϕ−1 dϕi ) ∧ θj + ϕ−1 ϕi θkj ∧ θk ϕk i ϕi k ϕj k ϕk j i k θ − θ ∧ θj + θ − θ + θk + ϕ ϕ ϕ ϕ ϕk ϕk i ϕjk ϕik ϕj ϕk ϕi ϕk θk ∧ θi − θk ∧ θj − θ ∧ θj , = Θij + −2 2 −2 2 ϕ ϕ ϕ ϕ ϕ2 that is, i = Θi + Θ j j −

ϕk ϕj ϕi ϕk ϕjk ϕik − 2 2 δti θk ∧ θt − − 2 2 δtj θk ∧ θt ϕ ϕ ϕ ϕ

ϕl ϕl i j k δ δ θ ∧ θt . ϕ2 k t

Hence, anti-symmetrizing the coeﬃcients of the wedge products on the right-hand side, and recalling the deﬁnition of the curvature tensor, we obtain i = Ri + ϕjk − 2 ϕk ϕj δ i − ϕjt − 2 ϕt ϕj δ i ϕ2 R (2.4) jkt jkt t k ϕ ϕ2 ϕ ϕ2 ϕit ϕi ϕk ϕi ϕt ϕik − 2 2 δtj + − 2 2 δkj − ϕ ϕ ϕ ϕ

ϕl ϕl i j i j − 2 δk δt − δt δk . ϕ Taking traces with respect to i and k we have jt = Rjt − (m − 2) ϕ2 R

ϕjt ϕl ϕ l ϕkk j ϕj ϕ t + 2(m − 2) 2 − (m − 3) 2 δtj − δ . ϕ ϕ ϕ ϕ t

(2.5)

Thus, denoting with |∇ϕ|, Hess(ϕ) and Δϕ respectively the length of the gradient, the Hessian and the Laplacian of ϕ in the metric , , and recalling that Rjt (resp.

2.1. The Yamabe equation

39

jt ) are the components of the Ricci tensor Ric with respect to the orthonormal R with respect to θi ), we have basis θi (resp. of Ric = Ric −(m − 2) 1 Hess(ϕ) + 2(m − 2) 1 dϕ ⊗ dϕ Ric ϕ ϕ2 2 |∇ϕ| Δϕ − (m − 3) 2 , − , . ϕ ϕ

(2.6)

A further tracing of (2.5) with respect to j and t yields ϕ2 S = S − 2(m − 1)

|∇ϕ|2 Δϕ − (m − 1)(m − 4) 2 . ϕ ϕ

(2.7)

In case m = dim M ≥ 3, we set 2

ϕ = u m−2

so that

4 , = u m−2 , .

(2.8)

In this case, (2.7) immediately gives m−2 = 0, cm Δu − Su + Su m+2

(2.9)

with cm = 4 m−1 m−2 . In case m = 2 we set ϕ = eu In this case (2.7) gives

so that

, = e2u , .

2u = 0. 2Δu − S + Se

(2.10)

(2.11)

Equations (2.9) and (2.11) are the classical Yamabe equations. We conclude this section with an immediate application of (2.7) in the compact case, improving on a result of Obata, [Oba62b]. First we need the next simple Lemma 2.1. Let (M, , ) be a compact Riemannian manifold. Then, every homothetic diﬀeomorphism is an isometry. Proof. Let ϕ : M → M be a diﬀeomorphism such that ϕ∗ , = c2 , for some constant c > 0. By contradiction, suppose c = 1. Without loss of generality, we can assume c > 1 (indeed, in case c < 1 it suﬃces to consider ϕ−1 ). Now, for every n ∈ N, let ϕ(n) : M → M be the n-th iterate of ϕ. Then

∗ ϕ(n) , = c2n , , proving that, for any ﬁxed p = q ∈ M ,

c2n d (p, q) = d ϕ(n) (p) , ϕ(n) (q) ≤ diam (M ) < +∞. Since c > 1, taking the limit as n → +∞, we obtain the desired contradiction.

40

Chapter 2. Pointwise conformal metrics We are now ready to prove the following

Theorem 2.2. Let (M, , ) be a compact manifold of dimension m ≥ 3 (for the ease of exposition) with scalar curvature s(x) ≤ 0. Let ψ : M → M be a conformal diﬀeomorphism with scalar curvature S(x). Thus ψ is an isometry if and only if S(x) = kS(x) for some k ∈ (0, +∞). Proof. If ψ is an isometry, clearly k = 1. Vice versa, to simplify notation let m+2 1−σ for some a ∈ (0, +∞). Since ψ is conformal cm = 4 m−1 m−2 , σ = m−2 and k = a and m ≥ 3, from equation (2.9) we deduce (2.12) cm Δu = S(x) u − a1−σ uσ with u > 0, u ∈ C ∞ (M ) such that 4

ψ ∗ , = u m−2 , . Deﬁne the vector ﬁeld

!

" u σ−1 − 1 ∇u; W = cm a

computing its divergence, using (2.12) and the divergence theorem we have

!

"2 u σ−1 1 u σ−1 2 − 1 S(x)u = cm (σ − 1) |∇u| . a u a M M

Since S(x) ≤ 0 it follows that |∇u| ≡ 0 and u ≡ a, that is ψ : M → M is a homothety. The result now follows from Lemma 2.1. We shall come back to this kind of problems again in Corollary 2.9 below and later in the complete noncompact case.

2.1.2

The Kazdan-Warner obstruction

Let now T denote the traceless Ricci tensor, that is (see equation (1.38)) T = Ric −

S , . m

T enables us to immediately ﬁnd an obstruction to the existence of a conformally deformed metric as in (2.8) or (2.10) with assigned scalar curvature S(x). We shall consider the case m ≥ 3 so that we shall provide a necessary condition for the existence of a positive solution on M of equation (2.9). Indeed, with respect to a local orthonormal coframe θi we have Tij = Rij −

S δij . m

2.1. The Yamabe equation

41

On the other hand, from (1.36), Rli,i =

1 Sl . 2

Thus, tracing the covariant derivative Tij,k = Rij,k −

Sk δij m

with respect to j and k yields Sk m−2 δik = Si . (2.13) m 2m Now, let X be a vector ﬁeld on M and consider the vector ﬁeld W associated to the 1-form T (X, ·) using the duality induced by the metric, namely, Tik,k = Rik,k −

W = T (X, ·) = X i Tij ej ,

Tij = Tij , (2.14) j where {ej } is the orthonormal frame dual to the coframe θ . Since the covariant derivative satisﬁes the Leibniz rule, the divergence of W is given by div W = Wkk = Xki Tik + X i Tik,k .

(2.15)

On the other hand, using the symmetry of Tik and (1.20), 1 1 Tik Xki = Xki + Xik Tik = (LX , )ik Tik . 2 2 It follows that 1 m−2 X(S). (2.16) div W = (LX , )ik Tik + 2 2m We now recall that a vector ﬁeld X is said to be conformal if it generates a local 1-parameter group ϕt of local conformal diﬀeomorphisms, that is (ϕ∗t , )p = ρt (p) , p ,

p∈M

for some positive function ρt . Using the formula which expresses the Lie derivative LX as a derivative along the ﬂow generated by X, namely, ϕ∗t , − , , t→0 t

LX , = lim

(see e.g. [Lee03], p. 472 ﬀ.) one proves that X is conformal if and only if LX , = λ , for some function λ = λ(p), which, using (1.19), is easily seen to be equal to 2 m (div X). Substituting into (2.16) and using the fact that T is traceless, we deduce that m−2 div W = X(S) (2.17) 2m whenever X is a conformal vector ﬁeld on M . Thus we have the next result which is known as the Kazdan-Warner obstruction (see [KW74a] and [BE87]):

42

Chapter 2. Pointwise conformal metrics

Theorem 2.3. Let (M, , ) be a compact manifold of dimension m ≥ 3 and , Then, for each a metric on M conformally related to , with scalar curvature S. conformal vector ﬁeld X on (M, , ) we have

+ = 0. dvol X(S)

(2.18)

M

Proof. Since X is conformal with respect to the metric , , it generates a ﬂow of local diﬀeomorphisms which are conformal transformations for the metric , and therefore also for the conformally related metric , . It follows that X is conformal also in the metric , . Considering the analogous (2.17) in the metric , and integrating with the aid of the divergence theorem we obtain (2.18). Remark. As a further application of the method of a moving frame, we give here , in terms of LX , and , , from which it can be an explicit formula for LX again deduced that a vector ﬁeld X conformal w.r.t. the metric , is conformal also in the metric , . Indeed we have the following Lemma 2.4. Let X ∈ X(M ) be a vector ﬁeld on the Riemannian manifold (M, , ), and let , = ϕ2 , , ϕ > 0, be a conformally deformed metric. Then LX , = ϕ2 LX , + 2ϕ X, ∇ϕ , .

(2.19)

Proof. We choose a local o.n. coframe θi with associated frame {ei }. First we observe that (2.2) implies that ei = ϕ−1 ei ; then, with the notation of section 1.1.2 we have i ei (2.20) X = X i ei = X and ∇X = Xki θk ⊗ ei ,

=X i θk ⊗ ei ∇X k

(2.21)

i θk = (dX i + X j θi ). From (2.20) we deduce with Xki θk = (dX i + X j θji ) and X j k i = ϕX i , X

(2.22)

while a computation using (2.21) and (2.3) shows that ki = Xki + ϕ−1 X i ϕk + X j ϕj δik − ϕi X k , X

(2.23)

ki = Xki + Xki + 2ϕ−1 X j ϕj δik . ki + X X

(2.24)

which implies

Equation (2.19) now follows easily from (2.24) and (1.20).

2.1. The Yamabe equation

43

Example. The m-dimensional standard sphere Sm , m ≥ 3, can be realized, outside the North pole N and the South pole S, as the model manifold (see also Chapter 4, section 4.3)

2 (0, π) × Sm−1 , dr ⊗ dr + (sin r) dθ2 , where dθ2 denotes the standard metric on Sm−1 . Using this isometric representation, a conformal vector ﬁeld X on Sm is given by XN = XS = 0.

X = sin r∇r,

Now, consider any smooth, radial function s (r (x)) on Sm . Then, by the co-area formula (1.87),

d s + X, ∇r dvol dr Sm π d s + dr = sin r dvol 0 ∂Br dr π d s + (∂Br ) . sin r vol = dr 0

+ = X( s) dvol Sm

,

Accordingly, if s (r) is monotonic and nonconstant, we deduce Sm

+ = 0. X( s) dvol

It follows from Theorem 2.3 that any nonconstant monotonic function s (r (x)) cannot be the scalar curvature of a pointwise conformal deformation of the canonical metric of Sm .

2.1.3

The Weyl and Cotton tensors

We now establish a relation between − S , T = Ric m

and

T = Ric −

S , , m

(2.25)

that will be used below in the proof of Theorem 2.8. Suppose, once again, that (2.1) holds, that is, , = ϕ2 , . Using (2.6) and (2.7), after some computations we obtain

44

Chapter 2. Pointwise conformal metrics

− S T = Ric , m ' & 1 1 Hess(ϕ) − 2 2 dϕ ⊗ dϕ = T − (m − 2) ϕ ϕ & ' (2.26) |∇ϕ|2 m − 2 Δϕ −2 2 , + m ϕ ϕ ' & Δ(ϕ−1 ) , . = T + (m − 2)ϕ Hess(ϕ−1 ) − m Next, we observe that using (2.4) and (2.5) we are able to detect a part of the curvature tensor which is naturally invariant with respect to a conformal change of the metric. Indeed, from (2.5) we have " ' & ! ϕt ϕj ϕjt jt − (m − 3) ϕl ϕl + ϕll δtj , −2 2 = Rjt − ϕ2 R (m − 2) ϕ ϕ ϕ2 ϕ and inserting into (2.4) gives !

" 1 j j 2 i i i Rik δt − Rjk δt + Rjt δk − Rit δk ϕ Rjkt − m−2

1 i Rik δtj − Rjk δti + Rjt δki − Rit δkj = Rjkt − m−2 & '

Δϕ |∇ϕ|2 i j 1 2 + (m − 4) 2 + δk δt − δti δkj . m−2 ϕ ϕ On the other hand, by (2.7) 2 and we obtain

|∇ϕ|2 1 Δϕ + (m − 4) 2 = − ϕ2 S − S ϕ ϕ (m − 1)

!

1 j jk δti + R jt δki − R it δ j Rik δt − R k m−2 .

S δki δtj − δti δkj + (m − 1)(m − 2)

1 i Rik δtj − Rjk δti + Rjt δki − Rit δkj = Rjkt − m−2

S + δki δtj − δti δkj . (m − 1)(m − 2) 2

ϕ

i jkt R −

It follows that the (1, 3)-tensor, called the Weyl tensor, deﬁned as !

1 Rik δtj − Rjk δti + Rjt δki − Rit δkj W = Riem − m−2

" S i j i j − δ δ − δt δk θk ⊗ θt ⊗ θj ⊗ ei (m − 1)(m − 2) k t

(2.27)

2.1. The Yamabe equation

45

is invariant under a conformal change of the metric. Remark. It is worth noting that the corresponding (0, 4)-version of W is not conformally invariant. Taking covariant derivatives we obtain 1 (Rik,t δjs − Ris,t δjk + Rjs,t δik − Rjk,t δis ) m−2 St (δik δjs − δis δjk ), + (m − 1)(m − 2)

i i = Rjks,t − Wjks,t

t , using so that taking the divergence with respect to the ﬁrst index, that is, Wjks,t (1.35) and (1.36) we get

1 1 1 1 Rtk,t δjs + Rts,t δjk − Rjs,k + Rjk,s m−2 m−2 m−2 m−2 Sk Ss + δsj − δjk (m − 1)(m − 2) (m − 1)(m − 2) 1 1 Rjk,s − Rjs,k = −Rjk,s + Rjs,k + m−2 m−2 1 1 1 1 1 1 Sk δjs − Ss δjk − − + m−2 m−1 2 m−2 m−1 2 1−m+2 −1 + m − 2 Rjk,s + Rjs,k = m−2 m−2 3−m 3−m 1 1 Sk δjs − Ss δjk + m − 2 2(m − 1) m − 2 2(m − 1) m−3 Cjsk , = m−2

t t Wjks,t = Rjks,t −

where Cjsk are the components of the Cotton tensor C, i.e., Cjsk = Rjs,k − Rjk,s +

1 (Ss δjk − Sk δjs ). 2(m − 1)

Note that, if m = 3, then div W ≡ 0; moreover, it is possible to prove that if m = 3, then W ≡ 0 (see for instance [Eis49]). The Cotton tensor can also be interpreted in the following way. Let A = Ric −

S , 2(m − 1)

be the Schouten tensor of components Aij = Rij −

S δij . 2(m − 1)

(2.28)

46

Chapter 2. Pointwise conformal metrics

Clearly A is symmetric, hence taking covariant derivatives Aij,k = Aji,k , but for the last two indices one immediately veriﬁes that Aij,k − Aik,j = Cijk . Hence we can think of the Cotton tensor as the obstruction for the Schouten tensor to be Codazzi; in other words, A is a Codazzi tensor if and only if the Cotton tensor C is identically null. Although we shall not make use of A in what follows, we recall that it enables to write, for m ≥ 3, the decomposition of the curvature tensor: denoting again with W the (0, 4) version of the Weyl tensor we have 1 A , , R = W+ m−2 where is the Kulkarni-Nomizu product; for the deﬁnition of this latter and more information on the Schouten tensor we refer to A. Besse’s treatise [Bes08]. The importance of the Weyl and the Cotton tensors will be emphasized in Theorem 2.6 below, which is a classical result due to Weyl (case m ≥ 4) and to Schouten (case m = 3). We ﬁrst recall the following Deﬁnition 2.5. A Riemannian manifold (M, , ) of dimension m ≥ 2 is said to be locally conformally ﬂat if, for every p ∈ M , there exist an open set U p and a function ϕ ∈ C ∞ (U ), ϕ > 0, such that the manifold (U, ϕ2 , ) is ﬂat. Remark. Recall that, by the classical Riemann theorem (see for instance [Spi79]), the ﬂat manifold (U, ϕ2 , ) is locally isometric to Rm . It follows that M is locally conformally ﬂat if each point x ∈ M has a coordinate chart (U, ξ) such that ξ ∗ , Rm is pointwise conformally related to , . Namely, ξ : U → Rm is a conformal imbedding. Obviuously, ﬂat spaces are locally conformally ﬂat. Here are some more interesting examples of both conformally ﬂat and nonconformally ﬂat manifolds. Example. Every 2-dimensional Riemannian manifold is locally conformally ﬂat. This fact is known as “the existence of isothermic coordinates” on any smooth surface and was established by Korn and Lichtenstein, [Kor14], [Lic16], assuming that the Riemannian metric at hand has H¨older continuous coeﬃcients. For a conceptually easier proof we refer the reader to the paper [Che55] by Chern. The case of real analytic Riemannian metrics goes back to Gauss. Some regularity condition on the coeﬃcients of the metric is needed as shown by Hartman and Wintner, [HW53]. Now, since every smooth manifold supports a smooth Riemannian metric, using isothermic coordinates one concludes that any orientable smooth surface possesses an underlying complex structure.

2.1. The Yamabe equation

47

Example. The standard sphere Sm and the standard hyperbolic space Hm are locally conformally ﬂat. As for the hyperbolic space, we are identifying Hm with either its Poincar´e or its half-space model. Example. An important class of m-dimensional, locally conformally ﬂat manifolds are those admitting a conformal immersion into the standard sphere Sm . In general, such an immersion does not exist. For instance, consider the ﬂat (hence conformally ﬂat) torus Tm . By standard topological arguments, if Tm were immersed into Sm , the immersion would be a covering map. Therefore, the fundamental group of Tm would inject into the fundamental group of Sm , and this is clearly impossible. A general obstruction to the existence of conformal immersions into spheres is represented by the value of the Yamabe invariant of the manifold. Let (M, , ) be a Riemannian manifold of dimension m ≥ 3 with scalar curvature S(x). The Yamabe invariant of M is the real constant # 2 m−2 |∇v| + 4(m−1) S (x) v 2 M Y (M ) = inf . #

m−2 v∈C0∞ (M )\{0} 2m m m−2 v M Note that, in case S (x) ≡ 0, Y (M ) reduces to the ordinary Sobolev constant. By a result of Schoen and Yau, if M has a conformal immersion into Sm , then Y (M ) = Y (Sm ) > 0; see Chapter 6 in [SY94]. By way of example, it is a simple matter to verify that the Riemannian product M = Tm−1 ×R satisﬁes Y (M ) = 0. Indeed, it is a complete, ﬂat manifold with sub-linear volume growth. Example. Suppose that m − k ≥ 2. Then, it can be shown that the Riemannian product Sm−k−1 × Hk+1 is conformally diﬀeomorphic to Sm \Sk , where Sk ⊂ Sm is an equatorial k-sphere. In particular, Sm−k−1 × Hk+1 is locally conformally ﬂat. Example. If M has constant sectional curvature, then the Riemannian products M × R and M × S1 are locally conformally ﬂat. Example. In general, the Riemannian product of locally conformally ﬂat manifolds is not locally conformally ﬂat. For instance, the product of standard spheres Sm × Sm , m ≥ 2, is an example of a compact, simply connected, non-locally conformally ﬂat manifold. A direct veriﬁcation is possible. However, note that this follows directly from Kuiper’s Theorem 2.7 below. Indeed, if Sm × Sm were locally conformally ﬂat, by Theorem 2.7 it would be conformally diﬀeomorphic to the standard sphere S2m . In particular, the m-th de Rham cohomology groups would satisfy 2m m m HdR (Sm × Sm ) HdR S , but this is impossible. Indeed, for instance, a Mayer-Vietoris argument (see for m S2m = 0. On the other hand, by the K¨ unneth example [Lee11]) shows that HdR formula, (see [GHV72]) m−k m k m HdR (Sm × Sm ) ⊕m (Sm ) = 0, k=0 HdR (S ) ⊗ HdR m 0 (Sm ) = HdR (Sm ) = R. because, by Poincar´e duality, HdR

48

Chapter 2. Pointwise conformal metrics

One of the most important results in Riemann surfaces theory is the Riemann-K¨ obe uniformization theorem (see for instance [For91]), according to which every simply connected Riemann surface is bi-holomorphic either to the complex plane C, to the open unit disk Bm ⊂ C or to the Riemann sphere S2 . Note that a bi-holomorphism is a (orientation preserving) conformal diﬀeomorphism. Therefore, recalling the existence theorem for isothermic coordinates alluded to above, we conclude e.g. that every 2-dimensional, compact, simply connected (hence orientable) Riemannian manifold is conformally diﬀeomorphic to S2 . In general, Riemannian manifolds of dimension m ≥ 3 do not enjoy any such uniformization property. An obstruction is given in the following (see [Eis49], [Bes08], [SY94]) Theorem 2.6. Let (M, , ) be a Riemannian manifold, dim M = m ≥ 3. A necessary and suﬃcient condition for (M, , ) to be locally conformally ﬂat is that C ≡ 0 if m = 3, W ≡ 0 if m ≥ 4. A complete classiﬁcation of locally conformally ﬂat manifolds is unknown. In general, in higher dimensions, uniformization does require curvature restrictions. However, in the compact setting, we have the following seminal result due to N. Kuiper, [Kui49]. Theorem 2.7. Let (M, , ) be a compact, simply connected, locally conformally ﬂat manifold of dimension m ≥ 3. Then, M is conformally diﬀeomorphic to the standard sphere Sm . Proof. The idea of the proof is as follows. Details can be found in Chapter 6 of Schoen-Yau’s book [SY94]. Since M is locally conformally ﬂat, every point x ∈ M has a neighborhood Uα conformally imbedded into Rm , hence into Sm . Let ξα be such a conformal imbedding. Consider the couple (Uα , ξα ). If (Uβ , ξβ ) is a second local conformal imbedding, the transition function ξα ◦ ξβ−1 is a local conformal automorphism of Sm . The classical Liouville theorem then shows that there exists a global conformal transformation ψ(α,β) ∈ Conf (Sm ) such that ψ(α,β) = ξα ◦ ξβ−1 , on ξβ (Uα ∩ Uβ ) . Note that, with the obvious meaning of the symbols, the following cycle conditions are satisﬁed: ψ(α,β) ◦ ψ(β,α) = id, ψ(α,β) ◦ ψ(β,γ) ◦ ψ(γ,α) = id.

(2.29)

We are now in the position to deﬁne a conformal immersion Φ : M → Sm . Indeed, let x0 ∈ M and let (Uα0 , ξα0 ) be a ﬁxed conformal imbedding with x0 ∈ Uα0 . For every x ∈ M let γ be a chosen path from γ (0) = x0 to γ (1) = x which is covered

2.2. Some applications in the compact case

49

by a ﬁnite chain of elements (Uα0 , ξα0 ) , . . . , (Uαn , ξαn ) such that Uαj ∩ Uαj+1 = ∅. We deﬁne Φ (x) = ξα0 (x) , on Uα0 , and Φ (x) = ψ(a0 ,α1 ) ◦ · · · ◦ ψ(αn−1 ,αn ) ◦ ξn (x) , on Uαn . Since M is simply connected, using the cycle conditions (2.29) together with a monodromy argument shows that Φ is well deﬁned and, by construction, Φ is a conformal immersion. It remains to show that Φ is a diﬀeomorphism. This follows from standard topological arguments. Indeed, since M is compact and Sm is connected, Φ is a covering map. But Sm is simply connected, hence Φ is a bijection.

2.2 2.2.1

Some applications in the compact case A rigidity result of Obata

We now prove a rigidity result for compact Einstein manifolds due to M. Obata, [Oba62a]. Its proof relies on the transformation laws (2.26) and on a rigidity result for complete Riemannian manifolds supporting nontrivial solutions of certain diﬀerential inequalities; see Theorem 2.10 below. Theorem 2.8. Let (M, , ) be a compact, Einstein manifold of dimension m ≥ 3 and let , be a pointwise conformal deformation of , . If (M, , ) has constant scalar curvature S, then (M, , ) is Einstein. Furthermore, if (M, , ) is not m 2 conformally diﬀeomorphic to the standard sphere S , then , = c , , for some constant c > 0. Remark. This result has been generalized by J. Escobar, [Esc90], to the case where M has a nonempty boundary ∂M = ∅. In this situation, one also requires that the inclusion ı : ∂M → (M, , ) is totally geodesic and that ı : ∂M → (M, , ) is minimal. Thus, in the conclusion of the theorem, the standard sphere Sm is replaced by the standard hemisphere Sm + . Escobar’s proof follows closely Obata’s original argument. We shall limit ourselves to proving Theorem 2.8, since in any case Escobar’s theorem will result as a consequence of Theorem 2.16 of section 2.2.3. Before proving the theorem we point out the following simple, interesting, consequence, to be compared with Theorem 2.2 . Later on, in the setting of complete, non-Einstein manifolds, we shall give a version of the next result assuming that the conformal diﬀeomorphism at hand preserves the scalar curvature (see Theorem 5.9). Corollary 2.9. Let (M, , ) be a compact Einstein manifold of dimension m ≥ 3 which is not conformally diﬀeomorphic to Sm . Let ϕ ∈ Conf (M ) be a conformal

50

Chapter 2. Pointwise conformal metrics

diﬀeomorphism such that ϕ∗ , has constant scalar curvature. Then ϕ ∈ Iso (M ), that is, it is a Riemannian isometry. Proof. By Theorem 2.8, ϕ∗ , = c2 , , for some constant c > 0. Therefore, the result follows from Lemma 2.1. Now we give the Proof of Theorem 2.8. For the sake of convenience, we set , = ϕ−2 , and denote with quantities that refer to the metric , . According to (2.26), / ϕ−1 −1 Δ T = T + (m − 2) ϕ Hess ϕ , . − m Since , is Einstein we have T ≡ 0, and ) *we deduce that, in components with respect to a local orthonormal coframe θj for , , −Tij = (m − 2) ϕ

−1

ϕ

ij

/ ϕ−1 Δ − δij . m

(2.30)

Since T is a traceless tensor, 2 ϕ−1 T = − (m − 2) ϕ−1 ij Tij . ,

On the other hand, according to equation (2.13), m−2 Si Tik,k = 2m and since S is constant, we obtain (2.31) Tik,k = 0. ) * + by Thus, if we let { ej } be the dual frame of θj , and deﬁne the vector ﬁeld W the formula + = ϕ−1 Tij ej , (2.32) W i −1 −1 i where d ϕ θ , a computation that uses (2.31) shows that = ϕ i +W + = ϕ−1 Tij + ϕ−1 Tik,k = ϕ−1 Tij , div ij i ij and we conclude that 2 +W +. ϕ−1 T = − (m − 2) div ,

2.2. Some applications in the compact case

51

Integrating on M, , and using the divergence theorem we deduce that

M

2 + = 0. ϕ−1 T dvol ,

Accordingly, T ≡ 0 and M, , is an Einstein manifold. Suppose now that the m-dimensional manifold (M, , ) is not conformally 4 , = u m−2 , , we shall diﬀeomorphic to Sm , m ≥ 3. Having set as usual prove that u is identically equal to a positive constant c2 . Let S be the the scalar curvature of , , which is constant since (M, , ) is Einstein. We ﬁrst claim that either S = S = 0 or S · S > 0. To see this, we recall that S and S are related by the Yamabe equation m+2 m−2 cm Δu − Su + Su = 0,

cm = 4

m−1 . m−2

Suppose that S ≤ 0 and S ≥ 0. Then, Δu ≥ 0, and since (M, , ) is compact we deduce that u is a positive constant. Therefore, inserting this information into the Yamabe equation, we conclude that 4 m−2 0 ≥ S = Su ≥ 0,

proving that S = S = 0, and the claim follows. The same conclusion holds if we assume instead that S ≥ 0 and S ≤ 0. Thus, we need to consider three possible cases. First case: S = S = 0. Using the Yamabe equation once more gives Δu = 0 and, therefore, u is a positive constant. Second case: S < 0 and S < 0. Up to rescaling , by a positive constant, we can assume that S = S < 0. Let u (x0 ) = max u, M

u (x1 ) = min u. M

Then, by the usual maximum principle, Δu (x0 ) ≤ 0 which implies, according to the Yamabe equation,

4 Su (x0 ) 1 − u m−2 (x0 ) ≤ 0. As a consequence, u (x) ≤ u (x0 ) ≤ 1. Similarly, since u achieves its minimum at x1 we have Δu (x1 ) ≥ 0. This latter, in turn, implies 1 ≤ u (x1 ) ≤ u (x) ,

52

Chapter 2. Pointwise conformal metrics

and, therefore, u ≡ 1. 4 Third case: S > 0 and S > 0. Let us observe that, since , = u m−2 , and both the metrics are Einstein, setting 2

v = u− m−2 , by (2.26) we have Δv , = 0 on M. m We deﬁne the vector ﬁeld X = ∇v so that div X = Δv and Hess(v) −

(2.33)

1 LX , = Hess(v). 2 From the above, we deduce div X 1 LX , = , , 2 m that is, X is a conformal vector ﬁeld. We now show, using again the moving frame formalism, that since M is Einstein, f = div X = Δv satisﬁes the equation Hess(f ) +

S f , = 0, m (m − 1)

(2.34)

where, we recall, s > 0. First we observe that the previous equation can be rewritten as S Hess(Δv) + Δv , = 0, m (m − 1) or, in components, (Δv)kt = viikt = −Δv

S δkt ; m (m − 1)

then we note that, by (2.33), we have vij =

vtt δij , m

(2.35)

vijk =

vttk δij m

(2.36)

vijkl =

vttkl δij . m

(2.37)

from which we deduce and By (1.54) applied to v we deduce

viikl = (Δv)kl = vikil + vsl Rsiik + vs Rsiik, l = vkiil − vsl Rsk − vs Rsk, l = vkiil − vsl Rsk ,

2.2. Some applications in the compact case

53

since Rsk,l = 0. Then, using (2.35), (2.36) and (2.37) we obtain (Δv)kl = vkiil −

vii vssil δki vii δsl Rsk = − Rkl , m m m

which easily implies (2.34). In case f is constant, we get f = Δv = 0 and, since M is compact, v itself must be constant. This implies that also u is constant, as desired. Finally, in case f is nonconstant, we can apply the next result to conclude that (M, , ) is isometric 2 to the sphere Sm k2 of constant curvature k = S/m (m − 1). But this implies that (M, , ) is conformally diﬀeomorphic to the standard sphere Sm , against our initial assumption. The following result is again due to Obata, [Oba62a] (for the analogous statement for Sm + , the upper hemisphere, see Theorem 2.14 below); see also [PR]. Theorem 2.10. Let (M, , ) be a complete Riemannian manifold of dimension m. Suppose that there exists a nonconstant, smooth function f : M → R such that, for some constant k > 0, Hess(f ) + k 2 f , = 0.

(2.38)

Then (M, , ) is isometric to the sphere Skm2 of constant sectional curvature k 2 . Proof. First, we show that f has a critical point. Indeed, by contradiction, suppose that ∇f = 0 on M . Consider the smooth vector ﬁeld X = ∇f / |∇f | on M . Since |X| ∈ L∞ (M ) and (M, , ) is geodesically complete, then X is complete (see e.g. [Lee03], Chapter 12). Let γ : R → M be an integral curve of X, namely Xγ(s) = γ˙ (s), for every s ∈ R. A direct computation that uses (2.38) shows that, for every vector ﬁeld Y , Dγ˙ γ, ˙ Y=

1 1 Hess (f ) (γ, ˙ Y)− Hess (f ) (γ, ˙ γ) ˙ γ, ˙ Y = 0. |∇f | |∇f |

Therefore, γ is a unit speed geodesic of M . Evaluating (2.38) along γ we deduce that the function y (s) = f ◦ γ (s) satisﬁes y = −k 2 y which is oscillatory. Let s0 ∈ R be such that y (s0 ) = 0. Then, recalling that γ is an integral curve of X, we conclude 0, 0 = y (s0 ) = ∇f (γ (s0 )) , γ˙ (s0 ) = |∇f (γ (s0 ))| = a contradiction. Let o ∈ M be a critical point of f and set r (x) = d (x, o). Note that f (o) = 0 for otherwise, having ﬁxed any unit speed geodesic γ issuing from o, we would have

54

Chapter 2. Pointwise conformal metrics

that y (s) = f ◦ γ (s) solves the Cauchy problem ⎧ ⎨ y = −k 2 y, y (0) = 0, ⎩ y (0) = 0, and, hence, y (s) ≡ 0. Since this would be true for every geodesic γ we should conclude that f ≡ 0, a contradiction. Thus, without loss of generality, we assume that f (o) = 1. We claim that, for every x ∈ M , it holds that f (x) = cos (kr (x)) .

(2.39)

Indeed, consider a unit speed, minimizing geodesic γ : [0, r (x)] → M from γ (0) = o to γ (r (x)) = x. As noted above, the smooth function y (s) = f ◦ γ (s) is the solution of the Cauchy problem ⎧ ⎨ y = −k 2 y, y (0) = 1, ⎩ y (0) = 0. Therefore, y (s) = cos (ks) . Evaluating at s = r (x) we conclude the validity of (2.39). Now, observe that the function cos (ks) is strictly decreasing on (0, π/k). It follows from (2.39) that r (x) = k −1 arccos (f (x)) is smooth on the geodesic ball Bπ/k (o) \ {o}. Applying the Bishop density result of Chapter 1 we therefore conclude that cut (o) ∩ Bπ/k (o) = ∅.

(2.40)

Therefore, the exponential map expo : Bm π/k (0) ⊂ To M → Bπ/k (o) is a diﬀeomorphism. Let us introduce geodesic polar coordinates (r, θ) on To M . Furthermore, let us consider a local orthonormal coframe {θα } on Sm−1 αwithαdual frame {Eα }. θ ⊗ θ . We extend both Thus, the standard metric of Sm−1 is written as dθ2 = {θα } and {Eα } radially. Then, by Gauss’ lemma, , = dr ⊗ dr + σαβ (r, θ) θα ⊗ θβ . Furthermore, since , is inﬁnitesimally Euclidean and the standard metric of Rm ≈ To M is written as , Rm = dr ⊗ dr + r2 δαβ θα ⊗ θβ , we have the further condition σαβ (r, θ) = δαβ r2 + o r2 , as r 0.

(2.41)

2.2. Some applications in the compact case

55

Now we use the fact that L∇r , = 2Hess (r) , on Bπ/k (o) \ {o} where L∇r is the Lie derivative in the radial direction ∇r. Thus, by the deﬁnition of Lie derivative we deduce that 2 Hess(r)(Eα , Eβ ) = L∇r , (Eα , Eβ ) = ∇r(σαβ ). Since ∇r = −

(2.42)

∇f , |∇f |

the Hessian of the distance function r can be expressed as , ∇f Hess (r) (Eα , Eβ ) = − DEα , Eβ |∇f | 1 Hess (f ) (Eα , Eβ ) =− |∇f | k 2 cos (kr) Eα , Eβ = k sin (kr) = k cot (kr) σαβ . Inserting into (2.42), and recalling (2.41) we obtain that σαβ are the (unique) solutions of the asymptotic Cauchy problems & ∂r σαβ = 2k cot (kr) σαβ , on Bπ/k (o) \ {o} , σαβ (r, θ) = δαβ r2 + o r2 , as r 0. Integrating, ﬁnally gives σαβ = k −2 sin2 (kr) δαβ . We have thus established that, in polar coordinates of Bm π/k (0) \ {o} = (0, π/k) × m−1 S ⊂ To M , it holds that , = dr ⊗ dr + k −2 sin2 (kr) dθ2 . Since

(0, π/k) × Sm−1 , dr ⊗ dr + k −2 sin2 (kr) dθ2

(2.43)

is isometric to the m-dimensional 2-punctured sphere Sm k2 \ {2 points} of constant curvature k 2 , we conclude that the geodesic ball Bπ/k (o) ⊂ M is isometric to Sm k2 \ {point}. To complete the proof it suﬃces to show that Bπ/k (o) = M \ {point} .

(2.44)

56

Chapter 2. Pointwise conformal metrics

Indeed, suppose we have already proved this fact. Then, by Seifert-Van Kampen’s theorem, (see [Lee11]) M is simply connected. Moreover, M \ {point} has constant curvature k 2 because it is isometric to Skm2 \ {point}. By continuity, M itself has constant curvature k 2 . Therefore, the Hopf classiﬁcation theorem (see [Pet06b]) tells us that M must be isometric to Skm2 , as desired. The proof of (2.44) combines Morse theoretic and cut-locus arguments. First of all, we observe that ∂Bπ/k (o) is made up entirely by nondegenerate critical points of f. Therefore, by Morse’s lemma, (see [Mil63], [Pet06b]) ∂Bπ/k (o) is a discrete, compact (hence ﬁnite) set. Indeed, for every x ∈ ∂Bπ/k (o), by continuity we have |∇f | (x) = −k sin (π) = 0, and, by assumption,

Hess (f ) (x) = −k 2 f (x) , x ,

where, f (x) = cos (π) = −1. In particular, Hess(f ) is (strictly) negative deﬁnite on ∂Bπ/k (o), as claimed. To conclude, we note that ∂Bπ/k (o) is connected. In fact,

∂Bπ/k (o) = expo ∂Bm (0) . (2.45) π/k To see this, we recall that

(0) ∩ T M \c (o) = ∂Bπ/k (o) ∩ M \cut (o) , expo ∂Bm o π/k

(2.46)

where c (o) ⊂ To M is the tangential cut-locus of o ∈ M . Let v¯ ∈ ∂Bm π/k (0) ∩ c (o). By deﬁnition, x ¯ = expo v¯ ∈ cut (o). Since, by (2.40), Bπ/k (o) ⊂ M \cut (o) , and, on a generic complete Riemannian manifold,

expo Bm (0) ⊆ Bπ/k (o), π/k we must conclude that x ¯∈ / Bπ/k (o), that is,

expo ∂Bm π/k (0) ∩ c (o) ⊆ ∂Bπ/k (o) ∩ cut (o) . Conversely, let x ∈ ∂Bπ/k (o) ∩ cut (o). By deﬁnition, there is a unit speed geodesic γ (t) = expo (vt) : [0, +∞) → M from γ (0) = o to γ (π/k) = x that does not minimize distances past π/k. Then, vπ/k ∈ ∂Bm π/k (0) ∩ c (o) and expo (vπ/k) = x. Summarizing,

expo ∂Bm (2.47) π/k (0) ∩ c (o) = ∂Bπ/k (o) ∩ cut (o) . From (2.46) and (2.47) we conclude the validity of (2.45).

2.2. Some applications in the compact case

2.2.2

57

A result by M. F. Bidaut-V´eron and L. V´eron

+ deﬁned in (2.32), with Tij given in (2.30), will suggest to us The vector ﬁeld W how to proceed to provide a proof of Theorem 2.12 below. First we give an intrinsic deﬁnition: by direct manipulation of the quantities involved we have & ' (ϕ−1 )kk −1 −1 + δij (m − 2)ϕ ej W = (ϕ )i −(ϕ )ij + m m−2 ϕ(ϕ−1 )kk (ϕ−1 )i ei − (m − 2)ϕ(ϕ−1 )ij (ϕ−1 )i ej = m −1 m − 2 −1 −1 ϕ−1 ∇ϕ = ,· ϕ(Δϕ )∇ϕ − (m − 2)ϕHess m / −1 −1 Δϕ −1 − Hess ϕ−1 ∇ϕ ∇ϕ = (m − 2)ϕ , m / −1 Δϕ 1 −1 2 −1 = (m − 2)ϕ ∇ϕ − ∇∇ϕ . m 2 ,

We set the following general Deﬁnition 2.11. Given u ∈ C 2 (M ), u > 0, the vector ﬁeld ' & Δu α 1 2 ∇|∇u| − ∇u , α ∈ R, Z=u 2 m

(2.48)

is called an Obata type vector ﬁeld. + above is an Obata vector ﬁeld (modulo the multiplicative constant Thus W 2 − m) with respect to the metric , . Furthermore, the function involving ϕ satisﬁes equation (2.7) with the roles of , and , interchanged, that is ϕ2 S = S − 2(m − 1)

Δϕ − (m − 1)(m − 4) ϕ

2 ∇ϕ

,

ϕ2

.

This, in turn, implies 2 + 2 ∇ϕ −1 = − 1 Δϕ Δϕ ϕ2 ϕ3 ,

=

S S m ϕ− ϕ−1 + 2(m − 1) 2(m − 1) 2

=−

2 ∇ϕ

,

ϕ3

S S m ϕ−1 + (ϕ−1 )−1 + 2(m − 1) 2(m − 1) 2

−1 2 ∇ϕ

,

ϕ−1

.

58

Chapter 2. Pointwise conformal metrics

Therefore, u = ϕ−1 is a positive solution of the diﬀerential equation 2 $ % ∇u S m S ,

−1 = u − u . + Δu 2 u 2(m − 1) S We have thus obtained the further suggestion to consider the vector ﬁeld Z deﬁned in (2.48), with u a positive solution of a diﬀerential equation of the type Δu = (β + 1)

|∇u|2 1 1+β(1−σ) − λu , β ∈ R \ {0}, σ > 1, λ ∈ R. (2.49) + u u β

Note that, if Δu = −f (u, |∇u|),

f = f (u, t) ∈ C 1 (R2 ),

a straightforward computation which exploits the Bochner-Weitzenb¨ ock formula (1.48) gives ' & (Δu)2 div Z = uα | Hess(u)|2 − m ' & 1 − m ∂f α + uα−1 f (u, |∇u|) + u (u, |∇u|) |∇u|2 m m ∂u + uα Ric(∇u, ∇u) + αuα−1 Hess(u)(∇u, ∇u) 1 − m α ∂f + u (u, |∇u|)|∇u|−1 Hess(u)(∇u, ∇u). m ∂t Hence, if f has the special form f (u, |∇u|) = δ

|∇u|2 + g(u), u

δ ∈ R,

the previous formula becomes ' & (Δu)2 α 2 div Z = u | Hess(u)| − m & ' 1 |∇u|2 1−m α α−1 +u g(u) + (α + m − 1)δ + ug (u) |∇u|2 m m u m + uα Ric(∇u, ∇u) ' & 1−m α−1 Hess(u)(∇u, ∇u). α + 2δ +u m To get rid of the last term containing Hess(u)(∇u, ∇u) we observe that, given γ ∈ R, div γuβ |∇u|2 ∇u = 2γuβ Hess(u)(∇u, ∇u) + γuβ |∇u|2 Δu + βγuβ−1 |∇u|4 .

2.2. Some applications in the compact case We choose β = α − 1, γ =

(m−1)δ m

V =Z+ we have

&

−

59

α 2

so that, for the vector ﬁeld α α−1 m−1 u δ− |∇u|2 ∇u, m 2

(2.50)

' (Δu)2 div V = u | Hess(u)| − m & ' m−1 α−1 α(m + 2) − 2(m − 1)δ g(u) − ug (u) |∇u|2 +u 2m m uα−2 3mαδ − mα(α − 1) − 2δ 2 (m − 1) |∇u|4 + 2m + uα Ric(∇u, ∇u). α

2

Now, if δ = −(β + 1) and g(u) = βλ u − β1 u1+β(1−σ) from the previous relation we obtain & ' (Δu)2 div V = uα | Hess(u)|2 − (2.51) m 1 − [2(m − 1)βσ + (m + 2)α]uα+β(1−σ) |∇u|2 2mβ 1 mα2 + (2 + 3β)mα + 2(m − 1)(β + 1)2 uα−2 |∇u|4 − 2m & ' 1 λ 2 + 2m Ric(∇u, ∇u) + [α(m + 2) + 2(m − 1)β]|∇u| uα . 2m β We are now ready to prove the following beautiful result ﬁrst due to Bidaut-V´eron and V´eron, [BVV91]. Theorem 2.12. Let (M, , ) be a compact manifold of dimension m ≥ 2 and Ricci curvature satisfying Ric ≥ k > 0. Let ϕ be a positive solution of Δϕ − λϕ + ϕσ = 0

on M

(2.52)

for some constant σ > 1, λ > 0. Then ϕ is constant provided (i) m = 2 and λ ≤ 2k/ (σ − 1) ; (ii) m ≥ 3,

λ ≤ mk/(m − 1)(σ − 1),

σ ≤ (m + 2) / (m − 2) and

(A) either at least one of the last two inequalities is strict or (B) (M, , ) has constant scalar curvature S and it is not isometric to Skm2 , the sphere of constant curvature k 2 = s/(m − 1)m.

60

Chapter 2. Pointwise conformal metrics

Proof. We set u = ϕ−1/β , β = 0. Then u is positive and satisﬁes equation (2.49), hence (2.51) holds. With the aid of the divergence theorem we obtain & ' 1 2 4 α 2 0= 2mu |Hess(u)| − (Δu) − A uα−2 |∇u| (2.53) m M M ) * 2 2 + uα 2m Ric (∇u, ∇u) + D|∇u| − B uα−β(σ−1) |∇u| , M

M

where, for ease of notation, we have set A = mα2 + (3β + 2)mα + 2(m − 1)(1 + β)2 , 1 λ B = [2(m − 1)βσ + (m + 2)α], D = [(m + 2)α + 2(m − 1)β]. β β Next we observe that, by Newton’s inequality 2

|Hess(u)| ≥

1 2 (Δu) , m

the ﬁrst integral on the right-hand side of the above is nonnegative. The idea of the proof is to ﬁnd, under the conditions listed in (i) and (ii), β = 0 and α ∈ R, such that D A ≤ 0, B ≤ 0, and Ric + ≥0 (2.54) 2m and at least one of the above inequalities is strict. Once this is achieved, then (2.53) implies that ∇u ≡ 0; thus u and therefore ϕ are constant. Let y = 1 + 1/β, δ = −α/β, so that y, δ ∈ R, y = 1. Rewriting A, B and D in terms of y and δ, the inequalities to be established become (a) (b) (c)

2 2 2 m−1 m y − 2δy + δ − δ ≤ 0, m−1 2σ m+2 ≤ δ,

m 2 m+2 Ric ≥ λ δ − 2 m−1 m+2 , ,

(2.55)

with at least one strict inequality. If either (i) or (ii) holds, then, for m ≥ 2, m−1 m−1 m , (2.56) Ric ≥ λ 2 σ−2 2 m+2 m+2 m+2 and setting δ = 2σ

m−1 > 0, m+2

inequalities (2.55) (b) and (c) are satisﬁed. In order to ﬁnd a value y = 1 which satisﬁes (2.55) (a) with strict inequality, it suﬃces that the quadratic polynomial in y on the left-hand side has two distinct solutions, which, taking into account our choice of σ, in turn amounts to the validity of the inequality (m+1)−(m−2)σ > 0.

2.2. Some applications in the compact case

61

This inequality being always trivially satisﬁed if m = 2, it remains to analyze the case where m > 2, σ = (m + 2) / (m − 2). We assume that (M, , ) has constant scalar curvature S and show that if ϕ is a positive nonconstant solution of (2.49), then (M, , ) is isometric to the sphere Sm k2 of constant sectional curvature k 2 = S/(m−1)m. Our assumption that ϕ is not constant implies that u is positive, and nonconstant; by the condition on σ, equality holds in (2.55) (b) and (a) with m y = m−2 , so that A = B = 0. Since the third and the fourth summands in (2.53) vanish, while the integrands in the ﬁrst and third integral are nonnegative, they must vanish identically. In particular we must have 2

|Hess(u)| =

(Δu)2 , on M m

and therefore, by the equality case in the Cauchy-Schwarz inequality, Hess(u) =

Δu , , on M. m

Now we proceed as in the proof of Theorem 2.8. We set X = ∇u and we observe that f = div X satisﬁes Hess(f ) +

S f , = 0. m(m − 1)

Note that f is not constant, for otherwise the above would imply Sf ≡ 0 and since S ≥ mk > 0 by the assumption on the Ricci curvature, we would conclude that f = Δu = 0 so that u is constant on M . Obata’s Theorem 2.10 implies the conclusion.

Remark. 1. Equation (2.52) is, of course, a normalized version of the more general case Δϕ − λϕ + μϕσ = 0 on M, (2.57) with λ, μ > 0 and ϕ > 0. In this situation the required condition on λ depends on μ. 2. Similarly to what happened in the proof of Theorem 2.8, if we consider equation (2.57) with λ ≤ 0, μ ≥ 0, then Δϕ ≤ 0 on the compact manifold (M, , ), and hence ϕ is a positive constant by the maximum principle. For λ ≥ 0, μ ≤ 0, then Δϕ − λϕ ≥ 0 and we obtain the same conclusion. Finally, for λ ≤ 0, μ ≤ 0, Δϕ + μϕσ−1 ϕ ≤ 0 and again ϕ is constant. 3. The above observation also shows that Theorem 2.8 is a consequence of Theorem 2.12 for m ≥ 3. Indeed, if S is the scalar curvature of (M, , ) and 4 = 0, with S the constant , = ϕ m−2 , , ϕ > 0, then cm Δϕ − Sϕ + Sϕ scalar curvature of , . However, the Einstein condition is not required for (M, , ) in Theorem 2.12.

62

Chapter 2. Pointwise conformal metrics

2.2.3

A version of Theorem 2.12 on manifolds with boundary

We shall now give a version of Theorem 2.12 in case M is compact with nonempty boundary ∂M . For simplicity we consider the case that M is oriented. In the next result we explicate the boundary term in applying the divergence theorem to (2.51). Lemma 2.13. Let (M, , ) be an m-dimensional oriented compact Riemannian manifold with boundary ∂M . Let II and h denote respectively the second fundamental tensor and the mean curvature of the embedding ∂M → M in the direction of the outward unit normal vector ﬁeld ν. Let u ∈ C 3 (M ) and let u = u|∂M . If V is the vector ﬁeld deﬁned in (2.50), that is ' & m−1 α α−1 Δu 2 α 1 u ∇|∇u| − ∇u − (β + 1) + |∇u|2 ∇u, V =u 2 m m 2 we have

2m

div V = M

∂u ∂ν ∂M ∂u 2 − [2(m − 1)(β + 1) + 3mα] uα−1 |∇ u| ∂ν ∂M 3 α−1 ∂u − [2(m − 1)(β + 1) + mα] u (2.58) ∂ν ∂M ∂u 2 2(m − 1)h uα + 2m uα II(∇ u, ∇ u), ν . + ∂ν ∂M 2 uα [(m − 1) Hess(u)(ν, ν) − (m + 1)Δ u]

Proof. Let {e1 , . . . , em−1 , em = ν} be a Darboux frame along ∂M → M . Set hij = II(ei , ej ), ν , so that h=

1 hkk , m−1

where 1 ≤ i, j, k ≤ m − 1. Then, with obvious meaning of the notation we have u i = ui ,

um =

∂u = ∇u, ν . ∂ν

It therefore follows that ⎧ ⎨u = u + um hij , ij ij ∂u ⎩ + = umi − u j hij . ∂ν i

2 Hence, using the identity ∇|∇u| , X = 2 Hess(u)(X, ∇u), we get

2.2. Some applications in the compact case

V, ν =

2m ∂M

−2 uα Δu

∂M

−

63

∂u + 2m uα Hess(u)(∇u, ν) ∂ν

[2(m − 1)(β + 1) + mα] uα−1 |∇u|

2 ∂u

∂ν ∂u = 2 uα [(m − 1) Hess(u)(ν, ν) − Δ u] ∂ν ∂M ∂u 2 + 2(m − 1)h uα ∂ν ∂M 0 ∂u 2 1 ∂u 2 u| + − [2(m − 1)(β + 1) + mα] uα−1 |∇ ∂ν ∂ν ∂M 2 3 + ∂u + u . 2m uα II(∇ u, ∇ u), ν + 2m uα ∇ , ∇ ∂ν ∂M ∂M

We use the ﬁrst Green formula (see e.g. [GT01]) to deal with the last integrand of the above formula, that is, 2 3 + ∂u ∂u 2 ∂u α . ∇ , ∇ u =− + 2mα uα−1 |∇ 2m u 2m uα Δ u u| ∂ν ∂ν ∂ν ∂M ∂M Formula (2.58) now follows at once.

In the next result we shall make use of the following extension of Obata’s theorem 2.10 due to Escobar [Esc90]. Its proof is similar to that of Theorem 2.10 and it will therefore be omitted. Theorem 2.14. Let (M, , ) be an m-dimensional Riemannian compact manifold with boundary ∂M . Assume that there exists a nonconstant function f on M satisfying & Hess(f ) + kf , = 0 on M , (2.59) ∂f ∂ν ≡ 0 on ∂M √

with k a positive constant. Then (M, , ) is isometric to Sm k , the upper + hemisphere of radius k −1/2 with the induced metric from Rm+1 . We are now ready for the next Lemma 2.15. Let (M, , ) be an m-dimensional, m ≥ 3, compact manifold with boundary ∂M such that the embedding ∂M → M is totally geodesic. Assume that (M, , ) is Einstein but not Ricci ﬂat. Let v be a nonconstant function satisfying & Hess(v) − Δv m , = 0 on M , (2.60) ∂v ≡ 0 on ∂M ∂ν with ν the outward unit normalto ∂M . Then the scalar curvature S is positive 4 m(m−1) m . and (M, , ) is isometric to S+ S

64

Chapter 2. Pointwise conformal metrics

Proof. Let X = ∇v. Then, because of the ﬁrst equation of (2.60) X is a conformal vector ﬁeld and hence, since S is constant, f = divX = Δv satisﬁes (2.34), that is Hess(f ) +

S f , m(m − 1)

on M .

(2.61)

We shall now show that

∂f ≡ 0 on M. (2.62) ∂ν Towards this end we observe that, with respect to a Darboux frame {ei , em = ν}, with 1 ≤ i ≤ m − 1 along the embedding ∂M → M , from conformality of X we have 2 Xim + Xmi = (div X)δim = 0. (2.63) m Thus diﬀerentiating in the ek direction, 1 ≤ k ≤ m − 1, Ximk + Xmik = 0.

(2.64)

∂v Now Xm = X, ν = ∂ν ≡ 0 on ∂M because of (2.60), thus Xmik = 0 on ∂M for 1 ≤ i, k ≤ m − 1. From (2.64) we then deduce

Ximk ≡ 0

on ∂M .

(2.65)

Now, since X is conformal Xab + Xba =

2 (divX)δab , m

1 ≤ a, b ≤ m

and for 1 ≤ k ≤ m − 1 Xkk =

divX m

(no sum over k).

Hence

1 (div X)m m From the Ricci identities (1.47), Xkkm =

(no sum over k).

Xkkm − Xkmk = Xs Rskkm

(2.66)

(no sum over k).

Thus using (2.65) m−1

Xkkm = −Xt Rtm

on ∂M .

k=1

But (M, , ) is Einstein and therefore Rsm = 0 for 1 ≤ s ≤ m − 1. Furthermore Xm = 0, hence m−1 Xkkm = 0. (2.67) k=1

2.2. Some applications in the compact case

65

Using (2.66) and (2.67) we ﬁnd m−1 m (divX)m = Xkkm = 0 m−1

on ∂M

k=1

S that is, the validity of (2.62). Thus (2.60) is satisﬁed by f = Δv with k = m(m−1) and the conclusion follows at once from Theorem 2.14 once we prove that f is nonconstant and S > 0. We reason by contradiction. If f were constant, by (2.61) since S = 0 we would have f ≡ 0 so that X = ∇v would be a Killing ﬁeld and 2 2 ∂v Δv ≡ 0. Since div(v∇v) = vΔv + |∇v| = |∇v| and ∂ν ≡ 0 the divergence theorem yields 2 |∇v| ≡ 0, M

that is, v is constant contradicting the assumption of the lemma. Thus f is nonconstant. Next, to show that S > 0 we trace (2.61) to obtain S f = 0 on M , Δf + m(m−1) ∂f ∂ν ≡ 0 on ∂M . From the divergence theorem S ∂f 2 2 = + |∇f | f div(f ∇f ) = f Δf + |∇f | = − f2 0= ∂ν m(m − 1) ∂M M M M and thus

# 2 |∇f | # S = m(m − 1) M > 0. f2 M

We are now ready to prove Theorem 2.16. Let (M, , ) be a compact manifold of dimension m ≥ 2 with boundary ∂M . Assume that the embedding ∂M → M is totally geodesic and that Ric ≥ k > 0. Let σ > 1, λ > 0 and ϕ a positive solution of & Δϕ − λϕ + ϕσ = 0 ∂ϕ ∂ν ≡ 0 on ∂M

(2.68)

on M ,

with ν the outward unit normal to ∂M . Then ϕ is constant provided (i) m = 2 and λ ≤ (ii) m ≥ 3, λ ≤

2k σ−1 ;

k m m−1 σ−1 ,

σ≤

m+2 m−2

and

(A) either at least one of the last two inequalities is strict or

(2.69)

66

Chapter 2. Pointwise conformal metrics (B) (M, , ) is Einstein and it is not isometric to an upper hemisphere of constant curvature.

Proof. We set u = ϕ−1/β with β = 0. Then u satisﬁes (2.49) and ∂u ∂ν ≡ 0 on ∂M . Using (2.51), (2.58), ∂u ≡ 0 and the fact that ∂M → M is totally geodesic we get ∂ν again (2.53). Now the proof proceeds exactly as in Theorem 2.12 up to Hess(u) = Since diction.

∂u ∂ν

Δu , m

on M.

≡ 0 on ∂M we can now apply Lemma 2.15 and we reach a contra

Remark. As it is apparent from the proof of Lemma 2.15, we may substitute the assumption that (M, , ) is Einstein in (B) with (M, , ) has constant scalar curvature and Ric(ν, X) = 0 for every X ∈ T ∂M ⊥ ⊆ T M on ∂M . Remark. 1. An observation similar to the Remark on page 61 holds also here. Point 1. applies verbatim to the more general equation Δϕ − λϕ + μϕσ = 0

on M

(2.70)

with λ, μ > 0 and ϕ > 0. 2. If λ and μ have diﬀerent sign, that is, λ ≤ 0 and μ ≥ 0 or λ ≥ 0 and μ ≤ 0, ϕ > 0, then from (2.69) ∂ϕ 0= = Δϕ = λϕ − μϕσ , ∂ν ∂M M M so that

ϕ λ − μϕσ−1 ≡ 0.

From this latter we easily deduce λ = μ = 0 and ϕ is constant from (2.69) and compactness of M , without requiring any further assumption. 3. Finally, if λ, μ < 0 we have Δϕ − λϕ = −μϕσ ≥ 0. Suppose ﬁrst that ϕ assumes its maximum at a point x0 ∈ ∂M . Then, by the boundary point Lemma (see [PW67] for more details) one has ∂ϕ ∂ν (x0 ) > 0, (x ) ≤ 0 on ∂M . Hence ϕ has to assume its maximum at contradicting ∂ϕ 0 ∂ν x0 ∈ M \ ∂M . Similarly ϕ attains its minimum at x1 ∈ M \ ∂M . Then 0 ≥ Δϕ(x0 ) = λϕ(x0 ) − μϕσ (x0 ), and therefore

μ σ−1 (x0 ) ≤ 1. ϕ λ

2.2. Some applications in the compact case

67

Analogously, since Δϕ(x1 ) ≥ 0 one gets μ σ−1 ϕ (x1 ) ≥ 1. λ Since μλ > 0 and ϕ is positive we conclude that ϕ(x0 ) = ϕ(x1 ) and therefore ϕ is constant. Remark. In Theorem 2.16 we can substitute the assumption that the embedding of the boundary in M is totally geodesic with that of a convex boundary (see for instance [PRS03a]). It seems also worth mentioning that using this result Ilias obtained some sharp Sobolev constants on M (see [Ili96] and [PRS03a], Theorem 3.5). With the above observation and with the aid of Lemma 2.19, from Theorem 2.16 we deduce the following result of Escobar (see [Esc90], Theorem 4.1): Theorem 2.17. Let (M, , ) be a compact Einstein manifold of dimension m ≥ 4 3 and with boundary ∂M totally geodesic in (M, , ). Let , = ϕ m−2 be a conformal change of metric with constant scalar curvature S and with ∂M → M, , minimal. Then , is Einstein and if (M, , ) is not isometric to an upper hemisphere of constant curvature, ϕ is constant, that is, the conformal change is an homothety.

2.2.4

A rigidity result of Escobar

In this section we prove a nice rigidity result due to Escobar, [Esc90]. Towards this aim we need some preliminary facts contained in the next two lemmas. Lemma 2.18. Let (M, , ) be a manifold with boundary, constant scalar curvature S and trace-free Ricci tensor T . Let ν be the outward unit normal to i : ∂M → M and let X be a conformal vector ﬁeld on M such that, for some ψ ∈ C ∞ (∂M ), X|∂M = ψν. Then ψT (ν, nu) = 0.

(2.71)

∂M

Proof. We let W = T (X, ) as in (2.14) so that, according to (2.15), div W = Xi,k Tik + Xi Tik,k . Since X is conformal, LX , = or, in other words,

2 div X , , m

Xik + Xki =

2 div X δik . m

(2.72)

68

Chapter 2. Pointwise conformal metrics

Hence Xik Tik

1 = (Xik + Xki )Tik = 2

1 div X Tkk = 0, m

since T is trace-free. Thus using (2.72) and the divergence theorem, T (X, ν) = (div T )(X). ∂M

M

However, since the scalar curvature S is constant, from (2.13) div T ≡ 0 and from the above we immediately deduce (2.71). Remark. For m = dim M = 2 there is no need to assume S constant, see equation (2.13). Lemma 2.19. Let (M, , ) be a manifold with boundary ∂M and , = ϕ2 , ∞ for some ϕ > 0, ϕ ∈ C (M ). Let ν be the unit outward normal of ∂M → (M, , ) form and hij the coeﬃcient of the second fundamental

in the direction of ν. Set hij for the analogous quantities for ∂M → M, , with respect to ν = ϕ−1 ν. Then, for 1 ≤ i, j ≤ m − 1, ∂ϕ δij , hij = ϕ−1 hij + ϕ−2 ∂ν

(2.73)

and for the mean curvature ∂log ϕ . (2.74) h = ϕ−1 h + ∂ν

In particular, if ∂M → (M, , ) and ∂M → M, , are both minimal, then ∂ϕ ∂ν

≡ 0 on ∂M .

Proof. A simple computation. Indeed, with the notation of Chapter 1 and section 2.1.1 we let θi be a Darboux frame along the inclusion map ı : ∂M → (M, , ). Thus θm = 0 on ∂M and hij are deﬁned by the requirement θim = hij θj ,

1 ≤ i, j, . . . ≤ m − 1.

, . Similarly, θi = ϕθi is a Darboux frame along the inclusion ı : ∂M → M, Thus θm = 0 and hij θj , 1 ≤ i, j, . . . ≤ m − 1. θm = i

hij we recall from (2.3) that To relate hij with ϕm i ϕi θim = θim + θm − θ, ϕ ϕ

dϕ = ϕj θj + ϕm θm .

2.2. Some applications in the compact case Hence

69

ϕm i j ϕ hij θj = hij θj − δ θ ϕ j

and it follows that

hij = ϕ−1 hij − ϕ−2 ϕm δij ,

from which (2.73), and thus (2.74), follow immediately.

Lemma 2.20. Let (M, g0 ) be a compact Einstein manifold such that ∂M → (M, g0 ) is totally geodesic and let g = ϕ−2 g0 , ϕ > 0, ϕ ∈ C ∞ (M ) be a conformally related metric with constant scalar curvature S and such that ∂M → (M, g) has constant mean curvature h with respect to the outward unit (with respect to g) normal ν. Let T be the trace-free Ricci tensor of g. Then 2 ϕ−1 |T | = −(m − 2) hϕ−1 T (ν, ν) (2.75) M

∂M

(where of course we are integrating with respect to the volume element of g on M and ∂M ). Proof. We set g for g0 so that g = ϕ2 g. From (2.26) −1 / −1 Δ ϕ δij , − Tij = Tij + (m − 2)ϕ ϕ ij m and since g by assumption is Einstein we get / −1 Δ ϕ−1 − δij . −Tij = (m − 2)ϕ ϕ ij m The fact that T is traceless yields 2 −|T | = (m − 2)ϕ ϕ−1 ij Tij . Integrating the above on (M, g) we obtain 2 ϕ−1 |T | = (m − 2) − M

M

ϕ−1

T . ij ij

(2.76)

Again from (2.13), since (M, g) has constant scalar curvature Tik,k = 0.

(2.77)

Similarly to what we did in the proof of Lemma 2.18 we consider the vector ﬁeld W = T ∇ϕ−1 .

70

Chapter 2. Pointwise conformal metrics

Taking its divergence in the metric g we have (see (2.72)) div W = ϕ−1 ij Tij + ϕ−1 i Tik,k . Hence, using (2.77) and the divergence theorem, −1 T ∇ ϕ−1 , ν = T . ϕ ij ij ∂M

(2.78)

M

We need now to write the left-hand side of (2.78) appropriately. First we show that S if Y ∈ T ∂M , then T (Y, ν) = 0. Since T = Ric − m , , this is clearly equivalent to showing that Ric(M,g) (Y, ν) = 0 (2.79) (and this is what we expect if we want to show that g is Einstein, see Theorem 2.21). By assumption ∂M → (M, g) is totally geodesic and therefore it follows from Lemma 2.19 that, since g and g are conformally related, ∂M → (M, g) is totally umbilical. Moreover, from (2.74) 0 = h + ϕ−1

∂ϕ , ∂ν

(2.80)

with h the constant mean curvature of ∂M → (M, g). In other words ϕh = −

∂ϕ . ∂ν

In this case we can rewrite (2.73) of Lemma 2.19, since ∂M → (M, g) is totally geodesic, as hij = hδij for 1 ≤ i, j, . . . , ≤ m − 1. Since h is constant, from the above we deduce hijk = 0. On the other hand, from the Codazzi equations, m . hijk − hikj = −Rijk

Tracing with respect to i and k we deduce Rmj = 0, where Rij are the components of the Ricci tensor of (M, g). This proves (2.79). It follows that (2.78) becomes −1 ∂ ϕ−1 T . ϕ T (ν, ν) = ij ij ∂ν ∂M M

2.2. Some applications in the compact case

71

On the other hand, we can rewrite (2.80) as ∂ ϕ−1 = ϕ−1 h. ∂ν Substituting into the above we get ϕ−1 hT (ν, ν) = ∂M

ϕ−1

M

T . ij ij

Hence (2.75) follows immediately from (2.81) and (2.76).

(2.81)

We are now ready to prove Escobar’s result ([Esc90]). Theorem 2.21. Let Bm be the (open) unit ball in Rm and Bm the closed unit ball of Rm ; let g be a metric on Bm conformally related to the Euclidean metric , . Assume that S, the scalar curvature of Bm , g , is constant and that ∂Bm → Bm , g has constant mean curvature h with respect to the outward unit (w.r.t. g) normal ν. Then Bm , g has constant sectional curvature and it is therefore isometric to a geodesic ball in a space form with an appropriate radius depending on h. Proof. We shall prove that Bm , g has constant sectional curvature: the remaining ´ Cartan (see e.g. [Car88]). We part of the conclusion is a well-known result of E. introduce on Bm the auxiliary metric g0 =

4 1 + |x|

2

2 , .

Note that g0 is obtained from the upper hemisphere Sm ¯ , g¯ denoting the stan+,g dard metric induced on Sm by the inclusion Sm → Rm+1 , by stereographic projection π from the South pole π : Bm → Sm + % $ 2 2x 1 − |x| π : x → 2, 2 . 1 + |x| 1 + |x| It follows that Bm , g0 is Einstein and ∂Bm → Bm , g0 is totally geodesic. Since the position vector ﬁeld X is conformal on Bm , , , it is conformal on Bm , g0 and, since g0 |∂Bm = , ∂Bm , X|∂Bm is the outward unit normal with respect to g0 . Since g is conformally related to , , it is conformally related to g0 . Let g = ϕ−2 g0 , ϕ > 0, ϕ ∈ C ∞ Bm . First we show that g is Einstein. Indeed, we are in the assumptions of Lemma 2.20 so that, with the same notation, we have the validity of (2.75). On the other hand, on ∂Bm , X = ϕ−1 ν

72

Chapter 2. Pointwise conformal metrics

where ν is the outward unit normal with respect to g. From Lemma 2.18, ϕ−1 T (ν, ν) = 0, Bm

and constancy of h and (2.75) give

2

ϕ−1 |T | ≡ 0,

M

that is, T ≡ 0 on Bm , g or, in other words, g is an Einstein metric. If m = 2 we are done; for m ≥ 3, to complete the proof we consider the decomposition of the i curvature tensor Rjkl given in (2.27) that we rewrite as 1 (Tik δjl − Til δjk + Tjl δik − Tjk δil ) m−2 S (δik δjl − δil δjk ) + m(m − 1)

i i Rjkl = Wjkl +

with respect to a local o.n. coframe for g. But g is conformally related to the Euclidean metric and therefore it is conformally ﬂat. Furthermore, g is Einstein and hence T ≡ 0 Thus the above formula becomes i = Rjkl

S (δik δjl − δil δjk ) m(m − 1)

and Bm , g has constant sectional curvature.

Chapter 3

General nonexistence results The aim of this chapter is to prove a number of very general nonexistence results for Yamabe-type inequalities of the form Δu + a(x)u − b(x)uσ ≥ 0 on complete, noncompact, Riemannian manifolds. Loosely speaking, triviality of the solutions is obtained under assumptions focusing on the nonnegativity of the spectral radius for the related Schr¨odinger operator and Lp -properties of the solutions themselves. The spectral condition is exploited and generalized via the existence on M of a positive solution ϕ of a diﬀerential inequality of the form Δϕ + Ha(x)ϕ ≤ −K

|∇ϕ|2 ϕ

with H, K parameters satisfying H > 0, K > −1. The function ϕ is used in two diﬀerent ways: in Theorem 3.2 one uses it to obtain an integral inequality involving u and its gradient from which one concludes that u is constant, and therefore necessarily identically zero; in a second group of results, we combine ϕ with the supposed solution u to give rise to a diﬀusion-type diﬀerential inequality for which we prove a Liouville theorem. Note that the limiting case K = −1 amounts to providing a solution on M of the Poisson equation Δψ + a(x) = 0 and it will be considered in Section 3.2 below. In the last section we give a reﬁned version of Theorem 3.2 by using a conformal transformation of the metric in m+2 case σ ≥ m−2 (see Proposition 3.10 and Theorem 3.11). As expected, the above geometrical limitation on σ plays an essential role with respect to the conformal transformation.

P. Mastrolia et al., Yamabe-type Equations on Complete, Noncompact Manifolds, Progress in Mathematics 302, DOI 10.1007/978-3-0348-0376-2_3, © Springer Basel 2012

73

74

3.1

Chapter 3. General nonexistence results

Some spectral considerations

First we recall some deﬁnitions and notation. From now on (M, , ) will always denote a complete, noncompact, connected Riemannian manifold. Let a(x) ∈ C 0 (M ) and for Ω ⊂ M , a bounded domain, deﬁne # |∇ϕ|2 − Ha(x)ϕ2 LH Ω # inf (3.1) λ1 (Ω) = ϕ∈C0∞ (Ω),ϕ ≡0 ϕ2 Ω so that LH denotes the operator LH = Δ + Ha(x),

H ∈ (0, +∞).

(3.2)

H If Ω has suﬃciently regular boundary, λL 1 (Ω) is achieved (see, e.g., [Dav95], [GT01]) by the nonzero solutions of the Dirichlet problem H Δu + Ha(x)u + λL 1 (Ω)u = 0 on Ω, (3.3) u≡0 on ∂Ω.

We then deﬁne the ﬁrst eigenvalue of LH on M as LH H λL 1 (M ) = inf λ1 (Ω), Ω

(3.4)

where Ω runs over all bounded domains in M . Observe that, due to the monoH with respect to Ω, that is tonicity of λL 1 Ω 1 ⊆ Ω2

implies

LH H λL 1 (Ω1 ) ≥ λ1 (Ω2 ),

(3.5)

we have LH H λL 1 (M ) = lim λ1 (BR ). R→+∞

◦

(3.6)

Note that in case Ω2 \Ω1 = ∅, that is the interior of Ω2 \Ω1 is nonempty, then the inequality in (3.5) becomes strict. Of course (3.5) is a trivial consequence of the variational characterization (3.1), while to prove the last statement we can proceed in an elementary way by using Picone’s identity, (3.7) below, as we are now going to show (see for instance [Swa75]). Let u, v ∈ C 2 (Ω), Ω ⊂ M a bounded domain, and suppose v = 0 on Ω. A direct computation yields , 2 u 2 u 2 u 2 , ∇v . (3.7) ∇v 0 ≤ v 2 ∇ − = = |∇u| − ∇ ∇u v v v In particular, ∇u − uv ∇v ≡ 0 on Ω if and only if u = Cv for some C ∈ R. Let now u and v be eigenfunctions on Ω1 and Ω2 respectively (Ω1 ⊆ Ω2 ), LH H with eigenvalues λL 1 (Ω1 ) and λ1 (Ω2 ) . Thus H Δu + Ha(x)u + λL 1 (Ω1 )u = 0 on Ω1 , (3.8) u≡0 on ∂Ω1 ,

3.1. Some spectral considerations

75

H Δv + Ha(x)v + λL 1 (Ω2 )v = 0 on Ω2 , v≡0 on ∂Ω2 .

(3.9)

Note that we can suppose v > 0 on Ω2 . Observing that, by the divergence theorem, ∀ f, g ∈ C 2 (Ω), f ∇g, ν = f LH g + ∇f, ∇g − f Ha(x)g, ∂Ω

Ω

Ω

Ω

with ν the outward unit normal to ∂Ω, integrating (3.7) on Ω1 and using (3.8) and (3.9) gives - , 2 u2 u 2 2 |∇u| − |∇u| + Ha(x)u2 , ∇v = LH v − ∇ 0≤ v v Ω1 Ω1 Ω1 Ω1 Ω1 0 1 2 LH LH LH 2 2 = |∇u| − Ha(x)u − λ1 (Ω2 )u = λ1 (Ω1 ) − λ1 (Ω2 ) u2 . Ω1

Ω1

Ω1

◦

LH H We now reason by contradiction by assuming Ω2 \Ω1 = ∅ and λL 1 (Ω1 ) = λ1 (Ω2 ). From the above inequalities it follows that, on the connected component of Ω1 , u = Cv for some C ∈ R (depending on the component). Choose one of the ◦ 1 , with ∂ Ω 1 ∩ Ω2 = ∅ (this is possible since Ω2 \Ω1 = ∅). Since component, say Ω 1 ∩ Ω2 , contradiction. u ≡ 0 on ∂Ω1 , we have that v ≡ 0 on ∂ Ω

For H = 1 we simply write L instead of L1 . It often happens, in geometrical problems, that the coeﬃcient a(x) of L is related to the Ricci curvature of M : precisely it is a lower bound for the radial Ricci curvature Ric (∇r, ∇r) once we ﬁx an origin o ∈ M , that is, Ric(∇r, ∇r) ≥ −(m − 1)A(r)

(3.10)

and a(x) ≥ A(r(x)) on M . We observe that the condition a(x) ≤ 0, or Ric(∇r, ∇r) ≥ 0 if equation (3.10) holds, implies (but it is not implied by) H λL 1 (M ) ≥ 0,

(3.11)

and therefore a request of the type of this latter may be interpreted as a weakening of the assumption of nonnegative Ricci curvature. It is also worth observing that (3.11) may be satisﬁed even if the entire sectional curvature is strictly negative. For 2 instance, on Hm −H 2 (the hyperbolic space of constant sectional curvature −H < 0) ∞ let a(x) = A(r(x)) with A(t) ∈ C ([0, +∞)), positive, nonincreasing and such that, for some ε > 0, 2 H 2 coth(Ht) on [ε, +∞), = (m−1) 4 (3.12) A(t) (m−1)2 ≤ H 2 coth(Ht) on [0, ε). 4

76

Chapter 3. General nonexistence results

Then, the problem sinhm−1 (Ht)α + A(t) sinhm−1 (Ht)α = 0, H > 0, on [0, +∞), α (0), α(0) > 0 has a solution α ∈ C 2 ([0, +∞)) satisfying (see Proposition 7.16) α(t) ≥ C t e−

m−1 2 Ht

on [0, +∞)

for some appropriate constant C > 0. It follows that u(x) = α(r(x)) is a positive, C 2 -solution on M of Δu + a(x)u = 0

(3.13)

and, by a result of Fisher-Colbrie and Schoen (see [FCS80], [PRS05c]), λL 1 (M ) ≥ 0.

(3.14)

We also note that, as shown in the aforementioned papers, the reverse implication is true, that is, the validity of (3.14) implies the existence of a positive solution of equation (3.13). Thus (3.11) is a genuine relaxation of the assumption Ric ≥ 0 when the coeﬃcient of the linear term in L provides a lower bound for Ric. We now relate (3.11) with the assumption on the existence of a positive solution ϕ of Δϕ + Ha(x)ϕ ≤ −K

|∇ϕ|2 , ϕ

K > −1, H > 0.

(3.15)

As already remarked, (3.11) yields the validity of (3.15) for some ϕ ∈ C ∞ (M ), ϕ = 0,with K = 0 and even with the equality sign. Thus if (3.11) holds, then (3.15) holds with K ≤ 0. On the other hand, assume that ϕ is a positive C 2 -solution of (3.15). Let ψ ∈ C0∞ (M ) and consider the vector ﬁeld ψ 2 ∇ log ϕ. We compute its divergence and we use (3.15), Schwarz’s and Young’s inequalities to obtain |∇ϕ|2 ψ2 ψ Δϕ − div(ψ 2 ∇ log ϕ) = 2 ∇ψ, ∇ϕ + ϕ ϕ ϕ 2 ψ |∇ϕ|2 ψ2 ≤ −Ha(x)ϕ − (K + 1) + 2 |∇ϕ|2 + |∇ψ|2 . ϕ ϕ ϕ Thus, integrating,

|∇ψ|2 − Ha(x)ψ 2 ≥ K M

ψ2 M

|∇ϕ|2 , ϕ2

3.1. Some spectral considerations

77

and from the variational characterization of the bottom of the spectrum we conclude that H if K ≥ 0, then λL 1 (M ) ≥ 0. The following simple proposition (see [BRS98]) motivates the nonexistence results of the next section. Proposition 3.1. Let (M, , ) be a complete manifold. a(x), b(x) ∈ C 0 (M ) and suppose that i) b(x) ≥ 0; ii) b(x) ≡ 0. (3.16) Assume that for some σ ∈ R there exists a positive C 2 -solution u ∈ L2 (M ) of the diﬀerential inequality Δu + a(x)u − b(x)uσ ≥ 0. (3.17) Then λL 1 (M ) < 0, with L = Δ + a(x). Proof. We reason by contradiction and assume that λL 1 (M ) ≥ 0. We ﬁx a geodesic ball BR and we consider a smooth cut-oﬀ function ϕ such that 0 ≤ ϕ ≤ 1 and supp ϕ ⊂ BR+1 ,

ϕ ≡ 1 on BR ,

|∇ϕ| ≤ 2,

(3.18)

where supp ϕ denotes the support of the function ϕ, i. e. the closure of the set where ϕ = 0. Let u be a positive solution of (3.17) and note that, according to the variational characterization of λL 1 (BR+1 ), we have # 2 |∇(uϕ)| − a(x)u2 ϕ2 BR+1 L # . (3.19) λ1 (BR+1 ) ≤ u2 ϕ 2 BR+1 Furthermore λL 1 (M ) ≥ 0 and the monotonicity property of eigenvalues yields (B ) ≥ 0. Next we consider the vector ﬁeld W = uϕ2 ∇u. Using the difλL R+1 1 ferential inequality (3.17) and computing we obtain 2

2

div W ≥ −a(x)u2 ϕ2 + b(x)uσ+1 ϕ2 + |∇(uϕ)| − u2 |∇ϕ| . Hence, by (3.19) and the divergence theorem, 2 L 2 2 2 u ϕ − u |∇ϕ| + 0 ≥ λ1 (BR+1 ) BR+1

BR+1

Rearranging and using (3.16) i) and (3.18)we obtain 2 σ+1 λL (B ) u + b(x)u ≤ 4 R+1 1 BR+1

BR+1

b(x)uσ+1 ϕ2 .

BR+1

u2 . BR+1 \BR

2

Now let R → +∞ and use u ∈ L (M ) to ﬁnally have 2 (M ) u + b(x)uσ+1 ≤ 0. λL 1 M

This contradicts

λL 1 (M )

≥ 0 and (3.16).

M

78

3.1.1

Chapter 3. General nonexistence results

The main nonexistence result

The next theorem, which is the main nonexistence result of the chapter, has been proved in [BRS98]. Theorem 3.2. Let σ > 1 and a(x), b(x) ∈ C 0 (M ) satisfy b(x) > 0 on M ;

a(x) ≤ Cb(x) for r(x) 1

(3.20)

and some constant C > 0. Given α > 0 and K > −1, suppose that, for some H≥

(1 + α)2 4α(K + 1)

(3.21)

there exists ϕ > 0 satisfying Δϕ + Ha(x)ϕ ≤ −K

|∇ϕ|2 ϕ

on M.

(3.22)

Then, the diﬀerential inequality Δu + a(x)u − b(x)uσ ≥ 0

(3.23)

has no bounded, nonnegative, nonidentically null C 2 -solutions u satisfying ⎧ ⎨i) ∈ L1 (M ); a(x)u1+α *−1 )# (3.24) ⎩ii) u1+α ∈ L1 (+∞). ∂Br Remark. Note that if a(x) is bounded, then (3.24) i), ii) are satisﬁed whenever u1+α ∈ L1 (M ). Remark. We recall that a suﬃcient condition for the validity of (3.24) ii) is r ∈ L1 (+∞) 1+α u Br

#

(3.25)

(see [RS01], Proposition 1.3). Proof. We reason by contradiction and we assume that u ∈ C 2 (M ) is a bounded, nonnegative, non-identically null solution of (3.23) satisfying (3.24). Because of (3.20) we can suppose that a(x) ≤ Cb(x)

on M

> 0. We deﬁne for some appropriate constant C ' & 1 1 σ−1 ,1 A = max sup u, C 2 M

3.1. Some spectral considerations

79

and we set v(x) = Then, supM v ≤

1 2

1 u(x). 2A

and v satisﬁes Δv + a(x)v ≥ ¯b(x)v σ ,

(3.26)

with

¯b(x) = (2A)σ−1 b(x) ≥ Cb(x) ≥ a(x) on M. Thus, it suﬃces to show that, if a(x) ≤ ¯b(x) on M , then (3.26) has no nonnegative, nontrivial solutions u satisfying (3.24) and the further requirement iii) sup u < 1. M

We proceed by contradiction and we divide the argument in four steps. Step 1. We claim that there exists a sequence {rk } ↑ +∞ such that |uα ∇u, ∇r | = 0. lim k→+∞

(3.27)

∂Brk

Indeed, let ε > 0. Deﬁne Wε = (u + ε)α ∇u: compute its divergence using (3.26) and apply the divergence theorem on the geodesic ball Br to obtain (u + ε)α ∇u, ∇r − α (u + ε)α−1 |∇u|2 (3.28) ∂Br Br ≥ (¯b(x)uσ − a(x)u)(u + ε)α ≥ a(x)(uσ − u)(u + ε)α Br

Br

where, in the last inequality, we have been using ¯b(x) ≥ a(x). Letting ε ↓ 0+ , the last term on the right-hand side of (3.28) converges to a(x)(uσ − u)uα Br

and the ﬁrst integral on the left-hand side tends to uα ∇u, ∇r . ∂Br

Thus, by B. Levi’s monotone convergence theorem, (u + ε)α−1 |∇u|2 −→ uα−1 |∇u|2 < +∞ Br

Br

as ε ↓ 0. In particular uα−1 |∇u|2 ∈ L1loc (M ). Moreover, uα ∇u, ∇r − α uα−1 |∇u|2 ≥ a(x)(uσ − u)uα . ∂Br

Br

Br

80

Chapter 3. General nonexistence results

Now, since u < 1, σ > 1, (3.24) i) implies that a(x)(uσ+α − u1+α ) ∈ L1 (M ) and, letting r → +∞, we obtain & ' lim inf uα ∇u, ∇r − α uα−1 |∇u|2 = B > −∞. r→+∞

∂Br

Br

We deﬁne

uα−1 |∇u|2 .

G(r) = Br

Since uα−1 |∇u|2 ∈ L1loc (M ), it follows from the co-area formula (1.87) that G(r) is absolutely continuous and that uα−1 |∇u|2 G (r) = ∂Br

is deﬁned almost everywhere and is locally L1 . Assume by contradiction that G(r) → +∞ as r → +∞; then, for large enough r > R, α α u ∇u, ∇r ≥ uα−1 |∇u|2 , 2 Br ∂Br and therefore, by the H¨older inequality, )α 2

*2 G(r)

&

'2

≤

u ∇u, ∇r

≤ G (r)

α

∂Br

uα+1 . ∂Br

We can also suppose to have chosen R so large that G(r) > 0 for r ≥ R. From the above we obtain & '−1 G (r) α2 α+1 ≥ u G2 (r) 4 ∂Br and integrating over [R, r), we get 1 α2 1 + ≥ − G(r) G(R) 4

r

&

'−1 u

R

α+1

dt.

∂Bt

Whence, letting r → +∞, we contradict assumption (3.24) ii). Hence uα−1 |∇u|2 ∈ L1 (M ) and therefore uα−1 |∇u|2 ∈ L1 ((0, +∞)). ∂Br

It follows that there exists a sequence {rk } ↑ +∞ such that $ %$ % uα−1 |∇u|2

uα+1

lim

k→+∞

∂Brk

∂Brk

= 0.

3.1. Some spectral considerations

81

By the H¨older inequality, / 12

|uα ∇u, ∇r ≤ ∂Brk

/ 12 uα−1 |∇u|2

uα+1 ∂Brk

→ 0 as k → +∞,

∂Brk

which implies the claim (3.27). Step 2. Let ϕ > 0 be a solution of (3.22), i.e., Δϕ + Ha(x)ϕ ≤ −K We claim that

|∇ϕ|2 . ϕ

uα+1 ϕ−2 |∇ϕ|2 ∈ L1 (M ).

Indeed, let V = uα+1 ϕ−1 ∇ϕ; since K > −1 we can choose θ > 0 such that 1 > 0. Using (3.22), a computation yields K + 1 − 2θ (3.29) div V = (α + 1)uα ϕ−1 ∇ϕ, ∇u + uα+1 ϕ−1 Δϕ − uα+1 ϕ−2 |∇ϕ|2 1 ≤ 2θ(α + 1)2 uα−1 |∇u|2 + uα+1 ϕ−2 |∇ϕ|2 − Ha(x)uα+1 2θ − (K + 1)uα+1 ϕ−2 |∇ϕ|2 1 2 α−1 2 α+1 − K − 1 uα+1 ϕ−2 |∇ϕ|2 , = 2θ(α + 1) u |∇u| − Ha(x)u + 2θ and applying the divergence theorem 1 α+1 −1 u ϕ ∇ϕ, ∇r + K + 1 − uα+1 ϕ−2 |∇ϕ|2 2θ ∂Br Br α+1 2 ≤ −H a(x)u + 2θ(α + 1) uα−1 |∇u|2 ≤ B ∈ R Br

Br

where, in the last inequality, we have used the fact that a(x)uα+1 , uα−1 ∈ L1 (M ). Setting for the ease of notation 1 F (r) = K + 1 − uα+1 ϕ−2 |∇ϕ|2 , 2θ Br the above inequality can be written in the form uα+1 ϕ−1 ∇ϕ, ∇r ≤ B − F (r). ∂Br

Now assume by contradiction that F (r) → +∞ as r → +∞. Thus, for each r ≥ R suﬃciently large, 1 uα+1 ϕ−1 ∇ϕ, ∇r ≤ − F (r). 2 ∂Br

82

Chapter 3. General nonexistence results

On the other hand, by the Schwarz inequality, uα+1 ϕ−1 ∇ϕ, ∇r F (r) ≤ −2 ∂Br

& ≤

u

α+1

'1/2 & 4

∂Br

that is,

u

α+1

ϕ

−2

|∇ϕ|

2

'1/2 ,

∂Br

&

'1/2

F (r) ≤

uα+1

1/2

{8F (r)}

.

∂Br

Thus, squaring, F (r) 1 ≥ F (r)2 8

&

'−1 u

α+1

r ≥ R.

,

∂Br

Integrating this latter over [R, r] gives F (R)

−1

− F (r)

−1

1 ≥ 8

r

&

'−1 u

R

α+1

dt,

∂Bt

thus, letting r → +∞, we contradict (3.24) ii). Step 3. We note that

uα+1 ϕ−1 ∇ϕ, ∇r = 0.

lim

r→+∞

∂Br

Indeed, because of Step 2, there exists a sequence {rk } ↑ +∞ such that / / α+1 α+1 −2 2 =0 lim u u ϕ |∇ϕ| k→+∞

∂Brk

and therefore,

∂Brk

/2 u

α+1

ϕ

−1

∂Brk

/

≤

u ∂Brk

∇ϕ, ∇r

α+1

/ u

α+1

ϕ

−2

|∇ϕ|

2

→0

as k → +∞.

∂Brk

It remains to show that the desired limit exists. According to (3.22), (3.29) and the divergence theorem, we have α+1 −1 u ϕ ∇ϕ, ∇r = (α + 1) uα ϕ−1 ∇ϕ, ∇u (3.30) ∂Br Br −H uα+1 a(x) − uα+1 ϕ−2 |∇ϕ|2 . Br

Br

3.1. Some spectral considerations

83

Thus, we are reduced to showing that each of the summands on the right-hand side of (3.30) has a ﬁnite limit, as r → +∞. This easily follows from assumption (3.24) i), and from the following fact from Step 1: α −1 u ϕ ∇ϕ, ∇u ≤ 2uα−1 |∇u|2 + 1 uα+1 ϕ−2 |∇ϕ|2 ∈ L1 (M ). 2 Step 4. Fix ε > 0 so small that supM u + ε < 1, and deﬁne the vector ﬁeld Zε = H −1

(u + ε)α uα+1 ∇ϕ − ∇u. ϕ 1 − (u + ε)σ−1

Then, using (3.22), (3.26) and ¯b(x) ≥ a(x), we have div Zε = H −1 (α + 1)uα ϕ−1 ∇ϕ, ∇u + H −1 uα+1 ϕ−1 Δϕ (u + ε)α−1 |∇u|2 1 − (u + ε)σ−1 (u + ε)α+σ−2 (u + ε)α 2 − (σ − 1) |∇u| − Δu 2 1 − (u + ε)σ−1 [1 − (u + ε)σ−1 ] − H −1 uα+1 ϕ−2 |∇ϕ|2 − α

≤ H −1 (α + 1)uα ϕ−1 ∇ϕ, ∇u − H −1 (K + 1)uα+1 ϕ−2 |∇ϕ|2 (u + ε)α−1 (u + ε)α+σ−2 2 |∇u|2 − (σ − 1) 2 |∇u| σ−1 σ−1 1 − (u + ε) [1 − (u + ε) ] (u + ε)α ¯b(x)uσ − a(x)u ≤ − 1 − (u + ε)σ−1 − a(x)uα+1 − α

≤ H −1 (α + 1)ϕ−1 uα ∇ϕ, ∇u − H −1 (K + 1)uα+1 ϕ−2 |∇ϕ|2 −α +

(u + εα−1 (u + ε)α+σ−2 2 2 |∇u| − (σ − 1) 2 |∇u| 1 − (u + ε)σ−1 [1 − (u + ε)σ−1 ]

(u + ε)α a(x)(1 − uσ−1 )u − a(x)uα+1 . 1 − (u + ε)σ−1

Integrating over Br and applying the divergence theorem gives (u + ε)α −1 α+1 −1 H u ϕ ∇ϕ, ∇r − ∇u, ∇r ≤ I + II + III, σ−1 ∂Br ∂Br 1 − (u + ε) where we have set H −1 (α + 1)ϕ−1 uα ∇ϕ, ∇u − (K + 1)uα+1 ϕ−2 |∇ϕ|2 ; I=

Br

(u + ε)α−1 (u + ε)α+σ−2 2 |∇u|2 − (σ − 1) 2 |∇u| ; σ−1 σ−1 1 − (u + ε) [1 − (u + ε) ] Br & ' α (u + ε) III = a(x) uα+1 − 1 − uσ−1 u . 1 − (u + ε)σ−1 Br II =

−α

84

Chapter 3. General nonexistence results

Letting ε ↓ 0+ and applying the dominated convergence theorem, the left-hand side of the above converges to uα H −1 uα+1 ϕ−1 ∇ϕ, ∇r − ∇u, ∇r σ−1 ∂Br ∂Br 1 − u while

II → −

α Br

uα−1 uα+σ−2 2 2 |∇u| + (σ − 1) 2 |∇u| 1 − uσ−1 [1 − uσ−1 ]

and ﬁnally III → 0. Then, using H −1

(α + 1)2 ≤ α, 4(K + 1)

we obtain H −1

uα uα+1 ϕ−1 ∇ϕ, ∇r − ∇u, ∇r σ−1 ∂Br ∂Br 1 − u & (α + 1)2 α−1 −1 ≤ −H (K + 1)ϕ−2 uα+1 |∇ϕ|2 + |∇u|2 u 4(K + 1) Br ' −1 α − (α + 1) ∇ϕ, ∇u ϕ u / 2 uα−1 uα+σ−2 −1 (α + 1) α−1 α |∇u|2 + (σ − 1) − u 2 −H σ−1 σ−1 1 − u 4(K + 1) [1 − u ] Br 2 u√ −1 α−1 α + 1 ≤ −H u 2√K + 1 ∇u − ϕ K + 1∇ϕ Br σ−1 1 + σ−1 2 α −u −α u(α−1)+(σ−1) 2 |∇u| . (1 − uσ−1 ) Br

Now, letting r → +∞ along the sequence {rk } constructed in Step 1 and using Step 3 yields 0≤H

−1

u Br

2 α+1 u√ 2√K + 1 ∇u − ϕ K + 1∇ϕ σ−1 1 + σ−1 α −u −α uα+σ−2 |∇u|2 ≤ 0. σ−1 )2 (1 − u Br

α−1

Let now A = ∅ be a connected component of the set {x ∈ M : u(x) > 0}.

3.1. Some spectral considerations

85

The above forces ∇u = 0 on A and thus u = c1 > 0 on A. In turn this implies ∇ϕ = 0 on A and ϕ = c2 > 0 on A. But, on all of M , hence on A, Δϕ + Ha(x)ϕ ≤ 0, then Ha(x)c2 ≤ 0. Because a(x)c1 − b(x)cσ1 ≥ 0 implies a(x)c1 ≥ 0, it follows that a(x) = 0 on A and therefore, since 0 = Δu ≥ −a(x)u + ¯b(x)uσ = ¯b(x)cσ1

on A

we have ¯b(x) = 0 on A contradicting (3.20).

Before commenting on Theorem 3.2 we prove the next result (see [PRS10]) in the same spirit. Theorem 3.3. Let a(x), b(x) ∈ C 0 (M ) and assume that b(x) ≥ 0. Let H > 0, K > −1, A ∈ R be constants satisfying max {0, A} ≤ H(K + 1) − 1,

(3.31)

and suppose that there exists a positive C 2 -solution ϕ of the diﬀerential inequality Δϕ + Ha(x)ϕ ≤ −K

|∇ϕ|2 ϕ

on M.

(3.32)

Then the diﬀerential inequality uΔu + a(x)u2 − b(x)uσ+1 ≥ −A|∇u|2 , σ ≥ 1,

(3.33)

has no nonnegative C 2 -solutions u on M satisfying supp u ∩ {x ∈ M : b(x) > 0} = ∅ and

& ϕ

β+1 H (2−p)

'−1 u

p(β+1)

∈ L1 (+∞)

(3.34)

(3.35)

∂Br

for some p > 1 and max {0, A} ≤ β ≤ H(K + 1) − 1. Remark. Note that if p = 2, then the nonintegrability assumption (3.35) involves u alone and reduces to & '−1 2(β+1) u ∈ L1 (+∞). (3.36) ∂Br

86

Chapter 3. General nonexistence results

Proof. Let u ≥ 0 be a solution of (3.33) on M satisfying (3.34) and (3.35). Fix ε > 0, α ∈ R and set v = ϕ−α (u2 + ε)

β+1 2

.

A straightforward computation that uses (3.32) and (3.33) yields |∇ϕ|2 + (β + 1)(u2 + ε)β b(x)uσ+1 v div(ϕ2α ∇v) ≥ α(K − α + 1)(u2 + ε)β+1 ϕ2 " ! u2 2 β+1 αH − (β + 1) 2 + a(x)(u + ε) u +ε " ! u2 + (β + 1)(u2 + ε)β 1 − A + (β − 1) 2 |∇u|2 . u +ε We choose α = H −1 (β + 1), so that our assumptions on β, H and K give 0 < α ≤ K + 1. Therefore α(K − α + 1) ≥ 0, and using b(x) ≥ 0 and β + 1 ≥ 0 we deduce that v div(ϕ2α ∇v) ≥ ε(β + 1)a(x)(u2 + ε)β (3.37) ! " 2 u + (β + 1)(u2 + ε)β 1 − A + (β − 1) 2 |∇u|2 . u +ε Let r(t) ∈ C 1 (R) and s(t) ∈ C 0 (R) satisfy the conditions ii) r(v) + vr (v) ≥ s(v) > 0 on [0, +∞)

i) r(v) ≥ 0,

(3.38)

and consider the vector ﬁeld Z = vr(v)ϕ2α . For ﬁxed t and δ > 0 let also ψδ be the Lipschitz function deﬁned by ⎧ ⎪ ⎨1, ψδ (x) =

t+δ−r(x) , δ ⎪

⎩

0,

if r(x) ≤ t, if t < r(x) < t + δ, if r(x) ≥ t + δ.

Using (3.37), (3.38) and the deﬁnition of ψδ we compute div(ψδ ) = ψδ div Z + ∇ψδ , Z

" ' u2 2 |∇u| χBt ≥ (β + 1)r(v)(u + ε) εa(x) + 1 − A + (β − 1) 2 u +ε 1 + s(v)ϕ2α |∇v|2 χBt + ∇r, Z χB¯t+δ \Bt δ 2

β

&

!

3.1. Some spectral considerations

87

whence, integrating and using the divergence theorem and Cauchy-Schwarz inequality we obtain " & ! ' u2 2 β 2 |∇u| (β + 1)r(v)(u + ε) ε r(x) + 1 − A + (β − 1) 2 u +ε Bt 1 + s(v)ϕ2α |∇v|2 ≤ |Z|. δ Bt+δ \Bt Bt By H¨ older’s inequality the integral on the right-hand side is bounded above by / 12 / 12 2 1 r(v) 1 v2 ϕ2α ϕ2α s(v)|∇v|2 . δ Bt+δ \Bt s(v) δ Bt+δ \Bt Inserting this into the above inequality, letting δ ↓ 0+ and using the co-area formula (1.87) we deduce that

' " u2 2 (β + 1)r(v)(u + ε) ε a(x) + 1 − A + (β − 1) 2 |∇u| u +ε Bt & ' 12 2 2α 1 2α 2 2α r(v) 2 +s(v)ϕ |∇v| ≤ ϕ ϕ s(v)|∇v|2 2 . v s(v) 2

&

!

β

(3.39)

As ε ↓ 0, v = vε → v0 = ϕ−α uβ+1 , therefore, using the dominated convergence theorem in (3.39) we get s(v0 )ϕ2α |∇v0 |2 + (β + 1)(β − A) r(v0 )u2β |∇u|2 (3.40) Bt

&

≤

ϕ2α

2

r(v0 ) 2 v s(v0 ) 0

' 12

Bt

ϕ2α s(v0 )|∇v0 |

Deﬁne

1 2 2

.

(3.41)

ϕ2α s(v0 )|∇v0 |2 ;

h(t) = Bt

by the co-area formula (1.87) h is Lipschitz and h (t) = ϕ2α s(v0 )|∇v0 |2 , ∂Bt

and noting that the coeﬃcient of the second integral on the left-hand side of (3.40) is nonnegative by the conditions imposed on β, we obtain & h(t) ≤

ϕ ∂Bt

2α r(v0 )

2

s(v0 )

v02

' 12

1

[h (t)] 2 .

(3.42)

88

Chapter 3. General nonexistence results

Our aim is to show that under assumption (3.35) v0 is constant. We reason by contradiction, then there exists R0 such that h(t) > 0 ∀ t ≥ R0 and therefore the right-hand side of (3.42) is positive for t ≥ R0 . Dividing through by h(t), squaring and integrating the resulting diﬀerential inequality between R and r with R0 ≤ R ≤ r yields '−1 r& 2 −1 −1 −1 2α r(v0 ) 2 v ϕ h(R) ≥ h(R) − h(r) ≥ dt. (3.43) s(v0 ) 0 R We choose a sequence f of functions rn (t) =

1 t + n 2

p−2 2

, sn (t) = min {p − 1, 1} rn (t) ∀ n ∈ N, p > 1.

Since condition (3.38) holds for all n, so does (3.43), hence, letting n → +∞ and using the Lebesgue and monotone convergence theorems we deduce that there exists c > 0 depending only on p such that '−1 '−1 & r & v0p−2 ϕ2α |∇v0 |2 ≥C ϕ2α v0p dt. (3.44) R

BR

∂Bt

Now, recalling that α = (β + 1)/H and the deﬁnition of v0 , we have ϕ2α v0p = ϕ(2−p)(β+1)/H u2(β+1) and the required contradiction is obtained by letting r → +∞ and using assumption (3.35). Thus v0 is constant, and we deduce that there exists a constant C ≥ 0 such that uH = Cϕ. Since u is not identically zero by (3.34), C > 0 and u is strictly positive on M . We insert the expression of ϕ in terms of u in (3.32), divide by CHuH−2 and subtract the result from (3.33) to obtain [A − H(K + 1)]|∇u|2 ≥ b(x)uσ+1 . Since the coeﬃcient of |∇u|2 is nonpositive, by (3.31) we conclude that b(x)uσ+1 ≤ 0 which contradicts (3.34).

Remark. Observe that the above proof actually shows that if A < H(K + 1) − 1, then ∇u = 0, so that u, and therefore ϕ, are necessarily constant. It follows from (3.32) and (3.33) that 0 ≥ a(x)u ≥ b(x)uσ+1 ≥ 0, so that, if a(x) does not vanish identically, then u ≡ 0 without any strict positivity assumption on b(x). On the other hand, if A = H(K + 1) − 1, then the conclusion depends on the fact that b(x) is positive somewhere.

3.1. Some spectral considerations

89

Remark. We also note that if u is assumed to be strictly positive, then the conclusion of Theorem 3.3 holds, with a much easier proof, if we assume that max {−1, A} ≤ H(K + 1) − 1 and that β > −1, β ≥ A, β ≤ H(K + 1) − 1. We observe that in both Theorems 3.2 and 3.3 we require some integrability type condition for the positive (nonnegative) solution u of (3.23) Δu + a(x)u ≥ b(x)uσ ,

(3.23)

respectively (3.24) ii), that is &

'−1 u

α+1

∈ L1 (+∞)

for some α > 0

∂Br

and (3.36), that is &

u2(1+β)

'−1

∈ L1 (+∞)

for some max{0, A} ≤ β ≤ H(K + 1) − 1.

∂Br

Sometimes this type of information is already contained in the diﬀerential inequality. This is the content of the next Proposition 3.4. Let (M, , ) be a complete Riemannian manifold and let a(x), b(x) ∈ C 0 (M ) with b(x) > 0 on M . Assume that u ≥ 0 is a C 2 -solution of the diﬀerential inequality uΔu + a(x)u2 − b(x)uσ+1 ≥ −A|∇u|2

(3.45)

for A ≤ 1 and σ > 1. Then, for every p ≥ 1, p > A + 2 there exist constants C1 , C2 > 0 which depend only on p, σ and R0 > 0 suﬃciently large, such that, ∀R ≥ R0 , b(x)u

p+σ−2

≤ C1 R

BR

−2 p+σ−2 σ−1

b(x)

p−1 − σ−1

+ C2

B2R

B2R

a+ (x) b(x)

p−1 σ−1

a+ (x), (3.46)

where a+ denotes the positive part of a. Proof. First we observe that we may suppose that u ≡ 0, for otherwise there is nothing to prove. Thus, there exists R0 > 0 such that u ≡ 0 on BR , ∀ R ≥ R0 . Next, ∀ R ≥ R0 let ψ = ψR : M → [0, 1] be a smooth cut-oﬀ function such that ψ ≡ 1 on BR ,

ψ ≡ 0 on M \B2R ,

|∇ψ| ≤

p−1 C p+σ−2 on B2R ψ R

(3.47)

90

Chapter 3. General nonexistence results

for some C > 0 which depends only on σ and p. Note that this is possible since the exponent p−1 < 1. p+σ−2 Having ﬁxed ε > 0, we let W be the vector ﬁeld deﬁned by W = ψ 2 (u + ε)p−3 u∇u; a computation that uses (3.45) yields & ' u div W ≥ ψ 2 (u + ε)p−3 −a(x)u2 + b(x)uσ+1 + 1 − A − (p − 3) |∇u|2 u+ε + 2ψ(u + ε)p−3 u ∇u, ∇ψ . We estimate the last term on the right-hand side using Cauchy-Schwarz inequality and Young inequality 2ab ≤ λa2 + λ−1 b2 with λ = p − 2 − A > 0 to obtain 1 div W ≥ ψ 2 (u + ε)p−3 −a(x)u2 + b(x)uσ+1 − u(u + ε)p−2 |∇ψ|2 . p−2−A We integrate the above inequality, apply the divergence theorem, rearrange, let ε → 0+ and use the dominated convergence theorem in this order, to deduce that 1 b(x)ψ 2 up+σ−2 ≤ up−1 |∇ψ|2 + ψ 2 a+ (x)up−1 . (3.48) p − 2 − A B2R B2R B2R If p = 1, the conclusion follows immediately using (3.47). If p > 1, we denote by I and II the two integrals on the right-hand side of (3.48), and use H¨older inequality with conjugate exponents p+σ−2 >1 p−1

and

p+σ−2 σ−1

and the assumption that b(x) > 0 to estimate p−1 & σ−1 ' p+σ−2 ' p+σ−2 & p−1 p−1 −2 σ−1 − σ−1 2 p+σ−2 2 p+σ−2 b(x)ψ u ψ b(x) |∇ψ| σ−1 I≤ B2R

and

&

B2R

2 p+σ−2

II ≤

p−1 & ' p+σ−2

b(x)ψ u

2

ψ a+ (x)

B2R

p+σ−2 σ−1

b(x)

p−1 − σ−1

σ−1 ' p+σ−2

.

B2R

Inserting into (3.48), noting that the integral on the left-hand side is strictly positive by the choice of R, and simplifying, we obtain & ! " p−1 p−1 1 −2 σ−1 − σ−1 2 p+σ−2 2 p+σ−2 b(x)ψ u ≤ ψ b(x) |∇ψ| σ−1 p − 2 − A B2R B2R p+σ−2 " σ−1 / σ−1 ! ψ 2 a+ (x)

+ B2R

p+σ−2 σ−1

p−1

b(x)− σ−1

p+σ−2

.

3.1. Some spectral considerations

91

The required conclusion follows again using (3.47) and the elementary inequality (a + b)τ ≤ 2τ (aτ + bτ ) valid for a, b, τ ≥ 0. For the sake of illustration let us consider the case b(x) ≥

b(x) > 0,

C r(x)μ

for r(x) 1

(3.49)

and some constants C > 0, μ ≥ 0. Then, from (3.46) we have up+σ−2 ≤ C1 Rμ−2

p+σ−2 p−1 σ−1 + σ−1 μ

BR

In particular, if

p+σ−2

vol(B2R ) + C2 B2R

a+ (x) b(x)

a+ (x) σ−1 a+ (x). (3.50) b(x)

is bounded above and a+ (x) ∈ L1 (M ), μ−2

up+σ−2 ≤ C2 + C1 R σ−1 (p+σ−2) vol(B2R ).

(3.51)

B2R

We observe now that (3.24) or (3.36) are implied by r

#

Br

and from (3.51)

uξ

∈ L1 (+∞),

&

'−1 u

p+σ−2

ξ≥0

∈ L1 (+∞)

∂Br

in case μ−2

C1 r σ−1 (p+σ−2) vol(B2R ) ≤ Cr2 log r

for r 1,

in other words if μ−2

vol(B2r ) ≤ Cr2− σ−1 (p+σ−2) log r

for r 1.

This is a slow growth for vol(Br ). This somehow forces us to ﬁnd a better way to guarantee integrability and this will be achieved with upper a priori estimates which however are obtained under lower bound type restrictions on the Ricci tensor. We shall return to this in Chapter 4. Using Proposition 3.4 we obtain the following Theorem 3.5. Let (M, , ) be a complete Riemannian manifold, and let a(x), b(x)∈ C 0 (M ) where b(x) > 0 on M and b(x) ≥

C r(x)μ

(3.52)

92

Chapter 3. General nonexistence results

for r(x) 1 and for some constants C > 0 and 0 ≤ μ ≤ 2. Assume that a+ (x) < +∞ and (ii) (i) sup a+ (x) = O r2−μ log r as r → +∞, b(x) M Br

(3.53)

and that, for some H ≥ 1, the operator LH = Δ + Ha(x) satisﬁes H λL 1 (M ) ≥ 0.

(3.54)

Finally, let A and σ be such that A ≤ 1, A < H − 1, 1 < σ ≤ 2H + 1 and σ < 2H − A and assume that

2H volBr = O r2+(2−μ) σ−1 log r as r → +∞. (3.55) Then the only nonnegative C 2 solution u of the diﬀerential inequality uΔu + a(x)u2 − b(x)uσ+1 ≥ −A|∇u|2

(3.56)

is u ≡ 0. Proof. If we set p = 2H + 2 − σ, the conditions imposed on the parameters imply that p satisﬁes the assumptions listed in the statement of Proposition 3.4. The lemma and condition (3.53) (i) show that there exist constants Ci > 0 such that 4H 2H b(x)u2H ≤ C1 r− σ−1 b(x)1− σ−1 + C2 a+ (x) (3.57) Br

B2r

B2r

for r > 0 suﬃciently large. We use condition (3.52) to estimate from below the integral on the left-hand side. On the other hand, since σ < 2H + 1, we may again use condition (3.52) to estimate from above the ﬁrst integral on the right-hand side, and (3.53) (ii) to estimate from above the second integral, and deduce that, for r suﬃciently large, 2H u2H ≤ C r(μ−2) σ−1 volB2r + r2 log r , Br

whence, using the volume growth condition (3.57) we conclude that u2H ≤ Cr2 log r for r 1. Br

This immediately implies that, for r large, #

1 r ≥C ∈ L1 (+∞), 2H r log r u Br

which in turn yields (see, e.g., [RS01] Proposition 1.3) #

1 ∈ L1 (+∞). u2H ∂Br

We may therefore apply Theorem 3.3 with K = 0 and β = H − 1 to deduce that supp u = ∅, that is, u ≡ 0.

3.2. The endpoint case K = −1 and the Poisson equation

93

Remark. The argument used in the proof shows that the condition that ab+ is bounded above may be removed provided we replace (3.57) with

2H 2H a+σ−1 = O r2−μ σ−1 log r as r → +∞. (3.58) Br

Note that since the integral on the left-hand side is a nondecreasing function of r this also imposes the further restriction μ ≤ (σ − 1)/H, with corresponding restrictions being imposed on the range of the other parameters.

3.2

The endpoint case K = −1 and the Poisson equation

Theorem 3.3 does not cover the “endpoint” case where K = −1 in (3.22), which we are going to consider presently. We therefore assume that there exists a positive solution ϕ of |∇ϕ|2 , H > 0. (3.59) Δϕ + Ha(x)ϕ ≤ ϕ If u is a C 2 solution of (3.33) with σ ≥ 0, we deﬁne v = ϕ−γ u, γ ≥ 0. A computation that uses (3.59), (3.33) and Young inequality yields vΔv = (γH − 1)a(x)ϕ−2γ u2 + b(x)ϕ−2γ uσ+1 + γ 2 ϕ−2γ−2 u2 |∇ϕ|2 − Aϕ−2γ |∇u|2 − 2γϕ−2γ−1 u∇u, ∇ϕ ≥ (γH − 1)a(x)ϕ−2γ u2 + b(x)ϕ−2γ uσ+1 1 + γ 2 (1 − )ϕ−2γ−2 u2 |∇ϕ|2 − (A + ε)ϕ−2γ |∇u|2 . ε Choosing γ = 1/H and ε = −A, the right-hand side reduces to b(x)ϕ−2/H uσ+1 +

1 1 (1 + )ϕ−2/H−2 u2 |∇ϕ|2 , H2 A

and we easily deduce that, if A ≤ −1, then the function v satisﬁes Δv ≥ b(x)ϕ(σ−1)/H v σ .

(3.60)

Using (3.60) we obtain the following version of Theorem 3.3. Theorem 3.6. Let a(x), b(x) ∈ C 0 (M ), with b(x) ≥ 0, and assume that ϕ is a positive C 2 solution of (3.59) satisfying ϕ(x) ≥ Cr(x)1/δ

(3.61)

for r(x) 1, and some constants C > 0 and δ > 0. Then the diﬀerential inequality uΔu + a(x)u2 − b(x)uσ+1 ≥ −A|∇u|2 ,

94

Chapter 3. General nonexistence results

with σ ≥ 0 and A ≤ −1, has no nonnegative C 2 solution satisfying supp u ∩ {x : M : b(x) > 0} = ∅

(3.62)

rδp ∈ L1 (+∞) p u ∂Br

(3.63)

and # for some p > 1. Proof. According to (3.60) above, the function v = ϕ−1/H u is sub-harmonic. Further, (3.61) and (3.63) imply that

−1 vp ∈ L1 (+ + ∞). ∂Br

An application of Theorem B in [RS01] shows that v is constant. The conclusion now follows as in the proof of Theorem 3.3. Note that, even in the case of Theorem 3.6, if A < −1, then the conclusion can be strengthened to assert that every nonnegative solution of (3.6) vanishes identically, unless a(x) = b(x) ≡ 0. In applying Theorem 3.6 it is of course crucial to be able to ﬁnd positive solutions of (3.59) satisfying the asymptotic lower bound (3.61). By contrast, in order to apply Theorem 3.3 one needs a positive solution of (3.22), whose existence, in typical applications like the one exempliﬁed by Theorem 3.5 above, is guaranteed by means of assumptions on the spectrum of a suitable operator. Observe now that if v is a solution of the Poisson equation Δv = a(x), then the function ϕ = e−v is a positive solution of Δϕ + a(x)ϕ =

|∇ϕ|2 , ϕ

and furthermore, an upper bound for v yields a lower bound for ϕ. The Poisson equation on complete Riemannian manifolds has been extensively studied using heat kernel techniques to obtain bounds on the Green kernel. To illustrate an application of Theorem 3.6, we consider the elementary case where the positive part of a(x) is integrable. We ﬁrst recall that a manifold is said to be parabolic if every positive bounded-above subharmonic function is constant, and nonparabolic otherwise (see [Gri99], [PRS05b]); we also recall that the deﬁnition of non-parabolicity is equivalent to the existence of a positive Green kernel (see again [Gri99]). Then we have the following lemma (see, e.g., the proof of Theorem 3.2 in [NST01]).

3.2. The endpoint case K = −1 and the Poisson equation

95

Lemma 3.7. Let (M, , )) be a complete, nonparabolic manifold, and let ρ ∈ C 0,α (M ) ∩ L1 (M ), 0 ≤ α < 1, be a nonnegative function. Then, there exists a solution v ∈ C 2 of the Poisson equation Δv = ρ satisfying v ≤ 0. Proof. Let G(x, y) be the Green kernel, i.e., the minimal positive fundamental solution of the Laplacian, which exists by the assumption that M is nonparabolic. ∞ The # Green kernel is symmetric and, if ψ ∈ Cc (M ), then the function u(x) = − M G(x, y)ψ(y) is smooth and satisﬁes Δu = ψ. # We claim that if ρ ∈ C 0 (M )∩L1 (M ), then the function v = − M G(x, y)ρ(y) is well deﬁned and locally bounded. Assuming the claim, for every ψ ∈ C0∞ (M ) we have vΔψ = − ψ(x) ΔG(x, y)ρ(y) M M M ρ(y) G(x, y)Δψ(x) = ρ(y)ψ(y), =− M

M

M

so that v satisﬁes the Poisson equation in distributional sense, and therefore, by standard elliptic regularity (see [Aub98]), Theorem 3.55), it is a classical solution. Clearly, v is nonpositive. To prove the claim, ﬁx R > 0 and for every x ∈ BR we write G(x, y)ρ(y) = G(x, y)ρ(y) + G(x, y)ρ(y). (3.64) M

B2R

M \B2R

Since G(x, y) is locally integrable uniformly for x ∈ BR , the ﬁrst integral on the right-hand side is bounded above by a constant independent of x ∈ BR . On the other hand, by the local Harnack inequality there exists a constant C independent of x ∈ BR and such that G(x, y) ≤ CG(o, y)

for every y ∈ M \ B2R .

Moreover, sup G(o, y) < +∞. M \B2R

Indeed, let Ωn be an exhaustion of M by open sets containing o and with smooth boundary and let Gn by the Green kernel of Ωn and recall that, by the standard construction of the Green kernel G(x, y), Gn (x, y) → G(x, y) locally uniformly in M \ {x}. Let C > sup∂B2R G(o, y), then, for every suﬃciently large n we have C > G(o, y) ≥ Gn (o, y) for y ∈ ∂B2R and clearly C > Gn (o, y) = 0 for y ∈ ∂Ωn . Thus, by the comparison principle, C > Gn (o, y) in Ω \ B2R , whence, letting

96

Chapter 3. General nonexistence results

n → +∞, G(o, y) ≤ C for y ∈ M \ B2R . It follows that there exists a constant C independent of x ∈ BR and y ∈ M \ B2R such that G(x, y) ≤ C . Since ρ is integrable, this implies that the second integral on the right-hand side of (3.64) is also bounded independently of x ∈ BR , as required to complete the proof of the claim. Corollary 3.8. Let (M, , ) be a complete, nonparabolic manifold, let the functions a(x) ∈ C 0,α (M ), and b(x) ∈ C 0 (M ) satisfy b(x) > 0 and a+ (x) ∈ L1 (M ),

(3.65)

and suppose that for some constants σ > 1, A ∈ (−∞, −1] and μ, p, q satisfying q > max{1, 3 − σ}, we have

0≤μ≤2

σ−1 , σ+q−2

p>

q+σ−2 , σ−1

p−1 as r → +∞ a+ (x)p = O r[2(σ−1)−μ(q+σ−2)] q−1 Br

σ+q−2 as r → +∞ volBr = O r2+(2−μ) σ−1 b(x) ≥

C r(x)μ

for r(x) 1.

(3.66) (3.67) (3.68)

Then there are no nonnegative, nonidentically zero C 2 (M ) solutions of the diﬀerential inequality uΔu + a(x)u2 ≥ b(x)uσ+1 − A|∇u|2

on M.

(3.69)

Proof. Since a+ is integrable, by Lemma 3.7 and the preceding discussion there exists a solution ϕ ≥ 1 of |∇ϕ|2 , Δϕ + a+ ϕ = ϕ and ϕ is a solution of the diﬀerential inequality (3.59). Now, let u be a nonnegative solution of (3.69). Noting that q > 1, and A ≤ −1 imply q > A + 2, applying Proposition 3.4 and using the lower bound (3.68) for b(x) we obtain q+σ−2 q+σ−2 σ+q−2 uq+σ−2 ≤ C1 r(μ−2) σ−1 volB2r + C2 rμ σ−1 a+ (x) σ−1 . (3.70) Br

B2r

We claim that (3.66) implies

σ+q−1 σ+q−2 a+ (x) σ−1 = O r2−μ σ−1 B2r

as r → +∞,

(3.71)

3.2. The endpoint case K = −1 and the Poisson equation

97

which, together with the volume growth assumption (3.67) yields uq+σ−2 = O r2 as r → +∞. Br

As in the proof of Theorem 3.5, it follows (see, e.g., [RS01] Proposition 1.3) that u satisﬁes condition (3.63) with δ = 0 and exponent q + σ − 2 which is greater than 1 by the conditions on q. Thus, Theorem 3.6 (with a(x) replaced by a+ (x)) applies and u vanishes identically. To conclude it remains to prove the claim. To this end, we set p = (q + σ − 2)/(σ − 1), and apply H¨older inequality with conjugate exponents (p − 1)/(p − p ) and (p − 1)/(p − 1) to estimate p−p p −1 a+ (x)p = a+ (x) p−1 +p p−1 Br

Br

≤

a+ (x) B2r

p−p p−1

a+ (x)

p

−1

pp−1

B2r

p −1 q+σ−2 = O r[2(σ−1)−μ(q+σ−2)] q−1 = O r2−μ σ−1 ,

as required.

Remark. Assume that b(x) satisﬁes the condition stated in the corollary, with μ < σ − 1, and that conditions (3.66) and (3.67) are replaced by

μ 2 a+ (x) σ−1 = O r2[1− σ−1 ] as r → +∞, Br

μ−2 as r → +∞. volBr = O r2[1− σ−1 ] It follows from (3.70) above with q + σ − 2 = 2, that every nonnegative solution of Δu + a(x)u − b(x)uσ = 0 satisﬁes

u2 ≤ Cr2 log r,

(3.72)

(3.73)

Br

and the same estimate is clearly satisﬁed by the diﬀerence of two solutions. An application of Theorem 4.1 in [BRS98] shows that (3.72) has at most one positive solution. We remark in this respect that if we replace u2 in (3.73) with up with p > 2, then the conclusion of Theorem 4.1 in [BRS98] fails, as the example described on pages 214-215 therein shows (see also Theorem 5.1 below). Clearly, whenever a solution to the Poisson equation is available one can obtain a version of Corollary 3.8 above. According to a recent result by O. Munteanu and N. Sesum, [MS10], if (M, , ) has Ricci curvature bounded from below by a

98

Chapter 3. General nonexistence results

negative constant, and its Laplace operator has positive bottom of the spectrum, and if ρ satisﬁes the decay condition |ρ(x)| ≤

C (1 + r(x))γ

for some γ > 1, then the Poisson equation Δv = ρ(x) has a solution satisfying |v(x)| ≤ AeBr(x) for some A, B > 0. Applying this result, and arguing as in the proof of Corollary 3.8 one obtains the following Corollary 3.9. Let (M, , ) be a complete manifold with Ricci curvature bounded from below by a negative constant, and positive bottom of the spectrum. Let a(x) ∈ C 0,α (M ), and 0 < b(x) ∈ C 0 (M ) and suppose that for some constants σ > 1, A ∈ (−∞, −1], μ > 0, p ≥ 1, γ ∈ (1, 2] and λ > 0 satisfying p > A + 2,

0≤μ≤2

σ−1 , σ+p−2

γ

p+σ−2 p+σ−2 ≤ λ ≤ 2 − (μ − γ) , σ−1 σ−1

we have a+ (x) ≤

C , (1 + r(x))γ

b(x) ≥

C and volBr = O rλ . μ (1 + r(x))

Then there are no nonnegative, nonidentically zero C 2 (M ) solutions of the diﬀerential inequality (3.69) on M .

3.3

A reﬁned version of Theorem 3.2

We shall now relax the assumption b(x) > 0 on M in Theorem 3.2. This is related to geometry and ﬁrst we need to derive the transformation law of the LaplaceBeltrami operator under a conformal change of the metric. We go back to Chapter 2 and we suppose to have a conformal change of the type , = ϕ2 , ,

ϕ > 0, ϕ ∈ C ∞ (M ),

(3.74)

with the notation in Chapter 2, θi = ϕθi

i = 1, . . . , m = dim M,

(3.75)

dϕ = ϕt θt ,

(3.76)

and, having set

3.3. A reﬁned version of Theorem 3.2

99

ϕj i ϕi j θ − θ . θji = θji + ϕ ϕ Let u ∈ C 2 (M ). Then

(3.77)

j ϕθj = uj θj , du = u j θj = u

that is, u j = uj ϕ−1 .

(3.78)

Furthermore, using (3.75) and (3.77), (3.76) ϕj k ϕk j t k −1 −1 k uj − u k θj = d(uj ϕ ) − (uk ϕ ) θj + u jt θ = d θ − θ ϕ ϕ = ϕ−1 duj − uj ϕ−2 ϕt θt − ϕ−1 uk θjk − ϕ−2 uk ϕj θk + ϕ−2 uk ϕk θj

= ϕ−1 ujt θt − ϕ−2 uj ϕt + ut ϕj − δtj uk ϕk θt 0

1 = ϕ−2 ujt − ϕ−1 uj ϕt + ut ϕj − δtj uk ϕk θt . It follows that u jt =

1 1 ujt − 3 uj ϕt + ut ϕj − δtj uk ϕk . 2 ϕ ϕ

(3.79)

Hence, 1 1 Hess(u) = Hess(u) − (du ⊗ dϕ + dϕ ⊗ du) + ∇u, ∇ϕ , ϕ ϕ and

= 1 Δu + m − 2 ∇u, ∇ϕ . Δu ϕ2 ϕ3

(3.80)

(3.81)

Note the simple form that equation (3.81) takes if m = 2: in this case, harmonic functions do depend only on the conformal class of the metric. Remark. Formula (3.81) shall also be used in Step 3 of the proof of Theorem 6.8. Proposition 3.10. Let (M, , ) be a complete Riemannian manifold of dimension m ≥ 3 and let b(x) ∈ C 0 (M ) be such that b(x) > 0

in M \BR

(3.82)

for some R > 0. Let v be a positive solution of Δv + a(x)v − b(x)v σ ≥ 0 with a(x) ∈ C 0 (M ) and σ ≥

m+2 m−2 .

(3.83)

Then, there exists ε = ε(v) > 0 such that, if

b(x) ≥ −ε

on M,

(3.84)

100

Chapter 3. General nonexistence results

then, there exist b(x) ∈ C 0 (M ) and C > 0 satisfying i) b(x) > 0 in M,

ii) b(x) = C 2 b(x) in M \BR+1

(3.85)

and a positive solution u on M of m+2 (m − 2)(σ − 1) a(x)u − b(x)u m−2 ≥ 0. 4 Furthermore, there exists C1 > 0 such that

Δu +

u(x) = C1 v(x)

(m−2)(σ−1) 4

for r(x) 1.

(3.86)

(3.87)

Proof. We set ϕ = v and we consider the conformally related metric , = 2 ϕ , . Let ψ be a solution of the Dirichlet problem = δ on BR+2 , Δψ (3.88) ψ≡0 on ∂BR+2 σ−1 2

for some constant δ > 0. We deﬁne β(x) = ξ(x)ψ(x) + β0 > 0 on M

(3.89)

where β0 > inf ψ is a constant that we will specify in the sequel and ξ(x) ∈ C ∞ (M ), ξ : M → [0, 1] is a cut-oﬀ function such that ξ(x) ≡ 1

on BR ,

ξ(x) ≡ 0 on M \BR+1 .

(3.90)

Finally we set u(x) = ϕ(x)

m−2 2

β(x).

(3.91)

Using the transformation law (3.81) we now compute Δu. To simplify the writing we set σ−1 m−2 , b= , (3.92) a= 2 2 and we observe that − (m − 2)ϕ−1 ∇β, ∇ϕ , Δβ = ϕ2 Δβ Δϕ = bv

b−1

Δϕ + b(b − 1)v

b−2

2

|∇v| .

(3.93) (3.94)

We then have Δu = βΔϕa + ϕa Δβ + 2aϕa−1 ∇ϕ, ∇β − (m − 2)ϕa−1 ∇β, ∇ϕ = βaϕa−1 Δϕ + βa(a − 1)ϕa−2 |∇ϕ|2 + ϕa+2 Δβ + 2aϕa−1 ∇β, ∇ϕ = βabϕa−1 v b−1 Δv + βabϕa−1 (b − 1)v b−2 |∇v| + βa(a − 1)ϕa−2 |∇ϕ|2 + (2a − m + 2)ϕa−1 ∇β, ∇ϕ + ϕa+2 Δβ ≥ −βabϕa−1 v b a(x) + βabϕa−1 v b−1+σ b(x) + ab(b − 1)βϕa−2 v 2b−2 |∇v| + (2a − m + 2)ϕa−1 ∇β, ∇ϕ + ϕa+2 Δβ.

3.3. A reﬁned version of Theorem 3.2

101

Therefore, + abβϕa−1 v b−1+σ b(x) Δu + aba(x)u ≥ ϕa+2 Δβ + ab[b − 1 + b(a − 1)]βϕa−2 v 2b−2 |∇v|

(3.95) (3.96)

+ (2a − m + 2)ϕ ∇β, ∇ϕ 0 1 a+2 σ−1−2b Δβ + abβv =ϕ b(x) + ab(ab − 1)βϕa−2 v 2b−2 |∇v|2 a−1

+ (2a − m + 2)ϕa−1 ∇β, ∇ϕ . Using (3.92) we get Δu +

(m−2)(σ−1) a(x)u 4

The choice σ ≥

m+2 m−2

1* ) m+2 0 m+2 + (m−2)(σ−1) βb(x) ≥ u m−2 β − m−2 Δβ 4 0 1 (m−2)(σ−1) (m−2)(σ−1) (m−2)(σ−1) 4 + − 1 βv |∇ log v|2 . 4 4

implies (m − 2)(σ − 1) − 1 ≥ 0. 4

Hence, having set " ! m+2 b(x) = β − m−2 + (m − 2)(σ − 1) βb(x) , Δβ 4 (3.95) implies the validity of (3.86). We also observe that, because of (3.89) and the choice of ξ(x), b(x) = C 2 b(x) on M \BR+1 , for some constant C > 0. Furthermore, since inf

B R+2 \BR

b(x) > 0

we can choose β0 suﬃciently large so that + (m − 2)(σ − 1) βb(x) > 0 on B R+2 \BR . Δβ 4

(3.97)

Finally, (3.89), (3.88), (3.84) imply that (3.97) holds on BR provided (m − 2)(σ − 1) (!ψ!∞ + β0 )ε < δ. 4 This proves that b(x) > 0 on M . (3.87) follows from (3.89) and (3.91).

102

Chapter 3. General nonexistence results

Remark. If in (3.83) we have the equality sign this does not hold in general in m+2 (3.86) unless σ = m−2 . Note that we can also choose β0 ≥ 1 so that u = β0 v is m+2 large for σ = m−2 . Hence supM v = v ∗ > 1 implies supM u = u∗ > 1. This is used in Theorem 5.9. Using Proposition 3.10 and Theorem 3.2 we obtain the following reﬁned version. Theorem 3.11. Let a(x), b(x) ∈ C 0 (M ) satisfy i) b(x) ≥ 0 on M ; ii) b(x) > 0 for r(x) 1; iii) a(x) ≤ Cb(x) for r(x) 1. Let σ ≥

m+2 m−2 ,

α > 0 and assume the existence of ϕ > 0 satisfying Lσ,α ϕ ≤ −K

|∇ϕ| ϕ

2

2

on M for K > −1, with Lσ,α = Δ + (m−2)(σ−1)(1+α) a(x). Then, the diﬀerential 16α inequality (3.98) Δu + a(x)u − b(x)uσ ≥ 0 has no positive bounded solutions on M satisfying a(x)u and

& u

(m−2)(σ−1)(1+α) 4

(m−2)(σ−1)(1+α) 4

∈ L1 (M )

'−1

∈ L1 (+∞).

(3.99)

(3.100)

∂BR

Remark. Theorem 3.11 will be used in the proof of Theorem 4.8 below. We would like to end the chapter with a further general nonexistence result which will be useful for the Yamabe problem once we will have obtained the a priori estimates from above in the next chapter; see Theorem 4.10. First we recall how the deﬁnition of λ1 (Ω) is extended in case Ω is not necessarily an open set. Thus, let S be an arbitrary bounded subset of M . We set L λL 1 (S) = sup λ1 (Ω),

where the supremum is taken over all open bounded sets with smooth boundary Ω such that S ⊂ Ω. Note that, by deﬁnition, if S = ∅, then λL 1 (S) = +∞. Finally, if S is an unbounded subset of M , we deﬁne L λL 1 (S) = inf λ1 (D ∩ S),

where the inﬁmum is taken over all bounded open sets D with smooth boundary. Note that the condition that λL 1 (S) > 0 means that the set S is small in suitable spectral sense. We are now ready to prove the following

3.3. A reﬁned version of Theorem 3.2

103

Theorem 3.12. Let a(x), b(x) ∈ C 0 (M ) satisfy b(x) ≥ 0

on M

and λL 1 (supp a+ ) > 0

(3.101)

with L = Δ + a(x). Then the diﬀerential equation Δu + a(x)u − b(x)uσ = 0,

σ > 1,

has no nontrivial nonnegative ground states. Remark. Recall that a ground state for the above equation is an entire solution u satisfying lim u(x) = 0. x→+∞

Proof. Let u be a nonnegative ground state. If u is not identically null it attains its absolute positive maximum at a point xo ∈ M . We claim that a(xo ) ≥ 0. Indeed, otherwise a(xo ) < 0 and 0 ≥ Δu(xo ) = −a(xo )u(xo ) + b(xo )u(xo )σ ≥ −a(xo )u(xo ) > 0, contradiction. Therefore, xo ∈ supp a+ . Using (3.101) we can ﬁnd a suﬃciently small bounded domain Ω xo such that λL 1 (Ω) > 0. Let ϕ be the positive corresponding eigenfunction, that is, Δϕ + a(x)ϕ + λL ϕ > 0 on Ω, 1 (Ω)ϕ = 0, ϕ=0 on ∂Ω. Thus, in particular, Δϕ + a(x)ϕ = −λL 1 (Ω)ϕ ≤ 0.

(3.102)

Next, we ﬁx 0 < γ < u(xo ) suﬃciently close to u(xo ) such that the connected component containing xo , Ωo , of {x ∈ M : u(x) > γ} is inside Ω. Note that u has an absolute maximum in Ωo and Δu + a(x)u = b(x)uσ ≥ 0

on Ωo .

(3.103)

Since ϕ satisﬁes (3.102) and ϕ > 0 on Ωo , by the generalized maximum principle (see [PW67], Section 5) u = Cϕ for some positive constant C. From (3.102) and (3.103) we have σ on Ωo . −λL 1 (Ω)u = b(x)u Thus at xo , contradiction.

0 ≤ b(xo )u(xo )σ−1 = −λL 1 (Ω) < 0,

Remark. If a+ (x) ≡ 0, supp a+ = ∅ and λL 1 (∅) = +∞. Thus in case a(x) ≤ 0 we have the validity of the conclusions of the theorem.

Chapter 4

A priori estimates In this chapter we determine a priori estimates on the behavior at inﬁnity of positive solutions of the equation Δu + a(x)u − b(x)uσ = 0,

σ>1

(4.1)

on M under assumptions on a(x) and b(x) related to the geometrical requirement Ric ≥ −(m − 1)H 2 (1 + r(x)2 ) 2 δ

on M

(4.2)

for some H > 0 and δ ∈ R. The estimates we shall provide are quite sharp and naturally divide into two types: from below and from above. In both cases the method we use in establishing their validity resembles the old idea in Ahlfors’ proof of the Schwarz lemma ([Ahl38]). Some further estimates, which cannot be obtained with the previous method, are provided by direct comparison with the aid of the maximum principle (see section 4.3). The chapter ends with some nonexistence result for the Yamabe problem, which complement those described in Chapter 3.

4.1

Estimates from below

We begin with an estimate from below for positive supersolutions (see [RZ07], Theorem 2.1). Theorem 4.1. Let (M, , ) be a complete manifold with Ricci tensor satisfying (4.2); let a(x), b(x) ∈ C 0 (M ) with b(x) > 0 on M \BR (o) for some R > 0 and lim inf r(x)→+∞

a(x) >0 r(x)α

(4.3)

with α > max{−2, δ}. Let ψ(t) be a positive, nondecreasing function deﬁned in a neighbourhood of inﬁnity such that, for some ε ∈ (0, 1),

P. Mastrolia et al., Yamabe-type Equations on Complete, Noncompact Manifolds, Progress in Mathematics 302, DOI 10.1007/978-3-0348-0376-2_4, © Springer Basel 2012

105

106

Chapter 4. A priori estimates ψ(t) = O ψ

t 1+ε

and assume that lim inf r(x)→+∞

as t → +∞

(4.4)

a(x) ψ(r(x)) > 0. b(x)

(4.5)

Then, any positive solution u ∈ C 2 (M ) of Δu + a(x)u − b(x)uσ ≤ 0,

σ > 1,

satisﬁes

on M

(4.6)

1

u(x) ≥ Cψ(r(x))− σ−1

(4.7)

for r(x) 1 and some constant C > 0. Proof. We ﬁx q ∈ M , with r(q) 1 and we set ρ(x) = dist(x, q). Fix T > 0 and consider on BT (q) the function F (x) =

u(x) [T 2

− ρ2 (x)]

ξ

for some ξ > 1. Note that, as x → ∂BT (q), F (x) → +∞, while F (x) > 0 on ¯ ∈ BT (q). Using a trick BT (q). Thus F attains a positive absolute minimum at x of Calabi, [Cal57], which enables us to suppose that ρ is smooth near x ¯, we deduce i) ∇ log F (¯ x) = 0,

ii) Δ log F (¯ x) ≥ 0.

(4.8)

Using (4.8) i) a computation yields ∇u ρ∇ρ (¯ x) = −2ξ 2 (¯ x), u T − ρ2

(4.9)

while from (4.8) ii) and (4.9) we have at x ¯, ρ2 Δu Δρ2 − 4ξ(ξ − 1) 2 + ξ ≥ 0. u [T − ρ2 ]2 T 2 − ρ2

(4.10)

We want to estimate Δρ2 ; since M is complete, we have Ric ≥ −(m − 1)Z 2

on BT (q)

(4.11)

for some constant Z > 0. Therefore, the Laplacian Comparison Theorem and a computation using Gauss’ lemma imply Δρ2 ≤ 2[m + (m − 1)Zρ]

on BT (q).

(4.12)

Inserting (4.6) and (4.12) into (4.10) we have at x ¯, 1 − σ−1

u≥b

&

ρ2 m + (m − 1)Zρ a + 4ξ(ξ − 1) 2 − 2ξ 2 2 [T − ρ ] T 2 − ρ2

1 ' σ−1

.

4.1. Estimates from below

107

Since x ¯ is the minimum of F on BT (q) we then deduce [T 2 − ρ2 (¯ x)]ξ u(y) ≥ [T 2 − ρ2 (y)]ξ u(¯ x) 1

x)− σ−1 ≥ [T 2 − ρ2 (y)]ξ b(¯ ' 1 & x) ρ2 (¯ m + (m − 1)Zρ(¯ x) σ−1 × a(¯ x) + 4ξ(ξ − 1) 2 − 2ξ [T − ρ2 (¯ x)]2 T 2 − ρ2 (¯ x) for each y ∈ BT (q). In particular, for y = q, !

a(¯ x) u(q) ≥ b(¯ x)

1 & " σ−1

ρ2 (¯ x) 4ξ(ξ − 1) x) 2ξ m + (m − 1)Zρ(¯ 1+ − a(¯ x) [T 2 − ρ2 (¯ x)]2 a(¯ x) T 2 − ρ2 (¯ x)

1 ' σ−1

. (4.13)

We set f (t) =

2ξ (m − 1)Zt 1 2ξ(ξ − 1) t2 − + 2 2 2 2 a(¯ x) [T − t ] a(¯ x ) T 2 − t2

and g(t) =

1 2ξ(ξ − 1) t2 m 2ξ + − 2 2 2 2 2 a(¯ x) [T − t ] a(¯ x ) T − t2

on [0, T ) and, considering the parabola (note ξ > 1) w=

2ξ(ξ − 1) 2 2ξ(m − 1)Z 1 y − y+ , a(¯ x) a(¯ x) 2

y ∈ [0, +∞),

we deduce that f (t) attains on [0, T ) its minimum value 1 1 ξ (m − 1)2 Z 2 f¯ = − . 2 2ξ−1 a(¯ x) As for g(t), we have g(0) =

1 2

−

2ξ m a(¯ x) T 2

and limt→T − g(t) = +∞; furthermore

4ξ t {(ξ − 1 − m)t2 + (ξ − 1 + m)T 2 } 2 a(¯ x) [T − t2 ]3 8 t3 ≥ ξ(ξ − 1) ≥ 0 a(¯ x) [T 2 − t2 ]3

g (t) =

on [0, T ) because of our choice of ξ. It follows that g(t) attains on [0, T ) its minimum value 1 2ξ m . g¯ = g(0) = − 2 a(¯ x) T 2 Going back to (4.13) we obtain !

a(¯ x) u(q) ≥ b(¯ x)

1 & " σ−1

ξ (m − 1)2 Z 2 2ξ m 1− − 2(ξ − 1) a(¯ x) a(¯ x) T 2

1 ' σ−1

.

(4.14)

108

Chapter 4. A priori estimates

We now choose T = εr(q) and we observe that, since ∀ x ∈ BT (q), r(q) − T ≤ r(x) ≤ r(q) + T , with our choice of T , using (4.3) we have on BT (q), a(x) ≥ C(ε)r(q)α , (4.15) δ Ric ≥ −(m − 1)H 2 1 + W (ε)2 r(q)2 2 for some constants C(ε), W (ε) > 0, depending only on ε > 0 and on the sign of α and δ respectively. Therefore, (m − 1)2 Z 2 2ξ m ξ − (4.16) 2(ξ − 1) a(¯ x) a(¯ x) T 2 ! " δ2 2mξ 1 1 ξ(m − 1)2 H 2 2 r(q)δ−α − r(q)−2−α . + W (ε) ≥1− 2(ξ − 1)C(ε) r2 (q) C(ε) ε2

Λ(¯ x)σ−1 = 1 −

Thus, using the assumption α > max{−2, δ} we can suppose to have chosen R > 0 suﬃciently large such that, ∀ q with r(q) > R, we have Λ(¯ x)σ−1 ≥ 12 . Hence u(q) ≥ and in turn u(q)ψ(r(q))

1 σ−1

1 5 . σ−1 a(x) 1 ≥ . ψ(r(q)) min 2 BT (q) b(x)

5

We claim that lim inf r(q)→+∞

! " 1 x) σ−1 1 a(¯ 2 b(¯ x)

. a(x) >0 ψ(r(q)) min BT (q) b(x)

(4.17)

(4.18)

so that (4.7) follows at once from (4.17). To prove the claim suppose that (4.18) is false, then there exists a sequence {yn } in M , with r(yn ) → +∞, such that . 5 a(x) lim = 0, (4.19) ψ(r(yn )) min r(yn )→+∞ BTn (yn ) b(x) where Tn = εr(yn ). Let {zn } ∈ BTn (yn ) realize the minimum, that is a(zn ) a(x) = . b(x) b(zn ) BTn (yn ) min

Fix η > 0 and choose n suﬃciently large so that, from (4.19), we have a(zn ) η < . b(zn ) ψ(r(yn ))

(4.20)

4.1. Estimates from below

109

Since zn ∈ BTn (yn ), r(zn ) 1+ε and therefore, since ψ is nondecreasing and (4.4) holds we have r(yn ) ≥

a(zn ) Cη η ≤ < r(zn ) b(zn ) ψ(r(z n )) ψ( 1+ε )

(4.21)

for some constant C > 0 independent of n. In other words a(zn ) ψ(r(zn )) < Cη b(zn ) for n 1. Since η > 0 was chosen arbitrarily this contradicts (4.5) and completes the proof of the theorem. Remark. If we assume ψ(t) nonincreasing, the conclusion of Theorem 4.1 holds substituting the requirement (4.4) with t as t → +∞, (4.22) ψ(t) = O ψ 1−ε for some ε ∈ (0, 1). Thus Theorem 4.1 is valid in particular with ψ(t) = Ctβ (log t)γ for t 1, β, γ ∈ R, C > 0 and ε ∈ (0, 1). Remark. Clearly, (4.4) or (4.22) are not satisﬁed if ψ(t) is of exponential type, for instance ψ(t) = Ceβt , C, β > 0. In this case condition (4.4) has to be substituted with ψ(t) = O(ψ(t − T0 )) as t → +∞ (4.23) for some T0 > 0. The result continues to hold with the requirement α > max{0, δ}

(4.24)

but the proof has to be modiﬁed as follows. Following the argument of Theorem 4.1 we arrive at (4.14). Now we choose T = T0 , then (4.15) holds with C(ε), W (ε) substituted with C(T0 ), W (T0 ), positive constants depending only on T0 and the sign of α and δ respectively. We then proceed, under the modiﬁed assumption (4.3) with α > max{0, δ}, directly to 1 5 . σ−1 1 a(x) 1 u(q)ψ(r(q)) σ−1 ≥ ψ(r(q)) min . (4.25) 2 BT0 (q) b(x) The remainder of the proof is the same as in Theorem 4.1 using (4.23) instead of (4.4). Clearly if ψ(t) is nonincreasing, then (4.23) has to be substituted with ψ(t) = O(ψ(t + T0 )) for some T0 > 0.

as t → +∞

(4.26)

110

Chapter 4. A priori estimates

Remark. In case α = δ the proof of Theorem 4.1 can be modiﬁed to obtain the same conclusion, provided a further condition on H is satisﬁed. Indeed, if case (4.4) or (4.22) holds, assume that the Ricci tensor of M satisﬁes (4.2) with δ > −2, and having set A = lim inf r(x)→+∞

a(x) > 0, r(x)α

suppose that H satisﬁes A 1 0 < H2 < 2 (m − 1)2

1−ε 1+ε

|δ| .

(4.27)

Under these assumptions, following the proof of Theorem 4.1, we arrive at (4.16) which takes the form ! " δ2 1 (m − 1)2 H 2 ξ 2mξ 1 2 + W (ε) − r(q)−2−δ Λ(¯ x)σ−1 ≥ 1 − 2(ξ − 1) C(ε) r2 (q) C(ε) ε2 where C(ε) and W (ε) are given by C(ε) = A(1 − (sgn(δ))ε)δ W (ε) = (1 + (sgn(δ))ε)

(4.28)

and sgn(δ) is the signum of δ. Using (4.27) and δ > −2 we can choose ξ > 1 and R > 0 both suﬃciently large that, for some η ∈ (0, 1), r(q) > R gives ! " δ2 1 ξ (m − 1)2 H 2 2 <η + W (ε) (ξ − 1) C(ε) r2 (q) and 1−

2mξ 1 r(q)−2−δ ≥ η. C(ε) ε2

Thus, Λ(¯ x)σ−1 ≥

η . 2

The remainder of the proof is as above. If ψ(t) satisﬁes (4.23) or (4.26), since δ > 0 because of (4.24) and α = δ, we require T0 δ (1 − R ) A 0 0 < H2 < T 2 (m − 1) (1 + R0 )δ 0

for R0 > T0 and the theorem is still valid. Remark. We underline that, unfortunately, in the geometric case of Yamabe’s equation, for m ≥ 3 say, under assumption (4.2) one has a(x) = −c−1 m S(x) ≤ hence (4.27) cannot be satisﬁed.

δ/2 m(m − 2) 2 , H 1 + r(x)2 4

4.2. Estimates from above

111

Remark. It is worth observing that the proof of Theorem 4.1 gives no uniform positive lower bounds on compact domains for the positive solutions of (4.1). As we shall see this contrasts with the case of the estimates from above.

4.2

Estimates from above

We now derive an estimate from above for nonnegative subsolutions in case δ ≥ −2. We recall that a subsolution (resp. a supersolution) of a Yamabe-type equation is a solution of the diﬀerential inequality Δu + a(x)u − b(x)uσ ≥ 0

(resp. ≤ 0).

First we need the following (see [RRV97], [PRS05b], Lemma 2.6) Lemma 4.2. Let a(x), b(x) ∈ C 0 (M ), σ > 1, 0 < T < T and Ω ⊂⊂ BT (q) ⊂ M . Assume that b(x) > 0 on BT (q). Then, there exists a constant C > 0 such that any nonnegative C 2 -solutions u on BT (q) of Δu + a(x)u − b(x)uσ ≥ 0

(4.29)

sup u ≤ C.

(4.30)

satisﬁes Ω

Proof. We let ρ(x) = dist(x, q) and, on the compact ball BT (q) we consider the continuous function 2 F (x) = T 2 − ρ(x)2 σ−1 u(x) (4.31) where u is any nonnegative C 2 -solution of (4.29). Note that F (x)|∂BT (q) ≡ 0. If u ≡ 0 on BT (q), there is nothing to prove; otherwise F has a positive absolute x) > 0. Again, using Calabi’s maximum at some point x ¯ ∈ BT (q). In particular, u(¯ trick, [Cal57], we can assume that ρ is smooth near x ¯. Then i) ∇ log F (¯ x) = 0;

ii) Δ log F (¯ x ≤ 0.

(4.32)

From (4.32) i) we obtain ∇u 4 ρ∇ρ = u σ − 1 T 2 − ρ2

at x ¯;

while from (4.32) ii) and (4.33) we deduce 2 Δu 2 8 ρ2 Δρ2 − 0≥ − +1 2 2 u σ−1 σ−1 σ − 1 T − ρ2 (T 2 − ρ2 ) Again,

Ric ≥ −(m − 1)A2

on BT (q)

(4.33)

at x ¯.

(4.34)

(4.35)

112

Chapter 4. A priori estimates

for some constant A ≥ 0, so that, by the Laplacian comparison theorem, Δρ2 ≤ 2[m + (m − 1)Aρ]

on BT (q).

Inserting (4.29) and (4.36) into (4.34) we obtain that, at x ¯, & ' 1 1 8(σ − 1) 4 m + (m − 1)Aρ σ−1 ρ2 + , u ≤ b− σ−1 a+ + (σ − 1)2 (T 2 − ρ2 )2 σ−1 T 2 − ρ2

(4.36)

(4.37)

where a+ is the positive part of a. Next, we recall the elementary inequality (v + w)δ ≤ C 2 (v δ + wδ ),

v, w ≥ 0

(4.38)

valid for any ﬁxed δ > 0 provided C = C(δ) > 0 is suﬃciently large. We use (4.38) 1 in (4.37) with δ = σ−1 to get, at x ¯, u≤C

2

a+ max BT (q) b

1 / σ−1

+C

2

1 /− σ−1 &

min b BT (q)

ρ2 m + (m − 1)Aρ + 2 (T − ρ2 )2 T 2 − ρ2

1 ' σ−1

whence 5 F (¯ x) ≤ C

2

1 1 . / σ−1 /− σ−1 2 1 a+ 4 2 T max min b +C T + AT 3 σ−1 . BT (q) b BT (q)

x) and therefore On the other hand, ∀y ∈ BT(q) , we have F (y) ≤ F (¯ 1 / σ−1 1 / σ−1 T4 a+ u(y) ≤ C max (4.39) 2 [T 2 − ρ2 (y)] BT (q) b 1 /− σ−1 1 / σ−1 T2 AT 3 2 +C min b . 2 + 2 [T 2 − ρ2 (y)] [T 2 − ρ2 (y)] BT (q)

2

Therefore, using the fact that ρ(y) ≤ T < T , from (4.39) we immediately obtain (4.30). We are now ready to prove the desired estimate (see [RRV97] and [PRS05b], Proposition 2.7). Proposition 4.3. Let a(x), b(x) ∈ C 0 (M ) and let σ > 1. Suppose that the Ricci tensor satisﬁes δ Ric ≥ −(m − 1)B 2 1 + r(x)2 2 (4.40) for some B > 0, δ ≥ −2. Let ω, ψ be nondecreasing positive functions deﬁned in a neighbourhood of +∞ with the following properties: there exists ε ∈ (0, 1) for which t t = O(ψ(t)); ii) ω = O(ω(t)) as t → +∞. (4.41) i) ψ 1−ε 1−ε

4.2. Estimates from above

113

Assume that, for some R0 > 0 and ∀ x : r(x) ≥ R0 , we have a+ (x) 1 ≤ ; b(x) ω(r(x))

i)

ii) b(x) ≥ ψ(r(x))r(x) 2 −1 . δ

(4.42)

If u is a nonnegative C 2 -solution on M of Δu + a(x)u − b(x)uσ ≥ 0,

1 1 u(x) = O ω(r(x))− σ−1 + ψ(r(x))− σ−1

then

(4.43) as r(x) → +∞.

(4.44)

Proof. Let q ∈ M \B2R0 (o) and set ρ(x) = dist(x, q). We ﬁx a radius T > 0 so small as to insure that BT (q) ⊂ M \B2R0 (o). Because of (4.40) we have 0 1δ 2 2 Ric ≥ −(m − 1)B 2 1 + (r(q) + sgn(δ)T )

on BT (q).

(4.45)

In (4.39) of Lemma 4.2 we choose y = q so that ρ(q) = 0 and 0 1δ 2 4 A = B 1 + (r(q) + sgn(δ)T ) .

(4.46)

In this way, we get 1 / σ−1 a+ max BT (q) b 1 /− σ−1 &

2

u(q) ≤ C

+C

2

min b BT (q)

(4.47) 1δ 1 B0 2 4 1 + (r(q) + sgn(δ)T ) + T2 T

1 ' σ−1

.

Set T = ε[r(q) − R0 ] in (4.47), and note that BT (q) ⊂ M \BR0 (o). Since δ ≥ −2 and r(q) ≥ 2R0 , it is easy to see that there exists a constant C > 0 independent of q such that &

1δ 1 B0 2 4 1 + (r(q) + sgn(δ)T ) + 2 T T

1 ' σ−1

1

≤ C 2 r(q)( 2 −1) σ−1 . δ

(4.48)

Now, recalling that b(x) satisﬁes (4.42) ii) and that ψ is nondecreasing, we obtain min b ≥ C 2 r(q) 2 −1 ψ(r(q) − T ) ≥ C 2 ψ((1 − ε)r(q)) r(q) 2 −1 δ

BT (q)

for some constant C = C(R0 ) > 0. On the other hand, (4.42) gives a+ 1 1 ≤ ≤ . b ω((1 − ε)r(q) + εR ) ω((1 − ε)r(q)) 0 BT (q) max

δ

114

Chapter 4. A priori estimates

Inserting these inequalities into (4.47) and using (4.41) we obtain ) * 1 1 u(q) ≤ C 2 ω((1 − ε)r(q))− σ−1 + ψ((1 − ε)r(q))− σ−1 ) * 1 1 ≤ C 2 ω(r(q))− σ−1 + ψ(r(q))− σ−1

as required. As a special case we have the following result (see [RZ07], Theorem 2.3) Theorem 4.4. Let (M, , ) be a complete manifold with Ricci tensor satisfying Ric ≥ −(m − 1)H 2 (1 + r2 ) 2 δ

for some constant H > 0 and δ ≥ −2. Let a(x), b(x) ∈ C 0 (M ) and satisfying a(x) ≤ Ar(x)α , b(x) ≥ Br(x)β ,

δ − 1, 2 δ β ≤1− +α 2

α≥

(4.49) (4.50)

for r(x) 1 and some constants A, B > 0. Then any nonnegative solution u ∈ C 2 (M ) of (4.51) Δu + a(x)u − b(x)uσ ≥ 0, σ > 1 on M satisﬁes β−α

u(x) ≤ Cr(x)− σ−1

(4.52)

for r(x) 1 and some constant C > 0. As in the case of Theorem 4.1, there are a number of versions of Proposition 4.3. We describe here (without proof) one of them that can be deduced from Proposition 4.1 of [RRV97]. Theorem 4.5. Let (M, , ) be a complete manifold with Ricci tensor satisfying Ric ≥ −(m − 1)H 2 (1 + r2 ) 2 δ

for some constants H > 0 and δ ≥ −2. Let a(x), b(x) ∈ C 0 (M ) and let ψ(t) be a nondecreasing positive function in a neighbourhood of +∞ such that there exists T0 > 0 for which ψ(t + T0 ) = O(ψ(t)) as t → +∞. (4.53) Assume, depending on the sign of δ, that either ⎧ b(x) ⎪ ⎪ i) for − 2 ≤ δ ≤ 0, lim sup a+ (x) < +∞ and lim inf =A>0 ⎪ ⎪ ⎪ ψ(r(x)) r(x)→+∞ r(x)→+∞ ⎨ or ⎪ ⎪ a+ (x) b(x) ⎪ ⎪ ⎪ lim inf = A > 0. ⎩ii) for δ > 0, lim sup r(x)δ < +∞ and r(x)→+∞ r(x)δ ψ(r(x)) r(x)→+∞

(4.54)

4.3. Sharpness of the previous results

115

If u is a nonnegative C 2 -solution on M of Δu + a(x)u − b(x)uσ ≥ 0, then

1 u(x) = O ψ(r(x))− σ−1

σ > 1,

as r(x) → +∞.

(4.55)

Furthermore, in case A = +∞ in (4.54), the conclusion (4.55) improves to

1 as r(x) → +∞. (4.56) u(x) = o ψ(r(x))− σ−1

4.3

Sharpness of the previous results

We now show by way of examples that the estimates we obtained are sharp. We begin with Theorem 4.1. Let g(r) ∈ C ∞ ([0, +∞]), g(r) > 0 for r > 0 be such that r on [0, 12 ], r g(r) = 2 δ 1 4 e m−1 0 (1+s ) ds on [1, +∞) for some δ ≥ −2. We deﬁne the model manifold Mg = Rm in the sense of Greene and Wu, [GW79], with metric on Mg \ {0} = (0, +∞) × Sm−1 given by , = dr2 + g(r)2 dθ2 where dθ2 is the canonical metric on the unit sphere Sm−1 . Note that, since g(r) = r on [0, 12 ], , can be smoothly extended to all of Mg . The Ricci curvature in the radial direction is given by Ric(∇r, ∇r) = −(m − 1)

g (r) g(r)

while, in the direction orthogonal to ∇r determined by the unit vector ν, Ric(ν, ν) = −

(m − 2) g (r) . (1 − g (r)2 ) − 2 g(r) g(r)

Thus, a simple computation shows that there exists an appropriate H > 0 such that δ Ric ≥ −(m − 1)H 2 (1 + r2 ) 2 on Mg . Next, we consider the function α−β

w(r) = (μ + P (r)) σ−1 > 0

on [0, +∞),

where μ > 0, β ≥ α > δ and P ∈ C 2 ([0, +∞)) satisﬁes P (r) = r, for r 1, P (r) ≥ 0 on [0, +∞), P (0) = 0.

116

Chapter 4. A priori estimates

We deﬁne Hw (r) = w

−σ

&

g (r) w + A(1 + r)α w w + (m − 1) g(r)

'

for some A > 0 constant. Computing we have, &! (μ + P (r))β−α P (r) (1 + r)−β Hw (r) = β−α (1 + r) (μ + P (r))(1 + r)α α−β−σ+1 P (r) + σ−1 (μ + P (r))2 (1 + r)α " ' P (r) g (r) + A . +(m − 1) g(r) (μ + P (r))(1 + r)α Next, we note that (m − 1)

δ g (r) = (1 + r2 ) 4 g(r)

on [1, +∞), and therefore δ P (r) g (r) P (r) = (1 + r2 ) 4 (1 + r)−α on [1, +∞). α g(r) (μ + P (r))(1 + r) (μ + P (r)) (4.57) Furthermore, the left-hand side of (4.57) is bounded near zero since P (0) = 0. It follows that, up to choosing μ 1, since P (r) = r for r 1, we can choose B > 0 such that Hw (r) ≤ B(1 + r)β

(m − 1)

on [0, +∞). Recalling the deﬁnition of Hw (r), we then have w + (m − 1)

g (r) w + A(1 + r)α w − B(1 + r)β wσ ≤ 0. g(r)

(4.58)

Setting u(x) = w(r(x)), a(x) = A(1 + r(x))α and b(x) = B(1 + r(x))β > 0 on Mg we then deduce Δu + a(x)u − b(x)uσ ≤ 0. Moreover, a(x) A = (1 + r(x))α−β b(x) B on Mg and we can therefore choose ψ(t) = tβ−α . Since β ≥ α, ψ is nondecreasing, satisﬁes (4.4), (4.5) and according to Theorem 4.1, α−β

u(x) ≥ Cr(x) σ−1

(4.59) α−β

for r(x) 1 and some constant C > 0. Since u(x) = (μ + r(x)) σ−1 for r(x) 1, estimate (4.7) cannot be improved.

4.4. Some further estimates

117

In the above example, let, as before, δ ≥ −2. Choose α > −2 and A > 0. Then, for every β ∈ R we can ﬁnd μ 1 so that (1 + r)−β Hw (r) ≥ B

∀ r > 0,

and for some B > 0. Therefore, u(x) = w(r(x)) satisﬁes the inequality Δu + a(x)u − b(x)uσ ≥ 0 on Mg .

(4.60)

Again, we observe that α−β

u(x) = (μ + r(x)) σ−1

on Mg \ B1 .

Thus, ﬁxing δ > −2 and choosing α ≥ δ/2 − 1 and β ≤ 1 − δ/2 + α, applying Theorem 4.4 to the diﬀerential inequality (4.60) we have α−β

u(x) ≤ Cr(x) σ−1

for some constant C > 0 and r(x) >> 1, showing that the upper bound provided by Theorem 4.4 cannot be improved.

4.4

Some further estimates

We now complement the analysis of the previous section by considering the situation where a(x) > −∞. (4.61) lim inf r(x)→+∞ r(x)α The technique used in the proof of Theorem 4.1 cannot be applied but it turns out that in this case an estimate similar to (4.7), depending directly on Ricci, can be obtained in an elementary way with the aid of the maximum principle. In fact we have (see [RZ07], Theorem 2.4) Theorem 4.6. Let (M, , ) be a complete manifold with radial Ricci curvature satisfying δ/2 Ric (∇r, ∇r) ≥ −(m − 1)H 2 1 + r(x)2 on M for some H > 0, δ > −2 and let a(x), b(x) ∈ C 0 (M ) and satisfy (4.61) and b(x)

lim sup r(x)→+∞

for some α < δ, β ≤ δ, γ >

r(x)β e

2H 2+δ (m

δ+2 (σ−1)γ 1+r(x) 2

< +∞

(4.62)

− 1), σ > 1. Then, any positive solution of

Δu + a(x)u − b(x)uσ ≤ 0

on M

(4.63)

satisﬁes lim inf

u(x)

δ+2 r(x)→+∞ e−γr(x) 2

> 0.

(4.64)

118

Chapter 4. A priori estimates

Proof. First of all, using Proposition 1.15, the Laplacian comparison theorem and the lower bound on Ricci, we deduce as r → +∞.

Δr ≤ (m − 1)Hrδ/2 (1 + o(1))

(4.65)

Next, we choose R > 0 suﬃciently large such that, using (4.61) and (4.62), a(x) ≥ −Ar(x) , b(x) ≤ Br(x) e α

β

δ+2 (σ−1)γ 1+r(x) 2

(4.66)

on M \BR for some A, B > 0. We let 0 < ξ < minBR u(x). To simplify the writing, set θ = 2δ + 1 and deﬁne θ v(x) = ξe−γ (1+r(x) ) − u(x).

Note that, since γ > 0, by our choice of ξ we have v(x) ≤ ξ − u(x) < 0

on BR .

(4.67)

We now reason by contradiction and suppose that lim inf r(x)→+∞

u(x) = 0. e−γ(1+r(x)θ )

This means that there exists a sequence {xn } ⊂ M , r(xn ) → +∞, such that u(xn ) θ −γ(1+r(x n) ) e

→0

as n → +∞.

(4.68)

Since γ, ξ > 0 and u(x) > 0, v ∗ = supM v(x) < +∞. Furthermore, because of (4.68), ' & u(xn ) −γ (1+r(xn )θ ) (4.69) v(xn ) = e ξ − −γ(1+r(x )θ ) > 0 n e for n suﬃciently large. Thus v ∗ > 0. Finally, γ, θ > 0 force that v ∗ has to be attained at some point x ¯ ∈ M . Let Ω be the level set Ω = {x ∈ M : v(x) > 0}; because of (4.67), Ω ⊂ M \ BR and θ u(x) < ξe−γ (1+r(x) )

on Ω.

(4.70)

On Ω we compute 0 1 θ θ Δv = ξ −γθe−γ (1+r(x) ) rθ−1 Δr + γθ γθr2θ−2 + (1 − θ)rθ−2 e−γ (1+r(x) ) − Δu so that, using (4.70), (4.65), (4.66) and (4.63),

4.4. Some further estimates

119

) 0 1 θ δ δ δ Δv ≥ ξr 2 −1+θ e−γ (1+r(x) ) γθ γθrθ1 − 2 + (1 − θ)r−1− 2 − (m − 1)H(1 + o(1)) * δ δ − Arα−θ+1− 2 − ξ σ−1 Brβ−θ+1− 2 & ! " δ+2 −γ 1+r(x) 2 δ −1− δ δ+2 δ+2 δ 2 γ γ− r = ξr e − (m − 1)H(1 + o(1)) 2 2 2 − A rα−δ − ξ σ−1 Brβ−δ . Note that, because of our assumptions on α, β, γ, δ we can choose R suﬃciently large and ξ > 0 suﬃciently small that Δv > 0

on Ω

contradicting the fact that x ¯ ∈ Ω.

The case δ = −2 is, loosely speaking, border line between Euclidean and hyperbolic geometry. As seen in Proposition 1.15 the bound −1 Ric (∇r, ∇r) ≥ −(m − 1)H 2 1 + r2

on M,

H>0

(4.71)

as r → +∞.

(4.72)

implies the estimate Δr ≤ (m − 1)r

−1 1

+

√

1 + 4H 2 (1 + o(1)) 2

Proceeding in a way similar to that of the argument of Theorem 4.6 we can prove the following (see [RZ07], Theorem 2.5) Theorem 4.7. Let (M, , ) be a complete manifold with radial Ricci curvature satisfying (4.71). Let a(x), b(x) ∈ C 0 (M ) satisfy (4.61) and b(x) < +∞ β r(x)→+∞ r(x)

(4.73)

lim sup

with α < −2, β ≤ γ(σ − 1) − 2 and γ > (m − 1)

1+

√

1 + 4H 2 − 1. 2

(4.74)

Then any positive solution u of Δu + a(x)u − b(x)uσ ≤ 0

on M,

σ>1

satisﬁes u(x) > 0. r(x)→+∞ r(x)−γ lim inf

(4.75)

120

Chapter 4. A priori estimates

Both estimates of Theorems 4.6 and 4.7 are quite sharp with respect to the range of the exponent γ. To simplify computations let us consider (4.75). Towards this aim, for a chosen R0 > 1, let Mg be the model with g(r) ∈ C ∞ (M ) positive on [0, +∞) and such that ⎧ ⎨r on 0, 12 , √ (4.76) g(r) = 1 ⎩r 2 1+ 1+4H 2 on [R , +∞) 0

> 0. Since Ric(∇r, ∇r) = −(m − 1) g (r) we have for some H g(r) 2 r−2 = −(m − 1)H 2 (1 + r−2 )(1 + r2 )−1 Ric(∇r, ∇r) = −(m − 1)H 2 (1 + R−2 ) so that on Mg \ BR0 . We let H 2 = H 0

−1 Ric(∇r, ∇r) ≥ −(m − 1)H 2 1 + r2

(4.77)

on Mg \ BR0 . It is not hard to show that g(r) can be deﬁned on way that (4.77) holds on all of Mg . Now we set

−1 ( m − 1 1 + 1 + 4H 2 γ= 2 and we choose A, B > 0 suﬃciently large that

1

2 , R0

− γ −1 g (r) 1 + r2 2 [0,R0 ] g(r) γ − γ α 2 − 2 −1 + 1 + R02 2 A inf (1 + r) ≤ γ 1 + R0

γ(γ + 2)R02 − γ(m − 1) inf r

σ−1

+ βξ0

σγ 2 − 2

1 + R0

in such a

(4.78)

[0,R0 ]

inf (1 + r)

β

[0,R0 ]

for some ﬁxed ξ0 > 0 and α, β ∈ R. We set α

a(x) = −A(1 + r(x)) ,

b(x) = B(1 + r(x))

β

and we deﬁne, with ξ ≥ ξ0 , − γ v(x) = ξ 1 + r(x)2 2 .

(4.79)

Then, a simple computation shows that on Mg , ) − γ −1 − γ −2 Δv + a(x)v − b(x)v σ = ξ γ(γ + 2)r2 (x) 1 + r2 (x) 2 − γ 1 + r2 (x) 2 − γ −1 g (r(x)) r(x) 1 + r2 (x) 2 − g(r(x)) − γ − A(1 + r(x)α ) 1 + r2 (x) 2 − − σγ * β − Bξ σ−1 (1 + r(x)) 1 + r2 (x) 2 . − γ(m − 1)

(4.80)

4.5. Nonexistence results for the Yamabe problem

121

Thus, since ξ ≥ ξ0 , (4.78) and inspection of (4.80) show that Δv + a(x)v − b(x)v σ ≤ 0

(4.81)

on BR0 . Next, we note that on M \ BR0 , (m − 1)

( g (r) m − 1 1 = 1 + 1 + 4H 2 = (1 + γ − ε) g(r) 2r r

(4.82)

for some ε = ε(R0 ) with ε → 0 as R0 → +∞. Rearranging the terms in (4.80) and using (4.82) we have Δv + a(x)v − b(x)v σ & − γ2 −2 2 2 εγ − γ(γ + 2 − ε)r(x)−2 = ξr (x) 1 + r (x)

(4.83)

2 1 r2 (x) β 2− γ2 (σ−1) ' 1 1 σ−1 β+2−γ(σ−1) 1+ 1+ 2 . − Bξ r(x) r(x) r (x)

− Ar(x)α+2 1 +

1 r(x)

α

1+

If β = (σ − 1)γ − 2 we can choose ξ ≥ ξ0 suﬃciently large that the above yields the validity of (4.81) on M \ BR0 . Note that, in this example, the range of α plays no role.

4.5

Nonexistence results for the Yamabe problem

In this section we apply the nonexistence results of Chapter 3 and the a priori estimates from above of Chapter 4 to give nonexistence results in the geometrical case of the Yamabe equation; similar results can, of course, be obtained for Yamabetype equations on the complete, noncompact, manifold (M, , ). We begin with the next result, that was ﬁrst proved in [BRS98]. Theorem 4.8. Let (M, , ) be a complete Riemannian manifold with scalar curvature S(x). Assume (4.84) Ric ≥ −(m − 1)B 2 for some B > 0 and that there exists a ϕ > 0 solution of Lϕ ≤ −K

|∇ϕ| ϕ

2

(4.85)

∞ for some K > −1, with L = Δ − m−1 4m S(x), m = dim M ≥ 3. Let K(x) ∈ C (M ) satisfy K(x) ≤ 0 on M, (4.86) 2 K(x) ≤ −Cr(x) m−1 +γ e2Br(x) for r(x) 1

122

Chapter 4. A priori estimates

and some constants C, γ > 0. Then the metric , cannot be pointwise conformally deformed to a metric , of scalar curvature K(x). Remark. Condition (4.86) is sharp only in the term e2Br . Indeed, for the Yamabe problem on hyperbolic space Hm −B 2 , the second of (4.86) can be relaxed to K(x) ≤ −C

e2Br r(x) log r(x)

for r(x) 1

(see Theorem 7.3 and the related Remark 2; see also the Remark after Theorem 6.15). 4 Proof. First of all we recall that if , = u m−2 , , then u > 0 has to satisfy m+2

cm Δu − S(x)u + K(x)u m−2 = 0, cm =

4(m − 1) , m−2

so that, with our notation for Yamabe-type equations, a(x) = −

m−2 S(x), 4(m − 1)

b(x) = −

m−2 K(x). 4(m − 1)

We want to apply Theorem 3.11 with the choices σ=

m+2 , m−2

α=

m . m−2

Indeed, in this case Lσ,α = Δ +

m−1 (m − 1)2 a(x) = Δ − S(x) = L, m(m − 2) 4m

L

so that (4.85) gives λ1 σ,α (M ) ≥ 0. From (4.86) we have b(x) ≥ 0 on M and b(x) > 0 for r(x) 1. We now have to show the existence of C > 0 such that a(x) ≤ Cb(x) Now, (4.84) implies sup a(x) ≤ M

for r(x) 1. m−2 mB 2 4

(4.87)

(4.88)

so that, since b(x) > 0 for r(x) 1, (4.87) is satisﬁed for C suﬃciently large. This also shows that in order to verify (3.99) and (3.100) it is enough to show that m−1

u2 m−2 ∈ L1 (M ). Towards this aim we apply Theorem 4.5 with the choice δ = 0 and 2

ψ(t) = t m−1 +γ e2Bt

(4.89)

4.5. Nonexistence results for the Yamabe problem

123

for some γ > 0 as in (4.86). Then assumptions (4.54) i) and ii) are satisﬁed because of (4.88) and (4.86). It follows that 1 m−2

u(x) ≤ Cr(x)− 2 m−1 −

m−2 4 γ

e−

m−2 2 Br(x)

(4.90)

for some constant C > 0 and r(x) 1. By Bishop’s volume comparison theorem (see Chapter 1) and (4.90) m−1 m−1 m−1 u(x)2 m−2 ≤ Cr−1− 2 γ e−(m−1)Br e(m−1)Br = Cr−1− 2 γ ∂Br

and since γ > 0 it follows that

m−1

u(x)2 m−2 ∈ L1 (+∞),

∂Br

that is, (4.89) is satisﬁed. Suppose now that we are able to solve Yamabe’s equation m+2

cm Δu − S(x)u + K(x)u m−2 = 0

on M, u > 0

(4.91)

4 on the complete manifold (M, , ); we then ask if the new metric , = u m−2 , , m = dim M ≥ 3 is still complete. The a priori estimates of Chapter 4 give us an answer, in the negative, with the following

Theorem 4.9. Let (M, , ) be a complete Riemannian manifold of dimension m ≥ 3 and scalar curvature S(x). Suppose that Ric ≥ −(m − 1)H 2 (1 + r(x)) 2 δ

(4.92)

for some H > 0, δ ≥ −2. Let K(x) ∈ C ∞ (M ) and assume that K(x) ≤ −K 2 r(x)δ+2 [log r(x)]

2(1+γ)

for r(x) 1

(4.93)

and some constants K, γ > 0. Then any metric conformal to , and with scalar curvature K(x) is incomplete. K(x) Proof. As above we let a(x) = − S(x) cm , b(x) = − cm . We apply Proposition 4.3 with the choices

ω(t) = Ct2 (log t)2(1+γ) , ψ(t) = Ct3+ 2 (log t)2(1+γ) δ

for t 1 and some C > 0. We note that ω and ψ satisfy (4.41) i), ii) ∀ ε ∈ (0, 1). Furthermore, due to (4.92) and (4.93), we respectively have b(x) ≥

δ K2 2(1+γ) r(x)δ+2 [log r(x)] ≥ ψ(r(x))r(x) 2 −1 Cm

for r(x) 1,

124

Chapter 4. A priori estimates a+ (x) 1 ≤ b(x) ω(r(x))

for r(x) 1

provided C > 0 is chosen appropriately. An application of Proposition 4.3 yields % $ − m−2 δ +1 1+γ − m−2 m−2 ] 2 r(x) 2 ( 2 ) 1+γ − 2 +[r(x)(log r(x)) u(x) = O r(x)(log r(x)) as r(x) → +∞. Since δ ≥ −2, from this latter we deduce that − m−2 2 for r(x) 1 u(x) ≤ C r(x)(log r(x))1+γ

(4.94)

and some C > 0. To show that the metric , = u m−2 , is not complete, we reason as follows. Let {xn } be a sequence diverging in M , that is, {xn } is deﬁnitely outside any ﬁxed compact set in M . Since (M, , ) is complete, r(xn ) → +∞ as n → +∞; in particular we can suppose that {xn } ⊂ M \BR on which (4.94) holds. For n ﬁxed, by the Hopf-Rinow theorem (see e.g. [Lee97]) there exists γn , a unit speed geodesic from o to xn , realizing the distance r(xn ). Thus if t is the arclength parameter of γn , then r(γn (t)) = t. 4

(o, x) we have For r(x) = dist ,

r(xn ) ≤

r(xn ) 0 2

≤ C m−2 =C

2

u m−2 (γn (t)) dt

2 m−2

r(xn )

0 r(xn )

r(γn (t))(log r(γn (t)))1+γ t(log t)

1+γ −1

0

dt ≤ C

−1

2 m−2

dt

+∞ 0

t(log t)1+γ

−1

6 dt ≤ C

6 an absolute constant. Thus, with C 6 r(xn )) ≤ C

∀ n ∈ N,

and the metric , cannot be complete since {xn } is a divergent sequence in M. In the next result we show that even a slow decay to 0 of K(x) at inﬁnity under some additional condition implies nonexistence. This will be an application of Theorems 3.12 and 4.4. Theorem 4.10. Let (M, , ) be a complete Riemannian manifold of dimension m ≥ 3 and scalar curvature S(x). Suppose that Ric ≥ −(m − 1)H 2 (1 + r(x)2 ) 2 δ

4.5. Nonexistence results for the Yamabe problem for some H > 0 and −2 ≤ δ < 2. For L = Δ −

125

1 cm S(x)

assume that

λL 1 (supp S− ) > 0,

(4.95)

where S− is the negative part of S. Let K(x) ∈ C ∞ (M ) be such that K(x) ≤ 0 on M and (4.96) K(x) ≤ −K2 r(x)β for r(x) 1 and some K > 0, δ < β ≤ 1 + 2δ . Then the metric , cannot be pointwise conformally deformed to a metric , of scalar curvature K(x). Proof. We reason by contradiction and suppose that , exists. Then we have a solution u > 0 of the diﬀerential equation m+2

Δu + a(x)u − b(x)u m−2

on M

with, as we saw in Theorem 4.9, a(x) = −

S(x) , cm

b(x) = −

K(x) . cm

The assumption on the Ricci tensor yields a(x) ≤ Ar(x)δ

for r(x) 1

and an appropriate constant A > 0, while from (4.96) we deduce b(x) ≥ Br(x)β . Note that, due to the choice of the range of the parameters δ and β, the assumptions of Theorem 4.4 are satisﬁed and we conclude that u(x) ≤ Cr(x)− 4 (β−δ) . m

Then u(x) → 0 as x → ∞. Using (4.95) and K(x) ≤ 0 on M we apply Theorem 3.12 to obtain a contradiction to the existence of u.

Chapter 5

Uniqueness The aim of this chapter is to prove some uniqueness results for positive solutions of Yamabe-type equations. Our ﬁrst theorem in this direction depends only on the sign of the coeﬃcient b(x) of the nonlinear term and, loosely speaking, on an L2 type estimate of the distance, at inﬁnity, of the two solutions under consideration. It is worth observing that this very general result is sharp and that the L2 -type condition cannot be substituted with a corresponding Lp condition with p > 2. Our second result, Theorem 5.2, is obtained via a comparison result whose proof requires a version of the weak maximum principle (see Theorem 5.3) which is interesting in its own. The strength of the theorem lies both in the fact that it requires that the solutions approach one another at inﬁnity in the weak sense of condition (5.13) below, and in the fact that the remaining assumptions are very general. Counterexamples are given after each result to prove sharpness. We end the chapter with a geometric application to the group of conformal diﬀeomorphisms of a complete manifold and to uniqueness for the Yamabe problem.

5.1

A sharp integral condition

The ﬁrst theorem gives a condition for two positive solutions of Δu + a(x)u − b(x)uσ = 0,

σ > 1,

on M

(5.1)

to coincide. The condition is expressed in integral form and, as we shall see, is quite sharp. Theorem 5.1. Let a(x), b(x) ∈ C 0 (M ) and assume that i) b(x) ≥ 0 on M ;

ii) b(x) ≡ 0 on M.

P. Mastrolia et al., Yamabe-type Equations on Complete, Noncompact Manifolds, Progress in Mathematics 302, DOI 10.1007/978-3-0348-0376-2_5, © Springer Basel 2012

(5.2)

127

128

Chapter 5. Uniqueness

Let u, v ∈ C 2 (M ) be positive solutions of (5.1) and suppose that &

(u − v)2

'−1

∈ L1 (+∞).

(5.3)

∂Br

Then u ≡ v on M . Note that condition (5.3) is implied by u − v ∈ L2 (M ) or even (u − v)2 = o(R2 ) as R → +∞. BR

Proof. Applying the divergence theorem on a geodesic ball BR of radius R to the vector ﬁeld v

, W = (v 2 − u2 )∇ log u using (5.1) and rearranging, we obtain 2 u 2 v W, ∇r − ∇u − ∇v − ∇v − ∇u v u ∂BR BR BR b(x)(v 2 − u2 )(v σ−1 − uσ−1 ). =

(5.4)

BR

Since the right-hand side is a nonnegative and nondecreasing function of R it tends to a nonnegative limit B ≤ +∞ as R → +∞. Next by Gauss’ lemma and Schwarz’s inequality

∇v ∇u ≤ |v − u| ∇v − v ∇u + ∇u − u ∇v |W, ∇r| ≤ |v − u||v + u| − v u u v and therefore '2 & & W, ∇r ≤ 2 ∂BR

(v − u) ∂BR

2

'&

' 2 u 2 v ∇v − ∇u + ∇u − ∇v . u v ∂BR (5.5)

We claim that

2 v u 2 ∇v − ∇u + ∇u − ∇v ∈ L1 (M ). u v Indeed, assume by contradiction that 2 v u 2 G(R) = ∇v − ∇u + ∇u − ∇v → +∞ as R → +∞. u v BR

Since

W, ∇r − G(R) → B ≥ 0 ∂BR

as R → +∞,

(5.6)

5.1. A sharp integral condition ¯ suﬃciently large, ∀R ≥ R

129

W, ∇r ≥ ∂BR

1 G(R) ≥ 0. 2

Therefore, using (5.5), '2 & '2 & & ' 1 2 W, ∇r ≤ 2G (R) (v − u) . G(R) ≤ 2 ∂BR ∂BR Arguing exactly as in Step 1 of the proof of Theorem 3.2 we contradict (5.3) and conclude that (5.6) holds. Now, because of (5.3) and (5.6) there exists an increasing sequence {rk } ↑ +∞ such that / / 2 2 v u =0 (u − v)2 lim ∇v − ∇u + ∇u − ∇v k→+∞ u v ∂Brk ∂Brk and then, using (5.5) we conclude that lim W, ∇r = 0. k→+∞

∂Brk

Next, we evaluate (5.4) along the sequence {rk } and let k → +∞ to obtain " ! 2 u 2 v 2 2 σ−1 σ−1 b(x)(v − u )(v −u )+ ∇v − ∇u + ∇u − ∇v = 0. (5.7) u v M M Due to the assumptions on b(x), ∇u − uv ∇v = 0 on M , so that u = Av for some constant A > 0; substituting into (5.7) yields b(x)v σ+1 = 0 (1 − A2 ) 1 − Aσ−1 M

and since v > 0, b(x) ≥ 0, b(x) ≡ 0 this forces A = 1, that is, u ≡ v on M .

Remark. Despite its generality, assumption (5.3) of Theorem 5.1 is quite sharp. Indeed suppose (M, , ) is Euclidean space Rm with its canonical metric; let a(x) and b(x) be nonnegative continuous functions satisfying (m − 2)2 −2 |x| , 4 (m−2)(σ+1) 2 |x| b(x) ≤ (log |x|)σ+1 (log log |x|)(log log log |x|)2 a(x) ≤

with equality holding for suﬃciently large |x|. We shall prove in the next section that the equation Δu + a(x)u − b(x)uσ = 0,

σ>1

on Rm

(5.8)

130

Chapter 5. Uniqueness

has a family of positive solutions uα (α > 0) satisfying uα (0) = α

uα (x) ∼ |x|−

and

m−2 2

log |x|

as |x| → +∞.

If α1 = α2 , then uα1 and uα2 are distinct solutions for which p

|uα1 − uα2 | rm−1 ≤ Crm−1− for r = |x| 1. Thus p

|uα1 − uα2 | ≤ Crm−1−

m−2 2 p

m−2 2 p

(log r)p

(log r)p .

∂Br

Hence, for p > 2,

& |uα1 − uα2 |

p

'−1

∈ L1 (+∞),

∂Br

while, for p = 2, (5.3) just falls short of being met. Note that this same example shows that the case p = 2 is special for the validity of the theorem.

5.2

A remark on the asymptotic behaviour of solutions: examples in Rm and Hm

Let b(x) ≥ 0 and let u and v be positive solutions of the equation Δu − b(x)uσ = 0,

σ > 1 on (M, , )

(5.9)

such that lim (u(x) − v(x)) = 0.

x→∞

(5.10)

Then, a simple application of the maximum principle shows that u and v coincide. Indeed, setting w = u − v, we have 2

Δ(w2 ) = 2wΔw + 2|∇w| ≥ 2wΔw = 2(u − v)b(x)(uσ − v σ ) ≥ 0 so that w2 is a nonnegative subharmonic function on M which tends to zero at inﬁnity. Thus it attains a nonnegative maximum and it is therefore identically zero by the maximum principle (see, [CN92]). This suggests that two positive solutions u and v of Δu + a(x)u − b(x)uσ = 0,

σ > 1 on (M, , )

may coincide if they have the “same” behaviour at inﬁnity. The following example shows that, in general, it may be that a simple limit condition like (5.10) is not enough to conclude. Let B be the unit ball in Rm and, for a > 1, deﬁne βa (r) =

m−2 1 (a2 − r2 )− 2 2 m(m − 2)a

r ∈ [0, 1), m ≥ 3.

5.2. Asymptotic behaviour of solutions: examples in Rm and Hm

131

A simple computation shows that β satisﬁes m+2 m−2 β = β , r ∈ [0, 1), βa + m−1 a a r 1 βa (0) = m(m−2)a , β m a (0) = 0, so that βa (|x|) is a family of positive radial solutions of (5.9) on B with b(x) ≡ 1, m+2 . Operating the change of variable σ = m−2 r(t) =

t (1 + tm−2 )

1/(m−2)

we obtain a solution αa (t) = βa (r(t)) of m+2 m−2 , t ∈ [0, +∞) αa + m−1 t αa = b(t)αa 1 αa (0) = m(m−2)am , αa (0) = 0, with

m−1 b(t) = 1 + tm−2 −2 m−2 .

Thus αa gives rise to a family of positive radial solutions ua (x) of m+2 −2 m−1 m−2 Δua − 1 + |x|m−2 uam−2 = 0

for which lim

|x|→+∞

ua (x) =

on Rm

m−2 1 (a2 − 1)− 2 . 2 m(m − 2)a

It is clear that two solutions ua1 and ua2 of the family coincide if and only if lim

|x|→+∞

ua 1 =

lim

|x|→+∞

u a2 ,

that is, a1 = a2 . Note that the geometrical meaning of these solutions is obvious: Rm can be conformally deformed to a complete Riemannian manifold of negative scalar curvature suﬃciently close to zero. Thus, in this case, uniqueness is expressed by (5.10). However, we may consider the same family of solutions on the hyperbolic space Hm of constant negative curvature −1 by identifying it with Rm with metric, in polar coordinates, ds2 = dr2 + (sinh r)2 dθ2 . In this case the situation changes completely: indeed, let γa (t) = βa

t tanh 2

5

2 1 − tanh2

.− m−2 2 t 2

.

Then, ⎧ ⎨γ + (m − 1) coth(t) γ + a a ⎩γa (0) = 0, γa (0) = (m−2) 2

2

m(m−2) γa 4 1 m(m−2)am

m+2

= γam−2

on [0, +∞),

132

Chapter 5. Uniqueness

and − m−2 "− m−2 ! 2 2 1 2 ρ(x) 2 ρ(x) 2 a 2 cosh − tanh , wa (x) = m(m − 2)a2 2 2 with ρ(x) = distHm (x, o), is a family of positive solutions of Δwa +

m+2 m(m − 2) wa − wam−2 = 0 4

on Hm .

We observe that m−2

wa (x) ∼

m−2 m−2 2− 2 (a2 − 1)− 2 e− 2 ρ(x) m(m − 2)a2

as ρ(x) → +∞.

Thus, wa (r) → 0

as r(x) → +∞

independently of a > 1 and with the same speed. In the next section we show that uniqueness of solutions converging to zero is indeed special.

5.3

Uniqueness via the weak maximum principle

In this section we prove a number of uniqueness results for nonnegative solutions of equation (5.1). We begin with the following (see [RZ07]). Theorem 5.2. Let (M, , ) be a complete manifold, a(x), b(x) ∈ C 0 (M ), σ > 1, τ ≥ 0, β + τ (σ − 1) > −2 and suppose that b(x) > 0 on M and i) b(x) ≥ Br(x)β for r(x) 1;

ii) sup M

a− (x) r(x)τ (1−σ) < +∞. b(x)

(5.11)

Suppose that u, v ∈ C 2 (M ) are two nonnegative solutions of Δu + a(x)u − b(x)uσ = 0 satisfying

on M

(5.12)

C −1 r(x)τ ≤ u(x), v(x) ≤ Cr(x)τ

(5.13)

for r(x) 1 and some constant C > 0. If lim inf r→+∞

log vol Br 2+β+τ (σ−1) r

< +∞,

(5.14)

then u ≡ v on M . Theorem 5.2 will follow immediately from a corresponding comparison result whose proof is based on the next weak maximum principle at inﬁnity.

5.3. Uniqueness via the weak maximum principle

5.3.1

133

A useful form of the weak maximum principle

The next result, which is a version of what is known in literature as the weak maximum principle at inﬁnity (see [RSV05], [PRS05b] and [MRS10] for a more general version), turns out to be quite useful in many contexts, both geometrical and analytical. It is worth observing that the function u under consideration is not necessarily bounded but is only required to satisfy some growth control (expressed in (5.17)). The other main assumption of the theorem is a control on the volume of geodesic balls (see (5.16)). Theorem 5.3. Let (M, , ) be a complete manifold, and let σ, μ ∈ R, be such that i) σ ≥ 0; Assume that lim inf r→+∞

ii) 2 − σ − μ > 0.

(5.15)

log vol Br = d0 < +∞. r2−σ−μ

(5.16)

Let u ∈ C 2 (M ) and suppose that u(x) < +∞. σ r(x)→+∞ r(x)

u 6 = lim sup

(5.17)

Then, given γ ∈ R such that Ωγ = {x ∈ M : u(x) > γ} = ∅, we have μ

u, 0} C(σ, η) inf [1 + r(x)] Δu ≤ d0 max {6 Ωγ

with

⎧ ⎪ ⎨0 C(σ, μ) = (2 − σ − μ)2 ⎪ ⎩ σ(2 − σ − μ)

if σ = 0; if σ > 0, μ + 2(σ − 1) < 0; if σ > 0, μ + 2(σ − 1) ≥ 0.

(5.18)

(5.19)

Proof. We begin by observing that if ν is any constant and we set uν = u + ν, then Δuν = Δu and {x ∈ M : u(x) > γ} = Ωγ = {x ∈ M : uν (x) > γ + ν}. Furthermore, if σ > 0, u 6=u 6ν so, in order to estimate μ

inf [1 + r] Δu Ωγ

we may replace u by a suitable translate. Next, ﬁx b > max {6 u, 0}; then, there exists a constant ν such that uν < b on M and uν (x0 ) > 0 for some x0 ∈ M. (5.20) (1 + r)σ

134

Chapter 5. Uniqueness

We will assume that a constant ν has been selected in such a way that (5.20) holds. In accordance with the observation made at the beginning of the proof, we are going to replace u with uν , and for ease of notation we will even suppress the subscript ν. We want to show (5.18) and we may suppose, without loss of generality, that γ ≥ 0. Next, let μ

K = inf [1 + r] Δu. Ωγ

Clearly, the theorem amounts to showing that K is bounded above by the righthand side of (5.19). If K ≤ 0 there is nothing to prove; so let us assume K > 0, in which case u is nonconstant on any connected component of Ωγ , and μ

[1 + r] Δu ≥ K > 0

on Ωγ .

(5.21)

We ﬁx θ ∈ 12 , 1 and we choose R0 > 0 large enough that |∇u| ≡ 0 on the nonempty open set BR0 ∩ Ωγ . Given R > R0 , let ϕ ∈ C ∞ (M ) be a cut-oﬀ function such that 0 ≤ ϕ ≤ 1,

ϕ ≡ 1 on BθR , ϕ ≡ 0 on M \BR , |∇ϕ| ≤

c R(1 − θ)

(5.22)

for some absolute constant c > 0. Let also λ ∈ C 1 (R) and F (v, r) ∈ C 1 (R2 ) be such that 0 ≤ λ ≤ 1; λ = 0 on (−∞, γ]; λ > 0, λ ≥ 0 on (γ, +∞)

(5.23)

and F (v, r) > 0;

∂F (v, r) < 0 on [0, +∞) × [0, +∞). ∂v

(5.24)

Finally, let W be the C 1 vector ﬁeld deﬁned on Ωγ by W = ϕ2 λ(u)F (v, r)∇u

(5.25)

v = α(1 + r)σ − u

(5.26)

where v is given by and α > b is a constant. Note that, according to (5.20), v > 0 on Ωγ , and, in fact, since u > γ > 0 on Ωγ , we have (α − b)(1 + r)σ ≤ v ≤ α(1 + r)σ

on Ωγ .

(5.27)

We note that W vanishes on ∂(Ωγ ∩ BR ) and it extends to a C 1 vector ﬁeld on the whole of M by setting it equal to 0 in the complement of Ωγ ∩ BR . A computation that uses the properties of ϕ, λ and F , the deﬁnition (5.26) of v, inequality (5.21) and Cauchy-Schwarz inequality yields

5.3. Uniqueness via the weak maximum principle

135

div W = ϕ2 λ(u)F (v, r)Δu + 2ϕλ(u)F (v, r) ∇ϕ, ∇u ∂F + ϕ2 λ (u)F (v, r)|∇u|2 + ϕ2 λ(u) (v, r) ∇v, ∇u ∂v ∂F (v, r) ∇u, ∇r + ϕ2 λ(u) ∂r ≥ ϕ2 λ(u)F (v, r)(1 + r)−μ K − 2ϕλ(u)F (v, r)|∇ϕ||∇u| ∂F (v, r) ∇u, ασ(1 + r)σ−1 ∇r − ∇u + ϕ2 λ(u) ∂v ∂F 2 (v, r) ∇u, ∇r + ϕ λ(u) ∂r ≥ −2ϕλ(u)F (v, r)|∇ϕ||∇u|+ ∂F 2 (v, r) + ϕ λ(u) ∂v 5 . / ∂F F (v, r) −μ 2 σ−1 ∂r ∂F K(1 + r) + |∇u| + ∂F − ασ(1 + r) × ∇u, ∇r , ∂v

∂v

that is, ∂F (v, r)B(∇u, r) div W ≥ −2ϕλ(u)F (v, r)|∇ϕ||∇u| + ϕ2 λ(u) ∂v

(5.28)

with 5 . ∂F F (v, r) −μ σ−1 ∂r B(∇u, r) = |∇u| + ∂F K(1 + r) + ∂F − ασ(1 + r) ∇u, ∇r . 2

∂v

∂v

(5.29) For notational convenience we set η = μ + 2(σ − 1) so that by (5.15), σ − η > 0, and we divide the proof into three cases. Case I. σ > 0, η < 0. We let F (v, r) = e−qv(1+r)

−η

where q > 0 is a constant that will be speciﬁed later. An elementary computation which uses the estimate for v given in (5.27) shows that ∂F ∂r (v, r) 0 ≥ ∂F − ασ(1 + r)σ−1 = ηv(1 + r)−1 − ασ(1 + r)σ−1 ≥ −α(σ − η)(1 + r)σ−1 (v, r) ∂v (5.30) and F (v, r) 1 ∂F (5.31) = (1 + r)η . (v, r) q ∂v

136

Chapter 5. Uniqueness

Inserting (5.30) and (5.31) into (5.29) and using Cauchy-Schwarz inequality we deduce B(∇u, r) ≥ |∇u|2 +

K (1 + r)2(σ−1) − α(σ − η)(1 + r)σ−1 |∇u|. q

(5.32)

At this point we need to estimate the right-hand side of (5.32) so as to have B(∇u, r) ≥ Λ|∇u|2 ,

(5.33)

with a constant Λ > 0 independent of ∇u and r. A simple computation shows that if we choose 4K q< (5.34) 2, [α(σ − η)] then (5.33) holds for every 2

0<Λ≤1−

q[α(σ − η)] . 4K

With this choice of Λ we insert (5.33) and the expression of deduce

(5.35) ∂F ∂v

into (5.28) to

div W ≥ −2ϕλ(u)F (v, r)|∇ϕ||∇u| + qϕ2 (1 + r)−η F (v, r)Λ|∇u|2 . We integrate this inequality once in Ωγ ∩ BR , apply the divergence theorem, and recall that W vanishes on ∂(Ωγ ∩ BR ) to get qΛ ϕ2 λ(u)F (v, r)(1 + r)−η |∇u|2 ≤ ϕλ(u)F (v, r)|∇ϕ||∇u|. 2 Ωγ ∩BR Ωγ ∩BR Applying H¨older’s inequality to the integral on the right-hand side and simplifying we ﬁnally obtain q 2 Λ2 ϕ1 λ(u)F (v, r)(1 + r)−η |∇u|2 ≤ λ(u)F (v, r)(1 + r)η |∇ϕ|2 . (5.36) 4 Ωγ ∩BR By the volume growth assumption (5.16), ∀ d > d0 there exists a divergent sequence Rk ↑ +∞ with R1 > 2R0 such that log vol(BRk ) ≤ dRkσ−η .

(5.37)

Noting that θRk > 12 Rk > R0 , we apply (5.36) with R = Rk , use the bound for |∇ϕ| and the fact that λ ≤ 1, to get 2 qλ λ(u)F (v, r)|∇u|2 (5.38) E= 2 Ωγ ∩BR0 −2 F (v, r). (5.39) ≤ (1 + θRk )η [(1 − θ)Rk ] Ωγ ∩(Brk \BθRk )

5.3. Uniqueness via the weak maximum principle

137

Now we observe that E > 0, since ∇u ≡ 0 on Ωγ ∩ BR0 . On the other hand, using the bound (5.27) for v and the expression of F we get F (v, r) ≤ e−q(α−b)(1+θRk )

σ−η

on Ωγ ∩ (BRk \BθRk ).

Inserting this into the right-hand side of (5.38), using (5.37) and rearranging we obtain the inequality σ−η

0 < E ≤ CRkη−2 edRk

−q(α−b)(1+θRk )σ−η

(5.40)

where C > 0 is a constant independent of k. In order for this inequality to hold for every k we must have d ≥ (α − b)qθσ−η , whence, letting θ tend to 1, d ≥ (α − b)q. We set α = tb, insert the deﬁnition (5.34) of q in the above inequality, solve with respect to K, and then let τ → 1 to obtain K≤d

b(σ − η)2 t2 , 4 t−1

whence minimizing with respect to t > 1 and letting d → d0 and b → max {6 u, 0}, we obtain 6, 0(σ − η)2 . (5.41) K ≤ d0 max u Case II. σ = 0 (and therefore η < 0 by (5.15)). We proceed exactly as in the previous case and conclude that for any C 2 function u bounded above (5.41) holds. But, as noted earlier, ∀ ν, lim inf (1 + r)μ Δu = Ωγ

inf

(1 + r)μ Δun u.

{x:uν (x)>γ+ν}

(5.42)

Thus we may choose ν in such a way that u 6ν ≤ 0, apply (5.41) to uν to conclude that the right-hand side of (5.42) is nonpositive and then deduce that K ≤ 0 as required. Case III. σ > 0, η ≥ 0. We choose F (v, r) = F (v) = e−qv

σ−η σ

where q > 0 is a constant to be speciﬁed later. Since σ − η > 0 by assumption, a computation gives ∂F σ − η −η (v, r) = −q v σ F (v) < 0, ∂v σ

138

Chapter 5. Uniqueness

while clearly ∂F ≡ 0. ∂r Using the above expression, recalling that v ≥ (α−b)(1+r)σ on Ωγ , and proceeding as in Case I, we estimate B(∇u, r) ≥ |∇u|2 − ασ(1 + r)σ−1 |∇u| +

Kσ(α − b4/σ (1 + r)2(σ−1) . q(σ − η)

It is now a simple matter to observe that, ∀ r ≥ 0 ﬁxed, the right-hand side of the above inequality is bounded from below by Λ|n∇u|2 provided Λ≤1−

qα2 σ 2 (σ − η) . 4Kσ(α − b)η/σ

(5.43)

Since the right-hand side of the above inequality is independent of r, for every such Λ we have B(∇u, r) ≥ Λ|∇u|2 . In particular, if τ ∈ (0, 1) and we choose q=

4τ Kσ(α − b)η/σ α2 σ 2 (σ − η)

and Λ = 1 − τ,

(5.44)

then Λ is positive and satisﬁes (5.43). Substituting into (5.28) and using the expression of ∂F ∂v we deduce that div W ≥ −2ϕ2 λ(u)F (v)|∇ϕ||∇u|Λq

σ−η 2 ϕ λ(u)v η/σ |∇u|2 . σ

We now proceed as in Case I, repeating, with minor adaptations, the arguments that lead to (5.40), to conclude that −2 λ(u)F (v)|∇u|2 ≤ C(1 + Rk )η [(1 − θ)Rk ] 0<E= Ωγ ∩BR0

×

F (v, r) Ωγ ∩(BRk \BθRk )

where C > 0 is a constant independent of k and θ. Using the inequality F (v) ≤ e−q(α−b)

σ−η (1+θRk )σ−η σ

valid on Ωγ ∩ (BRk \BθRk ) and the volume growth estimate (5.38), we deduce that 0 < E ≤ C(α − b)−q (1 + θRk )−qσ (1 + Rk )σ Rkd−2

5.3. Uniqueness via the weak maximum principle

139

with C > 0 independent of k. This forces σ + d − 2 ≥ qσ, and letting d ↓ d0 , σ + d0 − 2 ≥ qσ. Thus, if σ+d0 −2 < 2 we get a contradiction, on K ≤ 0. Otherwise, if σ+d0 −2 ≥ 0, substituting the value of q, solving for K, letting α = tb, t > 1, with τ ↑ 1 and b ↓ max {6 u, 0}, we obtain K≤

t2 1 max {6 u, 0}σ(σ + d0 − 2) ; 4 t−1

whence, minimizing over t > 1, K ≤ max {6 u, 0}σ(σ + d0 − 2). This completes the proof if σ > 0. The case σ = 0 is dealt with as in Case II. We note, for future use, the following corollary (see [PRS05b], Corollary 4.4): Corollary 5.4. Let (M, , ) be a complete manifold, and let b(x) ∈ C 0 (M ) satisfy b(x) ≥

C (1 + r(x))μ

for some μ ∈ R. Let u ∈ C 2 (M ) be a solution of the diﬀerential inequality Δu ≥ b(x)f (u)

(5.45)

and suppose that, for some 0 ≤ σ < 2 − μ, either σ > 0, lim inf t→∞ f (t) > 0 and u(x) = o (r(x)σ ) as r(x) → +∞, or σ = 0 and u is bounded above. If lim inf r→∞

vol Br < +∞, r2−σ−μ

then u∗ = sup u < +∞ and f (u∗ ) ≤ 0. Proof. Observe ﬁrst of all that under the stated assumptions it follows from Theorem 5.3 that, for every γ < u∗ , inf b(x)−1 Δu ≤ C −1 inf (1 + r(x))μ Δu ≤ 0. Ωγ

Ωγ

Now, if u were unbounded, then Ωγ would be nonempty for every γ and it would follow from (5.45) that (5.46) inf f (t) ≤ 0, t>γ

contradicting the assumption that f is bounded away from zero at inﬁnity. Thus u is necessarily bounded above and u∗ < +∞. Arguing as above shows that (5.45) holds for every γ < u∗, and therefore f (u∗ ) ≤ 0.

140

5.3.2

Chapter 5. Uniqueness

A comparison result

We are now ready to prove the desired comparison result (see [RZ07] and also [MR10] for the case of a diﬀusion-type operator), from which the proof of Theorem 5.2 follows at once. Theorem 5.5. Let a(x), b(x) ∈ C 0 (M ), σ > 1, τ ≥ 0, β + τ (σ − 1) > −2 and suppose that b(x) > 0 on M and i) b(x) ≥ Br(x)β , B > 0 for r(x) 1;

ii) sup M

a− (x) r(x)τ (1−σ) < +∞. (5.47) b(x)

Let u, v ∈ C 2 (M ) be nonnegative solutions of Δu + a(x)u − b(x)uσ ≥ 0 ≥ Δv + a(x)v − b(x)v σ

(5.48)

on M , satisfying i) v(x) ≥ C1 r(x)τ ;

ii) u(x) ≤ C2 r(x)τ

(5.49)

for r(x) 1 and some positive constants C1 , C2 . If lim inf r→+∞

log vol Br < +∞, r2+β+τ (σ−1)

(5.50)

then u(x) ≤ v(x) on M . Proof. First of all, let u(x) ≡ 0, otherwise there is nothing to prove. Next, we observe by (5.48), (5.49) and the strong maximum principle that v(x) > 0 on M . This fact, u(x) ≡ 0, and (5.49) i), ii) imply that ξ = sup M

u(x) v(x)

(5.51)

satisﬁes 0 < ξ < +∞. If ξ ≤ 1, then u(x) ≤ v(x) on M . Let us assume, by contradiction, ξ > 1 and deﬁne ϕ = u − ξv. Note that ϕ ≤ 0 on M . We claim sup r(x)−τ ϕ(x) = 0.

(5.52)

M

Indeed, let {xn } ⊂ M be a sequence realizing ξ; then & ' u(xn ) −ξ . r(xn )−τ ϕ(xn ) = r(xn )−τ v(xn ) v(xn )

(5.53)

5.3. Uniqueness via the weak maximum principle Now observe that

141

r(xn )−τ v(xn )

is bounded because otherwise (5.49) ii) would imply ξ = 0. Then, it follows from (5.53) that r(xn )−τ ϕ(xn ) → 0 as n → +∞ proving (5.52). We now use (5.48) to obtain Δϕ ≥ −a(x)ϕ + b(x)(uσ − (ξv)σ ) + b(x)v σ ξ(ξ σ−1 − 1). (5.54) We write (uσ − (ξv)σ )(x) = h(x)ϕ(x)

where h(x) =

σu(x)σ−1 σ u(x)−ξv(x)

# u(x) ξv(x)

(5.55)

if u(x) = ξv(x), t

σ−1

dt

if u(x) = ξv(x),

is continuous and nonnegative on M . Fix the level set Ω1 = {x ∈ M : ϕ(x) > −1}, and note that (1 + r(x))−τ v(x) is bounded above on Ω1 . Indeed, using (5.49) ii), for every y ∈ Ω1 we have ξ

v(y) (u − ϕ)(y) 1 2 + 2 + 1. = ≤C ≤C (1 + r(y))τ (1 + r(y))τ (1 + r(y))τ

Thus there exists a constant C3 > 0 such that v(x) ≤ C3 (1 + r(x))τ

on Ω1 .

(5.56)

Now, from the mean value theorem for integrals, for some y ∈ (ξv(x), u(x)) or y ∈ (u(x), ξv(x)), σ (u(x) − ξv(x))y σ−1 u(x) − ξv(x) ≤ σ uσ−1 (x) + (ξv(x))σ−1

h(x) =

≤ C(1 + r(x))τ (σ−1)

(5.57)

on Ω1

because of (5.49) ii) and (5.56). Now we note that, since b(x) > 0 on M we can rewrite (5.47) i) as + r(x))β on M, b(x) ≥ B(1 (5.58) > 0. Using b(x) > 0 and (5.55), from (5.54) and ϕ ≤ 0 we for some appropriate B deduce 1 a− (x) Δϕ ≥ + h(x) ϕ(x) + v σ (x)ξ(ξ σ−1 − 1) b(x) b(x) and therefore, from ξ > 1, (5.47) ii), (5.57), (5.49) i) and ϕ ≤ 0, (1 + r(x))−στ

1 Δϕ ≥ C(1 + r(x))−τ ϕ(x) + Dξ(ξ σ−1 − 1) b(x)

on Ω1

142

Chapter 5. Uniqueness

for some appropriate constants C, D > 0. Next, we choose 0 < ε < 1 suﬃciently small so that 1 (5.59) C(1 + r(x))−τ ϕ(x) ≥ − Dξ(ξ σ−1 − 1) 2 on Ωε = {x ∈ M : ϕ(x) > −ε} ⊂ Ω1 . Note that this is possible since τ ≥ 0. Then, on Ωε , Δϕ ≥ 0 so that (5.58) implies (1 + r(x))−β−στ Δϕ ≥ (1 + r(x))−στ and thus inf (1 + r)−β−στ Δϕ ≥ Ωε

1 Δϕ b(x)

1 Dξ(ξ σ−1 − 1) > 0 2

(5.60)

since ξ > 1. This fact, together with (5.50), contradicts Theorem 5.3.

Combining the a priori estimates given in Theorems 4.1 and 4.4, the volume growth estimates of Chapter 1 and Theorem 5.2, we obtain Theorem 5.6. Let (M, , ) be a complete manifold with Ricci tensor satisfying Ric ≥ −(m − 1)H 2 (1 + r2 )δ/2

(5.61)

for some H > 0, δ ≥ −2. Let a(x), b(x) ∈ C 0 (M ) with b(x) > 0 on M and assume that, for some A, B > 0, A−1 r(x)α ≤ a(x) ≤ Ar(x)α , B

−1

r(x) ≤ b(x) ≤ Br(x) β

(5.62)

β

(5.63)

for r(x) 1 and with α > δ, α ≥ max β, β + 2δ − 1 . Then, there exists at most one nonnegative, nontrivial solution u ∈ C 2 (M ) of Δu + a(x)u − b(x)uσ = 0,

σ > 1,

on M.

(5.64)

Remark. The geometric case of Yamabe’s equation for which α ≥ δ will be dealt with in Section 5.3. Proof. First observe, by the maximum principle, that if u ≥ 0 and u ≡ 0 is a solution of (5.64), then u > 0 on M . Thus if u ≡ 0, according to the a priori estimates of Theorems 4.1 and 4.4 we have (note that α ≥ 2δ − 1 because α > δ ≥ −2, while β ≤ 1 − 2δ + α because of our assumptions) β−α

β−α

C −1 r(x)− σ−1 ≤ u(x) ≤ Cr(x)− σ−1

(5.65)

for some constant C > 0 and r(x) 1. To apply Theorem 5.2, since a− (x) ≡ 0 for r(x) 1 because of (5.62) we only need to have α ≥ β and α > −2. This

5.3. Uniqueness via the weak maximum principle

143

latter is guaranteed by α > δ, while the ﬁrst is satisﬁed by assumption. It remains to check that (5.14) holds, which in this case becomes lim inf r→+∞

log vol Br < +∞. r2+α

(5.66)

Towards this aim, we need to estimate vol Br from above using assumption (5.61). Using Proposition 1.15, we see that if δ ≥ −2, then log vol Br ≤ Cr1+ 2 δ

as r → +∞

for some constant C > 0, and (5.66) is satisﬁed provided α≥ If δ = −2, then

vol Br ≤ C

r

s

1+

√

1+4H 2 2

δ − 1. 2

(m−1)

0

(5.67)

ds # r

1+

√

1+4H 2 2

(m−1)+1

,

and (5.66) is satisﬁed provided α > −2.

(5.68)

Note that α > δ implies (5.67) and (5.68).

5.3.3

Uniqueness of ground states

Our aim is now to prove uniqueness of positive solutions converging to zero at inﬁnity, or ground states (see also Chapter 3, section 3.2). As the above examples show we have to impose some extra condition. Surprisingly, this condition is again related to λL 1 (M ) ≥ 0, as we shall see below. We begin with the following general result proved in [RRV97]; we denote with W 1,2 (M ) the Sobolev space of functions which are in L2 (M ) and have L2 weak gradient. Theorem 5.7. Let (M, , ) be a complete manifold and let a(x), b(x) ∈ C 0 (M ), b(x) ≥ 0, b(x) ≡ 0. Let u1 and u2 be positive solutions of Δu + a(x)u − b(x)uσ = 0.

(5.69)

1,2 (M ), ϕ > 0 and such that If there exists ϕ ∈ C 0 (M ) ∩ Wloc

Δϕ + a(x)ϕ − σb(x)uσ−1 ϕ≤0 i

(5.70)

in the weak sense on M for i = 1, 2, and u1 (x) − u2 (x) = o(ϕ(x)) then u1 ≡ u2 .

as r(x) → +∞,

(5.71)

144

Chapter 5. Uniqueness

Proof. We divide the proof in two steps. Step I. For ε > 0 we set v = vε = u1 + εϕ. We claim that H(x) = Δv + a(x)v − b(x)v σ ≤ 0

(5.72)

in the weak sense on M . Indeed we ﬁnd H(x) = Δu1 + a(x)u1 − b(x)(u1 + εϕ)σ + εΔϕ + εa(x)ϕ.

(5.73)

From (5.69) we get " ! 1 H(x) = ε Δϕ + a(x)ϕ − b(x)((u1 + εϕ)σ − uσ1 ) . ε ϕ, by convexity we have Since (u1 + εϕ)σ − uσ1 ≥ εσuσ−1 1 H(x) ≤ ε Δϕ + a(x)ϕ − σb(x)uσ−1 ϕ 1

(5.74)

(5.75)

which implies (5.72) from (5.70). Step II. We claim that u 1 ≥ u2 .

(5.76)

Δv Δu2 − − v σ−1 ≥ 0 − b(x) uσ−1 2 u2 v

(5.77)

From (5.69) and (5.72) we have

weakly on M . Since (u22 − v 2 )+ ∈ C 0 (M ) ∩ W 1,2 (M ) and compactly supported, by (5.71)), it is an admissible test function and we have Δu2 Δv 2 2 u2 − v + − − b(x) uσ−1 − v σ−1 (u22 − v 2 )+ ≥ 0. (5.78) 2 u v 2 M M Suppose now that Ω = {x ∈ M : u2 (x) > v(x)} = ∅. Then, (5.78) yields 2 2 v u2 − v σ−1 (u22 − v 2 ) ≤ 0 (5.79) ∇u2 − ∇v + ∇v − ∇u2 + b(x) uσ−1 2 v u 2 Ω Ω and since b(x) ≥ 0, b(x) ≡ 0 we conclude, as in Theorem 5.1, that u2 ≤ u1 + εϕ

∀ ε > 0,

that is, (5.76). Similarly, u1 ≤ u2 and therefore u1 ≡ u2 .

5.3. Uniqueness via the weak maximum principle

145

Corollary 5.8. Let (M, , ) be a complete manifold and let a(x), b(x) ∈ C 0 (M ), b(x) ≥ 0, b(x) ≡ 0. Suppose ψ is a positive solution of Δψ + a(x)ψ ≤ 0

on M.

(5.80)

If u1 and u2 are positive solutions of (5.69) on M satisfying u1 (x) − u2 (x) = o(ψ(x))

as r(x) → +∞,

(5.81)

then u1 ≡ u2 . Proof. It is enough to show that ψ satisﬁes (5.70) of Theorem 5.7. But this is obvious, since b(x) ≥ 0, ui > 0 and (5.80) imply 0 ≥ Δψ + a(x)ψ ≥ Δψ + a(x)ψ − σb(x)uσ−1 ψ, i

i = 1, 2.

Remark. By a result of Fisher-Colbrie and Schoen, [FCS80], condition (5.80) is equivalent to λL 1 (M ) ≥ 0 with L = Δ + a(x) (see the discussion at the beginning of Chapter 3). We observe that if 2−m t > 0 on [0, +∞), ρ(t) = C cosh 2 then the function ψ(x) = ρ(r(x)) is a radial solution of

m(m − 2) ψ=0 4 on the m-dimensional hyperbolic space Hm , of constant negative curvature −1 (m ≥ 3). In this case condition (5.81) becomes Δ Hm ψ +

(u1 (x) − u2 (x))e

m−2 2 r(x)

→0

as r(x) → +∞.

In the case of the example in section 5.2 we have 5 . m−2 m−2 m−2 m−2 2− 2 (a22 − 1)− 2 (a21 − 1)− 2 r(x) 2 (wa1 (x) − wa2 (x))e ∼ − m(m − 2) a21 a22 and this converges to zero if and only if a1 = a2 . This shows that (5.81) cannot be relaxed to O(ψ(x)) as r(x) → +∞. It is clear that the applicability of Corollary 5.8 depends on the knowledge of the asymptotic behaviour of the solution ψ of (5.80). We shall come back to this later, in section 7.3.2.

146

5.4

Chapter 5. Uniqueness

Some geometric applications and further uniqueness

We apply the uniqueness results obtained so far to the problem of characterizing isometries into the group of conformal diﬀeomorphisms of the complete manifold M . We then go back to uniqueness for the Yamabe problem; this will be achieved via an L∞ a priori estimate of independent interest.

5.4.1

Conformal diﬀeomorphisms

Let (M, , ) be a Riemannian manifold with scalar curvature S(x). To simplify the writing we consider the case m = dim M ≥ 3, but one could treat the case m = 2 with few changes. Suppose that ϕ : M → M is a conformal diﬀeomorphism, that is, ϕ is a diﬀeomorphism such that, for some u ∈ C ∞ (M ), u > 0, 4

ϕ∗ , = u m−2 ,

(5.82)

(u is usually called the stretching factor of the diﬀeomorphism). We shall express this fact by writing ϕ ∈ Conf(M ), where Conf(M ) denotes the group of conformal diﬀeomorphisms. If u ≡ 1, then ϕ is an isometry. If we denote by Iso(M ) the group of isometries, then, clearly, Iso(M ) ⊆ Conf(M ). Note that if ϕ is an isometry, then it preserves the scalar curvature (in fact the sectional curvature). It is thus meaningful to investigate when a conformal diﬀeomorphism preserving the scalar curvature is an isometry. If ϕ preserves scalar curvature, then its stretching factor u is a positive solution of the (special) Yamabe equation ) 4 * m−2 Δu = − S(x)u u m−2 − 1 . (5.83) 4(m − 1) Since the inverse diﬀeomorphism ϕ−1 also preserves the scalar curvature and 4

(ϕ−1 )∗ , = u− m−2 , , we conclude that proving that ϕ is an isometry amounts to showing that u ≤ 1. After this preparation we prove the next (see [RRV94a]) Theorem 5.9. Let (M, , ) be a complete manifold with Ricci tensor satisfying Ric ≥ −(m − 1)H 2 (1 + r(x)2 )δ/2

on M

(5.84)

for some H > 0, δ ≥ −2. Suppose furthermore that the scalar curvature S(x) satisﬁes S(x) ≤ 0 on M and S(x) ≤ −

d2 (1 + r(x))μ

for r(x) 1

(5.85)

5.4. Some geometric applications and further uniqueness

147

and some d > 0 and μ < 1 − 2δ . Then any conformal transformation of (M, , ) which preserves the scalar curvature is an isometry. m−2 Proof. Let a(x) = b(x) = − 4(m−1) S(x). In Proposition 4.3 we chose ω(t) = C1 > 0 constant and δ δ ψ(t) = C2 t1− 2 −μ , μ ≤ 1 − . 2

Then, it follows from (5.83) and (5.85) that the solution u > 0 of (5.83) satisﬁes u∗ = sup u < +∞. M

According to the discussion above, in order to show that ϕ is an isometry it is enough to prove that u∗ ≤ 1. We observe that from Theorem 3.10 and the Remark thereafter we can suppose S(x) < 0 on M without loss of generality. Now we reason by contradiction and we suppose u∗ > 1. We ﬁx n ∈ N suﬃciently large such that 1 < u∗ − n1 < u∗ and we set ' 1 . x ∈ M : u(x) > u − n

& Ωn =

∗

Fix R > 0 suﬃciently large that (5.85) holds on M \B R . Then, using (5.85) and the fact that u∗ − n1 > 1 we have (1 + r(x))μ Δu ≥

4 m − 2 2 m−2 d u u −1 ≥C >0 4(m − 1)

on Ωn ∩ (M \B R )

for some constant C while (1 + r(x))μ Δu ≥ C > 0

on Ωn ∩ B R

since S(x) < 0 on M . Now, if lim inf r→+∞

log vol Br < +∞ r2−μ

(5.86)

we can apply Theorem 5.3 to obtain the desired contradiction. Proceeding as in the proof of Theorem 5.6 we see that (5.84) implies (5.86) in the assumptions that μ < 1 − 2δ . Remark. In case δ > −2 the conclusion in the above theorem is reached under the milder assumption μ ≤ 1 − 2δ . Indeed, as shown in [RRV94a], Corollary 1, the results holds for ν ≤ 1 − 2δ in the whole range δ ≥ −2. Similarly to what we did above we can use the uniqueness result contained in Theorem 5.1 to obtain the following version:

148

Chapter 5. Uniqueness

Theorem 5.10. Let (M, , ) be a complete manifold of dimension m ≥ 3 and scalar curvature S(x) satisfying i) S(x) ≤ 0 on M ;

ii) S(x) ≡ 0 on M.

Let ϕ : M → M be a conformal diﬀeomorphism whose stretching factor u satisﬁes & (u − 1)

2

'−1

∈ L1 (+∞).

∂Br

Then ϕ is an isometry.

5.4.2

Uniqueness for the Yamabe problem

As we pointed out in Section 4.1 , see the remark after the proof of Theorem 4.1, the lower estimates on u obtained there do not apply to solutions of the Yamabe problem since condition (4.27) is not satisﬁed under the usual assumption (4.2). This implies that Theorem 5.6 cannot be applied to deduce uniqueness for the Yamabe problem. However, the next result where no assumptions on the Ricci tensor are required does hold (see [RZ07], Theorem 3.4). Theorem 5.11. Let (M, , ) be a complete manifold of dimension m ≥ 3 and scalar curvature S(x) satisfying sup S(x) < +∞.

(5.87)

M

Let K(x) ∈ C ∞ (M ), K(x) < 0 on M and suppose that K(x) ≤ −

B r(x)β

for r(x) 1,

(5.88)

some constant B > 0, β < 2 and inf M

S(x) > −∞. K(x)

(5.89)

log vol Br < +∞. r2−β

(5.90)

Assume lim inf r→+∞

4

Then, there exists at most one conformal deformation to a new metric g = u m−2 , with scalar curvature K(x) and such that u∗ = inf M u > 0. Proof. Suppose such a metric exist. Thus u > 0 satisﬁes m+2

cm Δu − S(x)u + K(x)u m−2 = 0

5.4. Some geometric applications and further uniqueness

149

on M . We claim that (5.87), (5.88), (5.89) and (5.90) imply u∗ = sup u < +∞.

(5.91)

M

Postponing for the moment the proof of the claim, we have 0 < u∗ ≤ u ≤ u∗ < +∞. A direct application of Theorem 5.5 with τ = 0 gives uniqueness of u.

It remains to show that u is bounded above under the sole assumption of the volume growth condition (5.90): this is precisely the content of the next remarkable result.

5.4.3

An L∞ a priori estimate

We prove here the following result of very general interest, which was obtained in [PRS03b]. Theorem 5.12. Let (M, , ) be a complete manifold and a(x), b(x) ∈ C 0 (M ). Assume sup a+ (x) < +∞ (5.92) M

and that b(x) satisfy b(x) > 0

on M,

b(x) ≥

B r(x)β

(5.93)

for r(x) 1, B > 0 and some 0 ≤ β < 2. Assume further that a+ (x) ≤E b(x)

on M.

(5.94)

Let u ∈ C 1 (M ) be a nonnegative solution of Δu + a(x)u − b(x)uσ ≥ 0

on M

(5.95)

with σ > 1. If lim inf r→+∞

log vol Br < +∞, r2−β

(5.96)

then 1

u(x) ≤ E σ−1

on M.

To prove Theorem 5.12, we ﬁrst need two technical results.

(5.97)

150

Chapter 5. Uniqueness

Lemma 5.13. Let a(x), b(x) ∈ C 0 (M ), a(x) = a+ (x) − a− (x), with a± ≥ 0, b(x) > 0, and assume that, for some E > 0 we have a+ (x) ≤E b(x)

on M.

(5.98)

Assume that u ∈ C 2 (M ) and γ > 0 are such that Ωγ = {x ∈ M : u(x) > γ} = ∅

(5.99)

and that u satisﬁes Δu ≥ b(x)uσ − a(x)u +

D 2 |∇u| , u

(5.100)

on Ωγ , for some constants D ∈ R and σ > 1. Let λ : R → [0, +∞) be a C 1 , nondecreasing function such that l(t) = 0 for t ≤ γ. Then, there exists R > 0 suﬃciently large, and a constant C > 0 such that, for every r > R, and for every α > max 1 − D, 1, &

b(x)λ(u) ≤ Br

(α + σ − 1)2 C 1 + σ−1 2 γ r inf B2r b D + α − 1 1

E γ σ−1

' α+σ−1 σ−1 b(x)λ(u). B2r

(5.101)

Proof. Let R > 0 large enough that BR ∩ Ωγ = ∅, and ﬁx ζ > 1 such that 2+

1+δ 1 − 1 > 0. σ−1 ζ

(5.102)

We choose a C ∞ cut-oﬀ function ψ : M → [0, 1] such that, for every r ≥ R, i) ψ ≡ 1 on Br ;

ii) ψ ≡ 0 on M \ B2r ;

iii) |∇ψ| ≤

C0 1/ζ ψ , r

(5.103)

for some constant C0 = C0 (ζ) > 0. Note that this is possible since ζ > 1. Finally, we ﬁx α and consider the vector ﬁeld W deﬁned by W = ψ 2(α+σ−1) λ(u)uα−1 ∇u. Note that the properties of λ and ψ imply that u vanishes oﬀ B2r ∩ Ωγ . A computation that uses (5.100), λ ≥ 0, α ≥ 1 − D and the Cauchy–Schwarz inequality, yields div W ≥ −2(α + σ − 1)ψ 2(α+σ−1)−1 λ(u)uα−1 |∇u||∇ψ| 0 1 2 + λ(u)ψ 2(α+σ−1) b(x)uα+σ−1 − a(x)uα + ((D + α − 1))uα−2 |∇u| .

5.4. Some geometric applications and further uniqueness

151

Recalling that W is compactly supported, the divergence theorem yields ψ 2(α+σ−1) λ(u)b(x)uα+σ−1 − ψ 2(α+σ−1) λ(u)a(x)uα 2 ≤ −(D + α − 1) ψ 2(α+σ−1) λ(u)uα−2 |∇u| + 2(α + σ − 1) ψ 2(α+σ−1)−1 λ(u)uα−1 |∇u||∇ψ|. (5.104) p p

Now we apply the inequality ab ≤ ε pa + conjugate exponents, with the choices p = q = 2,

bq εq q ,

valid for a, b ≥ 0, ε > 0, with p, q 1/2

ε = (2(D + α − 1))

to the second integral on the right-hand side of (5.104) to obtain 2(α + σ − 1) ψ 2(α+σ−1)−1 λ(u)uα−1 |∇u||∇ψ| 2 ≤ (D + α − 1) ψ 2(α+σ−1) λ(u)uα−2 |∇u| (α + σ − 1)2 2 ψ 2(α+σ−1)−2 λ(u)uα |∇ψ| . + D+α−1 Inserting this into (5.104) and using λ ≥ 0 gives (5.105) ψ 2(α+σ−1) λ(u)b(x)uα+σ−1 ≤ ψ 2(α+σ−1) λ(u)a+ (x)uα

2 (α + σ − 1)2 + ψ 2(α+σ−1)−2(1−1/ζ) λ(u)uα ψ −1/ζ |∇ψ| . D+α−1 For ease of notation, we denote by I and II the integrals on the right-hand side. We use (5.98), multiply and divide by b(x), and use H¨ older inequality with conjugate exponents p and q to be chosen later, to estimate I≤E

ψ

2(α+σ−1)

1/p λ(u)b(x)u

αp

ψ

2(α+σ−1)

1/q λ(u)b(x)

.

(5.106)

Similarly, using (5.103) iii), we see that II ≤

C02 r2

ψ 2(α+σ−1) λ(u)b(x)uαp ×

ψ

1/p

2(α+σ−1)+2q(1/ζ−1)

λ(u)b(x)

1−q

1/q ,

(5.107)

152

Chapter 5. Uniqueness

where C0 = C0 (ζ) is the constant appearing in (5.103) iii). In the above inequalities, it is assumed that p and q are chosen in such a way that 2(α + σ − 1) + 2q(1/ζ − 1) > 0 is positive. To proceed we need to estimate the powers of u in the above integrals. We choose p in such a way that αp is equal to the exponent of u in the integral on the left-hand side of (5.105), namely α + σ − 1. This implies p=

α+σ−1 , α

q=

α+σ−1 . σ−1

Note that, since σ > 1, we have p, q > 1, and that, by our choice of ζ, ! " 2 1 1 = (α + σ − 1) 2 + > 0. 2(α + σ − 1) + 2q ζ −1 σ−1 ζ −1 Since ψ ≤ 1, and ψ = 0 oﬀ B2r , we have I≤E

ψ 2(α+σ−1) λ(u)b(x)uα+σ−1

α α+σ−1

×

λ(u)b(x)

σ−1 α+σ−1 ,

(5.108)

B2r

and II ≤

C02 γ δ−1 r2 inf B2r b

ψ 2(α+σ−1) λ(u)b(x)u(α+σ−1)

α α+σ−1

σ−1 α+σ−1

×

λ(u)b(x)

.

(5.109)

B2r

Substituting (5.108) and (5.109) into (5.105), and rearranging yield

ψ 2(α+σ−1) λ(u)b(x)uα+σ−1 ≤

(α + σ − 1)2 1 C E+ 2 r inf B2r b D + α − 1

α+σ−1 σ−1 λ(u)b(x) B2r

with C = C02 C1 . We estimate from below the integral on the left-hand side using ψ = 1 on Br , and λ(u) = 0 if u ≤ γ, and arrive at

λ(u)b(x) ≤ Br

E γ σ−1

C γ 1−σ (α + σ − 1)2 + 2 r inf B2r b D + α − 1

α+σ−1 σ−1 λ(u)b(x), B2r

which is the required conclusion.

Lemma 5.14. Let G, Q : [R, +∞) → [0, +∞) be nondecreasing functions such that, for some constants 0 < Λ < 1 and B, θ > 0, G(r) ≤ ΛBr

θ

/Q(2r)

G(2r)

∀r ≥ R.

(5.110)

5.4. Some geometric applications and further uniqueness

153

Then there exists a constant S = S(θ) > 0 such that, for every r ≥ 2R, Q(r) Q(r) log G(r) ≥ θ log G(R) + SB log(1/Λ). rθ r

(5.111)

Proof. Let r0 = R and rk = 2k r0 . Then, for every r ≥ 2r0 , there exists k such that rk ≤ r ≤ rk+1 . Applying inequality (5.110) k-times, we obtain G(r0 ) ≤ ΛB

k−1 j=0

rjθ /Q(2rj )

G(rk ).

(5.112)

Using the deﬁnition of rk and the fact that Q is nondecreasing we estimate k−1 j=0

rjθ /Q(2rj ) ≥

k−1 θ rk+1 1 − 2−kθ −θ r0θ rθ 2 ≥S 2jθ = θ Q(2rk−1 ) j=0 Q(2rk−1 ) 2 − 1 Q(r)

with S = 2−θ /(2θ − 1). Substituting into (5.112), and recalling that 0 < Λ < 1 and that G is nondecreasing, we conclude that G(r0 ) ≤ ΛBSr

θ

/Q(r)

G(r),

whence (5.111) follows by taking logarithms. We are now ready for the

Proof of Theorem 5.12. First of all, we note that since !a+ !∞ < +∞, we may assume that !b!∞ < +∞. We let R, λ and γ be as in Lemma 5.13, and set, for r ≥ R, G(r) = λ(u)b(x). Br

Applying Lemma 5.13 with D = 0, the lower bound b(x) ≥ B/r(x)β , and the inequality (α + σ − 1)2 ≤ 23 α α−1 valid for α ≥ max{σ − 1, 2}, we deduce that there exists a constant C3 depending only on σ such that, for every r ≥ R, γ < u∗ and α > max{σ − 1, 2}, G(r) ≤

E γ σ−1

C3 α + σ−1 2−β γ r

α+σ−1 σ−1 G(2r).

(5.113)

Now, assume by contradiction that u∗ = +∞, so that Ωγ = ∅ for every γ > 0. Fix 0 < ρ < 1 and let γ0 > 0 be such that, for every γ ≥ γ0 , E ≤ 1 − 2ρ. γ σ−1

(5.114)

154

Chapter 5. Uniqueness

Next, we choose α of the form α = α(r) = ρ

γ σ−1 2−β r , C3

and observe that, since β < 2, there exists R1 ≥ R such that α(r) satisﬁes the condition stated before (5.113) for every r ≥ R1 and C3 α E + σ−1 2−β α ≤ 1 − ρ. γ σ−1 γ r Setting Λ = 1 − ρ, (5.113) gives G(r) ≤ ΛBγ

σ−1 2−β

r

G(2r)

∀r ≥ R1 ,

(5.115)

where B > 0 is a constant depending only on ρ and σ. We apply Lemma 5.14 to deduce that there exists a constant S such that, for every r ≥ R1 , 1 1 log b(x)λ(u) ≥ 2−β log b(x)λ(u) + SBγ σ−1 log (1/Λ). (5.116) r2−β r Br BR 1 To reach the required contradiction, we choose λ satisfying sup λ = 1/ supM b, so that b(x)λ(u) ≤ 1, and let r → +∞ in (5.116). Since 2 − β > 0, we conclude that 1 1 b(x)λ(u) ≥ SBγ σ−1 log (1/Λ). lim inf 2−β log vol Br ≥ lim inf 2−β log r→+∞ r r→+∞ r Br Since σ > 1, by taking γ suﬃciently large, this contradicts (5.96). To complete the proof of the theorem we observe that the diﬀerential equation (5.95) yields Δu ≥ b(x)u(uσ−1 − E), whence the required conclusion follows applying Corollary 5.4 with f (u) = u · (uσ−1 − E). Remark. One might wonder whether a corresponding result on u∗ holds. This is the case under further restrictions on the nonlinearity of the equation, see for instance [PRS05a]. However, this does not apply to Yamabe’s equation, so that the assumption that u∗ > 0 in the statement of Theorem 5.11 is necessary. See also the ﬁnal remark of Section 4.1. We conclude this section with a last application of Theorem 5.12. Towards this goal, let (M, g) be a complete Riemannian manifold with scalar curvature S(x), and, as usual, let r(x) be the Riemannian distance function from a ﬁxed reference point o. Assume that ϕ : (M, g) → (N, h) is a conformal immersion, so that, assuming it will simplify matters, m = dim(M ) ≥ 3, 4

ϕ∗ h = u m−2 g

(5.117)

5.4. Some geometric applications and further uniqueness

155

for some positive function u ∈ C ∞ (M ). Let also K(x) be the scalar curvature of the conformal metric ϕ∗ h. Note that if u ≤ 1 the length of curves in M with respect to the conformal metric ϕ∗ h is less than or equal to their lengths with respect to the original metric g, and therefore the Riemannian distance induced by ϕ∗ h is smaller than or equal to the original distance function. In this situation we say that ϕ is weakly distance decreasing . Theorem 5.15. Assume that inf S(x) > −∞ M

and that K(x) < 0, K(x) ≤ S(x) on M

and

K(x) ≤ −

A r(x)β

for r(x) >> 1, and for some constants A > 0 and β < 2. If lim inf r→∞

log (vol Br ) < +∞, r2−β

then ϕ is weakly distance decreasing. Proof. Recall that the stretching factor u of ϕ∗ h with respect to g satisﬁes the Yamabe equation m+2

cm Δu − S(x)u + K(x)u m−2 = 0

on M.

K(x) Setting S(x) = − S(x) cm and b(x) == − cm , the conditions imposed on S(x) and K(x) imply that all the assumptions of Theorem 5.12 are satisﬁed with E = 1. Hence u ≤ 1 and ϕ is weakly distance decreasing.

Remark. Theorem 5.15 has its roots in the work started, in the compact case, by A. Lichnerowicz, [Lic58], Obata, [Oba62b] (see Section 2.1.1), and K. Yano and T. Nagano, [YN59], and extended to the noncompact case by Yau, [Yau73] where it was assumed that K(x) = S(x) ≤ −ε < 0 and that the sectional curvature of (M, g) was bounded from below. We also remark that some negativity is necessary for the conclusion of Theorem 5.15 to hold, as the following elementary example shows: let (M, g) be a manifold of dimension m ≥ 3 with nonnegative scalar curvature S(x), and for 2 every ﬁxed a > 1 let (N, h) = (M, a2 g). Then K(x) = a− m−2 S(x) ≤ S(x) since a > 1 and the identity map is a conformal diﬀeomorphism of (M, g) onto (N, h) which is not weakly distance decreasing.

Chapter 6

Existence In this chapter we provide existence results for nonnegative solutions of the Yamabe-type equation Δu + a(x)u − b(x)uσ = 0, σ > 1

(6.1)

on the complete, noncompact, manifold M . Existence is obtained by various versions of the monotone iteration scheme (see the appendix at the end of the chapter) and we consequently concentrate on the construction of (global and local) superand subsolutions for the problem. The ﬁrst are obtained with some assumptions on the sign of b(x) and on that of the ﬁrst eigenvalue of L = Δ + a(x) on appropriate domains. In the situation at hand, subsolutions are generally harder to ﬁnd and in what follows we give a number of suﬃcient conditions that allow their construction. We mention in this respect Theorem 6.15, in which existence is guaranteed under a very weak growth condition on b(x), and also Theorem 6.16, where a further weakening the condition on the sign of b(x) is balanced by the necessity of imposing a constant negative lower bound on the Ricci curvature. We note that the assumptions of our existence theorems match those of the nonexistence results in the previous chapters. Existence can be in particular guaranteed for the Yamabe problem, but we limit ourselves to explicitly state only one of the possible consequences at the end of the chapter, in Theorem 6.18, leaving the derivation of further results to the interested reader. The chapter begins with some introductory material, in particular a useful comparison result and some facts from basic spectral theory; then we prove a result of Li, Tam and Yang (see [LTY98]) concerning the nonnegativity of the ﬁrst eigenvalue of Schr¨odinger operators. These steps are used to outline a general procedure that reduces the existence problem to that of the existence of a positive subsolution; next, we provide diﬀerent suﬃcient conditions for this latter. We then deal with a case where the existence of a supersolution is no longer a consequence of a spectral assumption and depends on a rather delicate construction. Finally, we

P. Mastrolia et al., Yamabe-type Equations on Complete, Noncompact Manifolds, Progress in Mathematics 302, DOI 10.1007/978-3-0348-0376-2_6, © Springer Basel 2012

157

158

Chapter 6. Existence

consider speciﬁcally the case of the geometric Yamabe equation. The chapter ends with an appendix in which we describe the method of sub- and supersolutions, and where we also prove that the maximum of two subsolutions (or the minimum of two supersolutions) is again a subsolution (resp. supersolution).

6.1

A general procedure

The aim of this section is to describe the general procedure mentioned above. The various steps towards this goal, achieved in Theorem 6.16, are explained in the next subsections. The next proposition is a useful comparison result that shall be used throughout the chapter.

6.1.1

Another comparison result

Proposition 6.1. Let D ⊂ M be a bounded open set with boundary ∂D. Assume ¯ and that b(x) is nonnegative. Let u, v ∈ C 0 (D) ¯ ∩ C 2 (D) be a(x), b(x) ∈ C 0 (D), solutions on D of Δu + a(x)u − b(x)uσ ≥ 0 (6.2) and Δv + a(x)v − b(x)v σ ≤ 0

(6.3)

with u ≥ 0, v > 0 and σ ≥ 1. If u ≤ v on ∂D, then u ≤ v on D. Proof. The proof is modeled on that of the generalized maximum principle (see for example [PW67]). Set w = u/v. A computation shows that Δw ≥ b(x) uσ−1 − v σ−1 w − 2 ∇w, ∇ log v . Assume by contradiction that u > v somewhere in D, so that for a suﬃciently small ε > 0, Ωε = {x ∈ D : w(x) > 1 + ε} = ∅. Then Ωε ⊂ D, and since u > v on Ωε , σ ≥ 1 and b(x) is nonnegative, Δw + 2 ∇w, ∇ log v ≥ 0

on Ωε .

Since w = 1 + ε on ∂Ωε , by the maximum principle w ≤ 1 + ε on Ωε , and this yields the required contradiction.

6.1.2

More basic spectral theory and a result of Li, Tam and Yang

In this section we collect some other simple results (see also Chapter 3) from the spectral theory of Schr¨ odinger operators . For more details, we refer to [PRS08] and references therein.

6.1. A general procedure

159

Let a(x) ∈ C 0 (M ) and L = Δ+a(x). As we already know, if Ω is a nonempty open set, the ﬁrst Dirichlet eigenvalue λL 1 (Ω) is variationally deﬁned by means of the formula & ' 1,2 2 2 2 λL (Ω) = inf |∇ϕ| − a(x)ϕ : ϕ ∈ W (Ω), ϕ = 1 , 1 0 Ω

Ω

and, if Ω is bounded and both Ω and a are suﬃciently regular, the inﬁmum is attained by a unique normalized eigenfunction v deﬁned on Ω and satisfying on Ω; Δv + a(x)v + λL 1 (Ω)v = 0, v > 0 on Ω, v ≡ 0 on ∂Ω. We extend the above deﬁnition to an arbitrary bounded subset Υ of M by setting L λL 1 (Υ) = sup λ1 (Ω),

(6.4)

where the supremum is taken over all open bounded sets with smooth boundary Ω such that Υ ⊂ Ω (see also Chapter 3). Note that, by deﬁnition, if Υ = ∅, then λL 1 (S) = +∞. Finally, if Υ is an unbounded subset of M , we deﬁne L λL 1 (Υ) = inf λ1 (D ∩ Υ),

where the inﬁmum is taken over all bounded open sets D with smooth boundary. Note that if {Dn } is an increasing sequence of open sets with smooth boundaries which exhausts M , then, by domain monotonicity, L λL 1 (Υ) = lim λ1 (Dn ∩ Υ). n→+∞

Since the ﬁrst Dirichlet eigenvalue of the Laplacian of a ball Br grows like r−2 as r → 0 (see for instance Chavel [Cha84]), λL 1 (Br ) > 0 provided r is suﬃciently small, and one may think that the condition λL 1 (Υ) > 0 expresses the fact that Υ is small in a spectral sense. This notion of smallness is appropriate for our purposes. Indeed, P. Li, L. F. Tam and D. Yang, [LTY98], have established the following relationship between the ﬁrst eigenvalue of B0 = {x ∈ M : b(x) = 0}

(6.5)

and the existence of a nontrivial supersolution of (6.1). We provide here a mildly improved proof of the original result. Theorem 6.2. Let a(x), b(x) ∈ C 0,α (M ), 0 < α ≤ 1, and suppose that b(x) ≥ 0 on M and that the set B0 deﬁned in (6.5) is bounded. Let Ω be a bounded open domain containing B0 . If Δu + a(x)u − b(x)uσ = 0,

σ>1

(6.6)

L has a positive supersolution on Ω, then λL 1 (B0 ) ≥ 0. Conversely, if λ1 (B0 ) > 0, then (6.6) has a positive supersolution on Ω.

160

Chapter 6. Existence

Proof. Suppose (6.6) has a positive supersolution u on Ω and, by contradiction, assume that λL 1 (B0 ) = −a for some constant a > 0. By the deﬁnition of λL 1 (B0 ) we can ﬁnd a sequence of open sets Ωi , i ∈ N, with smooth boundaries such that, ∀ i ∈ N, Ωi ⊂ Ω,

+∞ 7

Ωi+1 ⊂ Ωi ,

Ωi = B0

i=0

and λi = λL 1 (Ωi ) → −a as i → +∞ monotonically. Corresponding to the eigenvalues λi there is a sequence of positive eigenfunctions vi such that Δvi + a(x)vi + λi vi = 0 on Ωi , (6.7) vi > 0 on Ωi , vi = 0 on ∂Ωi . Note that, since vi > 0 on Ωi , ∇vi , ν ≤ 0

on ∂Ωi ,

(6.8)

where ν is the outward unit normal to ∂Ωi . By assumption, u > 0 satisﬁes Δu + a(x)u ≤ b(x)uσ

on Ω;

(6.9)

using the second Green’s formula and (6.7), (6.8), (6.9) we have 0≥ u ∇vi , ν = u ∇vi , ν − vi ∇u, ν = uΔvi − vi Δu ∂Ωi ∂Ωi Ωi ≥ −a(x)uvi − λi uvi + a(x)uvi − b(x)vi uσ = − vi u λi + b(x)uσ−1 , Ωi

Ωi

that is,

Ωi

vi u(λi + b(x)uσ−1 ) ≥ 0.

(6.10)

Now, but b(x) ≡ 0 on

8 i

λi + b(x)uσ−1 ≤ −a + b(x)uσ−1 , Ωi = B0 , hence, for i suﬃciently large, λi + b(x)uσ−1 < 0

on Ω1

contradicting (6.10). Conversely, suppose λL 1 (B0 ) > 0. Let D and D be bounded open domains such that B0 ⊂⊂ D ⊂⊂ D ⊂⊂ Ω

6.1. A general procedure

161

and λL 1 (D) > 0. Let u1 be a positive solution of Δu1 + a(x)u1 + λL 1 (D)u1 = 0, u1 = 0,

on D; on ∂D.

(6.11)

Since b(x) > 0 on M \B0 , and Ω\D ⊂⊂ M \B0 , β = inf b > 0. Ω\D

We claim that a suﬃciently large positive constant u2 is a supersolution of (6.6) on Ω\D . Towards this end let α = sup a(x), Ω\D

and note that α < +∞ since Ω is bounded. Then we have ) ≤ u2 (α − βuσ−1 )≤0 Δu2 + a(x)u2 − b(x)uσ2 = u2 (a(x) − b(x)uσ−1 2 2 on Ω\D provided

1 σ−1 α u2 ≥ . β

Now let ψ ∈ C ∞ (M ) be a cut-oﬀ function such that 0 ≤ ψ ≤ 1, ψ ≡ 1 on D , supp ψ ⊂ D. Fix any constant γ > 0 and deﬁne u = γ(ψu1 + (1 − ψ)u2 ).

(6.12)

Since b(x) ≥ 0 and λL 1 (D) > 0, on D , we have

Δu + a(x)u − b(x)uσ = Δγu1 + a(x)γu1 − b(x)(γu1 )σ σ−1 = − λL γu1 < 0 1 (D) + b(x)(γu1 ) independently of the value of γ > 0. Moreover, in Ω\D, Δu + a(x)u − b(x)uσ = Δγu2 + a(x)γu2 − b(x)(γu2 )σ = γ a(x)u2 − b(x)γ σ−1 uσ2 . Now, for γ ≥ 1,

b(x)γ σ−1 ≥ b(x),

thus, on Ω\D, Δu + a(x)u − b(x)uσ ≤ γ(Δu2 + a(x)u2 − b(x)uσ2 ) ≤ 0 because of our choice of u2 . It remains to analyze the situation on D\D . On this set (Δ + a(x))(ψu1 + (1 − ψ)u2 ) ≤ C (6.13)

162

Chapter 6. Existence

for any constant C > 0 suﬃciently large. Now σ

inf b(x)(ψu1 + (1 − ψ)u2 ) > 0

D\D

and we can therefore choose C > 0 suﬃciently large that (6.13) holds together with σ b(x)(ψu1 + (1 − ψ)u2 ) > C −1 (6.14) on D\D . On this set we then have Δu + a(x)u − b(x)uσ = γ(Δ + a(x))(ψu1 + (1 − ψ)u2 ) σ − b(x)γ σ (ψu1 + (1 − ψ)u2 ) ≤ γC − γ σ C −1 = γ C − γ σ−1 C −1 ≤ 0 2

provided γ ≥ C σ−1 . Thus u is a supersolution of (6.6) on Ω.

6.1.3

Two useful lemmas

In the following lemmas we proceed to construct a solution of (6.6). Lemma 6.3. Let a(x), b(x) ∈ C 0,α (M ), 0 < α < 1, and suppose that b(x) ≥ 0 on M and that B0 is bounded. Let Ω ⊃ B0 be a bounded open domain. Having set L = Δ + a(x), assume that (6.15) λL 1 (B0 ) > 0. Then, for every n ∈ (0, +∞), there exists a solution of the problem Δu + a(x)u − b(x)uσ = 0, u > 0 on Ω, σ > 1, u=n on ∂Ω.

(6.16)

Proof. By the deﬁnition of λL 1 (B0 ) and (6.15) there exists an open domain D with ∞ smooth boundary such that B0 ⊂ D ⊂⊂ Ω and λL 1 (D) > 0. Let ψ ∈ C (M ) be a cut-oﬀ function such that 0 ≤ ψ ≤ 1 and ψ≡1

ψ ≡ 0 on M \Ω. Δ Fix N ≥ max supΩ |a(x)| + 1, λΔ 1 (M \Ω) + 1 , where λ1 (M \Ω) ≥ 0 is the bottom of the spectrum of the Laplacian on M \ Ω. Deﬁne on D,

a ¯(x) = ψ(x)a(x) + N (1 − ψ(x)) ¯ =Δ+a and consider the operator L ¯(x). Since a ¯(x) ≡ a(x) on D, ¯

L λL 1 (B0 ) = λ1 (B0 ) > 0.

(6.17)

6.1. A general procedure

163 ¯

Furthermore, since N ≥ λ1 (M \ Ω) + 1, we have λL 1 (M \ Ω) ≤ −1 and it follows that there exists R > 0 suﬃciently large such that Ω ⊂ BR (o)

¯

λL 1 (BR (o)) < 0.

and

¯ on BR (o) relative to the eigenvalue Let ϕ be the normalized eigenfunction of L ¯ (B (o)), so that λL R 1 ¯ + λL¯ (BR (o))ϕ = 0 on BR (o), Lϕ 1

ϕ=0

on ∂BR (o)

and ||ϕ||L2 (BR (o)) = 1. We ﬁx γ > 0 suﬃciently small that

¯

|∇ϕ|2 − a ¯(x)ϕ2 + γb(x)ϕ2 = λL 1 (BR (o)) + γ BR

b(x)ϕ2 < 0, BR

= Δ+a showing that the operator L ¯(x) − γb(x) satisﬁes λL 1 (BR (o)) < 0. Let ψ (B (o)). Then ψ satisﬁes be a positive eigenfunction corresponding to λL R 1 ¯ − γb(x)ψ = −λL (BR (o))ψ ≥ 0 on BR (o), Lψ 1 ψ≡0 on ∂BR (o). If we choose

%−1

$ 0<μ≤γ

1 σ−1

sup ψ

,

BR (o)

then the function v− = μψ satisﬁes ¯(x)v− − b(x)v− σ ≥ 0, v− > 0 on BR (o), Δv− + a v− ≡ 0 on ∂BR (o). ¯

On the other hand, since λL 1 (B0 ) > 0, by Theorem 6.2 there exists v+ > 0 on BR (o) satisfying Δv+ + a ¯(x)v+ − b(x)v+ σ ≤ 0, v+ > 0 on BR (o), v+ ≥ 0 on ∂BR (o). Thus, by the monotone iteration scheme (see section 6.5), we ﬁnd a solution w of the problem Δw + a ¯(x)w + −b(x)wσ = 0, w > 0 on BR (o), w≡0 on ∂BR (o).

164

Chapter 6. Existence

¯(x) ≥ a(x) on Ω, Note that inf ∂Ω w > 0 since Ω ⊂ BR (o). Thus, recalling that a it is easily veriﬁed that if α > is suﬃciently large, then the function w+ = αw satisﬁes Δw+ + a(x)w+ − b(x)w+ σ ≤ 0, w+ > 0 on Ω, w+ ≥ n on ∂Ω. Finally, since w− ≡ 0 is a subsolution of the problem, by the monotone iteration scheme (see section 6.5) we deduce the existence of a solution u ≥ 0 of (6.16). To conclude, note that since u satisﬁes Δu + a(x) − b(x)uσ−1 u = 0 it follows from the strong maximum principle on page 35 of [GT01] that u > 0. Now we produce a solution blowing up at the boundary of Ω. Lemma 6.4. In the assumption of Lemma 6.3 there exists a solution of the problem Δu + a(x)u − b(x)uσ = 0, u > 0 on Ω, σ > 1, (6.18) u = +∞ on ∂Ω. Proof. By standard regularity theory (see [GT01]) it is enough to show that the sequence {un }, with un solution of (6.16), is bounded on any compact subset K of Ω. If K ⊂ Ω\Bo , then we can ﬁnd a ﬁnite covering of balls Bi for K, i = 1, . . . , t such that b(x) > 0 on each Bi . Applying Lemma 4.2 we deduce the existence of a constant C1 > 0 such that un (x) ≤ C1

∀ x ∈ K, ∀ n ∈ N.

(6.19)

It remains to ﬁnd an upper bound of un on a neighbourhood of Bo . Towards this aim, for η > 0 we let Nη = {x ∈ M : d(x, Bo ) < η} where η > 0 is small enough that Nη ⊂ Ω. Furthermore, by the deﬁnition of L λL 1 (Bo ) and the fact that λ1 (Bo ) > 0, we can also suppose to have chosen η so small that λL 1 (Nη ) > 0. Now ∂Nη/2 is closed and bounded, hence compact by the completeness of M , therefore (6.19) holds on ∂Nη/2 for some constant C2 > 0. Let ϕ be a positive eigenfunction corresponding to λL 1 (Nη ). Then, there exists a positive constant μ such that μϕ > C2 on ∂Nη/2 . On Nη/2 we have Δ(μϕ) + a(x)(μϕ) = −λL 1 (Nη )(μϕ) < 0 while, by (6.16), Δun + a(x)un = b(x)uσn ≥ 0. It follows from the generalized maximum principle, [PW67] Theorem 2.10, that un cannot attain an interior positive maximum unless it is constant, the function μϕ and since, un ≤ C2 < μϕ on ∂Nη/2 ,

6.1. A general procedure

165

we conclude that un ≤ μϕ ≤ C3 on Nη/2 with C3 independent of n. This proves the lemma.

6.1.4

Existence of a maximal solution

With the aid of the previous lemma we are ready to prove the following general result; we denote with C 0,α (M ) the space of locally H¨older continuous functions on M with exponent α. 0,α (M ) for some 0 < α ≤ 1. Assume that b(x) ≥ 0 Theorem 6.5. Let a(x), b(x) ∈ Cloc and it is strictly positive outside a compact set, and that

B0 = {x ∈ M : b(x) = 0} 1,2 0 satisﬁes λL 1 (B0 ) > 0 with L = Δ + a(x). If u− ∈ C (M ) ∩ Hloc (M ), u− ≥ 0, u− ≡ 0, is a global subsolution of

Δu + a(x)u − b(x)uσ = 0, σ > 1,

(6.20)

on M , then (6.20) has a maximal positive C 2 -solution . Proof. We ﬁx an exhausting sequence {Dk } ⊂ M by open domains with smooth boundaries such that B0 ⊂ Dk ⊂ Dk ⊂ Dk+1 ∀ k, and for every k we denote by u∞ k the solution of the problem Δu + a(x)u − b(x)uσ = 0 on Dk , u = +∞ on ∂Dk obtained by applying Lemma 6.4. It follows from Proposition 6.1 that ∞ u− ≤ u∞ k+1 ≤ uk

¯ k. on D

(6.21)

Thus {u∞ k } converges monotonically to a function u which solves (6.20), and because of (6.21) u ≥ u− ≥ 0, u− ≡ 0. Thus u ≥ 0 and u ≡ 0. By the maximum principle (see [GT01], page 35) it follows that u > 0 on M . Let now u1 > 0 be a second solution of (6.20) on M . Again by Proposition 6.1, u1 ≤ u∞ k on Dk ∀ k, and therefore u1 ≤ u, as required to show that u is a maximal positive solution. The next result is an immediate consequence of the above reasoning and it will be used in the proof of Theorem 6.8. Theorem 6.6. Let a(x), b(x) ∈ C ∞ (M ) and assume b(x) > 0. If u− ≥ 0, u− ≡ 0 is a subsolution of Δu + a(x)u − b(x)uσ = 0,

σ>1

on M \ BR0 ,

then (6.22) has a maximal positive smooth solution u ≥ u− on M \ BR0 .

(6.22)

166

6.2

Chapter 6. Existence

Subsolutions and existence

In this section we give suﬃcient conditions to guarantee the existence of a positive subsolution of equation (6.20). An application of Theorem 6.5 then yields the existence of a maximal solution.

6.2.1

Existence with λL1 (M ) < 0

The ﬁrst assumption we consider to provide an entire (i.e., global), nonnegative and nontrivial subsolution of (6.20) is the spectral condition (6.23) below. 0,α (M ) for some 0 < α ≤ 1. Assume that Theorem 6.7. Let a(x), b(x) ∈ Cloc b(x) ≥ 0, b(x) > 0 outside a compact set, and that λL 1 (B0 ) > 0 where B0 = {x ∈ M : b(x) = 0} and L = Δ + a(x). Furthermore assume

λL 1 (M ) < 0.

(6.23)

Δu + a(x)u − b(x)uσ = 0, σ > 1

(6.24)

Then the equation possesses a minimal and a maximal (possibly coinciding) positive solutions. Proof. According to Theorem 6.5 it is enough to prove that there exists a minimal solution. Since λL 1 (M ) < 0, we can ﬁnd a relatively compact domain with smooth boundary Ω0 ⊃ B0 such that λL 1 (Ω0 ) < 0. Arguing exactly as in the proof of Lemma 6.3, we see that if γ > 0 is suﬃciently small, then

λL 1 (Ω0 ) < 0, = Δ + a(x) − γb(x) so that, if ψ is a positive eigenfunction corresponding where L L to λ1 (Ω0 ), then Δψ + a(x)ψ ≥ γb(x)ψ on Ω0 , ψ≡0 on ∂Ω0 −1 1 and, choosing μ ≤ γ σ−1 supΩ0 ψ , the function v− deﬁned by v− = μψ satisﬁes Δv− + a(x)v− ≥ b(x)v− σ on Ω0 , (6.25) v− ≡ 0 on ∂Ω0 . Since λ1 (B0 ) > 0, Theorem 6.2 guarantees the existence of a supersolution v+ > 0 to (6.25), and possibly multiplying v+ by a large positive constant we can also suppose that v+ ≥ v− on Ω0 . By the monotone iteration scheme (see section 6.5), we deduce the existence of a positive solution u0 ∈ C 2 (Ω0 ) of Δu0 + a(x)u0 − b(x)uσ0 on Ω0 , (6.26) u0 = 0 on ∂Ω0 .

6.2. Subsolutions and existence

167

Now let Ω1 ⊃ Ω0 ⊃ Ω0 . By domain monotonicity L λL 1 (Ω1 ) ≤ λ1 (Ω0 ) < 0

and the above procedure yields the existence of a positive solution u1 of (6.26) on Ω1 . Since ∂Ω0 ⊂ Ω1 , u0 = 0 < u1 on ∂Ω0 , it follows by Proposition 6.1 that u0 ≤ u1 on Ω0 . Choosing an increasing exhaustion of M by relatively compact domains +∞ with smooth boundaries {Ωi }i=0 , the above procedure produces a sequence {ui } of solutions of (6.26) on Ωi satisfying ui ≤ ui+1

on Ωi .

Using the procedure of Lemma 6.4 we see that {ui } is uniformly bounded on Ωk for i ≥ k + 1; therefore ui converges to u > 0, solution of (6.24). The argument is any other solution of used to prove the monotonicity of {ui } shows that if u (6.24), then u ≥ ui on Ωi ∀ i ≥ 0. Thus u ≥ u and u is the required positive minimal solution of (6.24). In the same vein we prove the following theorem (see [BRS98]). Theorem 6.8. Let (M, , ) be a complete manifold of dimension m ≥ 3 satisfying δ/2 Ric ∇r, ∇r ≥ −(m − 1)H 2 1 + r2

on M,

(6.27)

for some constants H > 0 and δ ≥ −2. Let a(x), b(x) ∈ C ∞ (M ), b(x) > 0 on M and suppose that, for some constants γ ≤ 0, μ < 1 − 2δ , a(x) >0 r(x)→+∞ r(x)−μ lim inf

(6.28)

and b(x)

lim sup

4γ

r(x)→+∞

Finally, given 1 < σ ≤

m+2 m−2 ,

r(x)−μ− m−2

let Lσ = Δ +

< +∞.

(m−2)(σ−1) a(x) 4

(6.29) and assume that

σ λL 1 (M ) < 0.

(6.30)

Δu + a(x)u − b(x)uσ = 0

(6.31)

Then equation has a positive solution u ∈ C ∞ (M ) on M satisfying lim inf r(x)→+∞

u(x) 4γ

r(x) (m−2)(σ−1)

> 0.

(6.32)

168

Chapter 6. Existence

Remark. Two comments are in order. First of all, the theorem establishes the existence of a solution to (6.31) satisfying an explicit lower bound at inﬁnity. In this respect it is worth noticing that, if δ > −2 and μ < −δ, then any positive solution u of (6.31) satisﬁes the estimate (6.32) by the a priori estimates of Theorem 4.1. However in case δ ≤ μ < 1 − 2δ , which is allowed since δ > −2, the lower bound (6.32) does not follow from Theorem 4.1, and in this case one has to construct the appropriate subsolution as in the proof of Theorem 6.8. The second observation concerns the validity of the spectral assumption (6.30). Note that in general, if 0 ≤ σ1 ≤ σ2 , then, by the variational characL L terization of the bottom of the spectrum, λ1 σ1 (M ) ≥ σσ12 λ1 σ2 (M ) (see the proof ∞ of Theorem 2 in [FCS80]). Indeed, for every φ ∈ C0 (M ), σ1 σ1 |∇ψ|2 − σ1 a(x)ψ 2 = |∇ψ|2 − σ2 a(x)ψ 2 ≥ |∇ψ|2 − σ2 a(x)ψ 2 σ σ 2 2 M M M and the claim follows taking the inﬁmum over all such ψ. Thus assumption (6.30) implies the condition λL 1 (M ) ≥ 0 considered in Theorem 6.7. Proof. Again according to Theorem 6.5 it suﬃces to ﬁnd a nonnegative subsolution of (6.31) satisfying (6.32). We divide the proof into several steps. For the sake of simplicity we assume that the M has a pole o, i.e., a point such that exp : To M → M is a diﬀeomorphism (see for instance [dC92], [GW79]). The general case is dealt with noting that, applying Lemma 1.12, the radial functions which are shown to be pointwise super-, respectively subsolutions within the cut locus, are in fact global super-, respectively subsolutions in the weak sense, and this is all that is needed to carry out the argument. Step 1: reduction to the case where σ =

m+2 m−2 .

1,2 (M ) which It suﬃces to show that there is a positive function v− ∈ C 0 (M ) ∩ Hloc satisﬁes weakly on M ,

Lσ v− −

m+2 (m − 2)(σ − 1) b(x)v−m−2 ≥ 0 4

(6.33)

and v− (x) ≥ Cr(x)γ

for r(x) 1

(6.34)

and some constant C > 0. Indeed, since 1 < σ ≤ a straightforward computation shows that if v− satisﬁes (6.33), then the function u− deﬁned by m+2 m−2 ,

4

u− = v− (m−2)(σ−1) satisﬁes Δu− + a(x)u− − b(x)uσ− 4 4 4 2 ≥ − 1 v− (m−2)(σ−1) −2 |∇v− | ≥ 0 (m − 2)(σ − 1) (m − 2)(σ − 1)

6.2. Subsolutions and existence

169

weakly on M , and therefore u− is a subsolution of (6.31) satisfying (6.32). Step 2: construction of a subsolution outside a compact set. Consider w(x) = r(x)γ = β(r(x)) for r(x) ≥ 1. Clearly w(x) satisﬁes the lower bound (6.34). We claim that there exist δ0 > 0 and R0 suﬃciently large such that, ∀ 0 < δ ≤ δ0 , the function v6 = δw satisﬁes (6.33) weakly on M \ BR0 . Indeed, from (6.27), the Laplacian comparison > H of Proposition 1.15, there theorem and Proposition 1.15, having ﬁxed H suﬃciently large such that exists R0 = R0 (H) δ/2 Δr(x) ≤ (m − 1)Hr(x)

on M \ BR0 .

Using (6.28) and (6.29) we can also suppose to have chosen R0 so large that a(x) ≥ A1 r(x)−μ

4γ

b(x) ≤ A2 r(x)−μ− m−2

and

(6.35)

on M \BR0 for some constants A1 , A2 > 0. Since β ≤ 0 it follows that, on M \BR0 , (m − 2)(σ − 1) a(x)w 4 (m − 2)(σ − 1) a(x)β = β + (Δr)β + 4

Lσ w = Δw +

γ−1+δ/2 + ≥ γ(γ − 1)rγ−2 + (m − 1)Hγr

(m − 2)(σ − 1) A1 rγ−μ. 4

Thus, in case μ < 1 − 2δ we deduce, up to choosing R0 suﬃciently large, that there exists a constant C > 0 such that Lσ w ≥ Crγ−μ

on M \ BR0 .

To conclude note that, having set m+2

bw = w− m−2 Lσ w, we have

4γ

bw ≥ Cr−μ− m−2 Since

on M \ BR0 . 4

bδw = δ − m−2 bw , it follows from (6.35) that, if δ is suﬃciently small, bδw ≥ b(x)

(m − 2)(σ − 1) 4

on M \ BR0

(6.36)

170

Chapter 6. Existence

as required to show that there exists δo suﬃciently small that for every δ < δo , v6 solves (6.33) weakly on M \ BR0 , and satisﬁes (6.34). Step 3: construction of a weak subsolution v− on M . σ Since by assumption λL 1 (M ) < 0, we can ﬁnd a relatively compact set with smooth σ boundary Ω ⊃ BR0 such that λL 1 (Ω) < 0. Let ϕ be a corresponding eigenfunction, so that Δϕ + (m+2)(σ−1) a(x)ϕ = −λϕ, ϕ > 0 in Ω, 4 ϕ = 0 on ∂Ω.

Arguing as above, it follows that if η > 0 is suﬃciently small, then ϕη = ηϕ is a subsolution of (6.33) on Ω. Finally let δ < δ0 be small enough that the function v6 < ϕη in BR0 + \ BR0 . Note that since v6 > 0 on ∂Ω there exists an open set D with D ⊂ Ω such that v6 > ϕη in Ω \ D. We claim that the function v− deﬁned by ⎧ ⎪ in BR0 + , ⎨ϕη v− = max {6 v , ϕη } in Ω \ BR0 , ⎪ ⎩ v6 in M \ D is the required subsolution. Note ﬁrst of all that v− is well deﬁned by construction, and it is clearly a subsolution in BR0 + and in M \ D. On the other hand, v− is a subsolution in Ω \ BR0 as maximum of two subsolutions, by Theorem 6.20.

6.2.2 λL1 (M ) < 0: some suﬃcient conditions The previous Theorems 6.7 and 6.8 point out the relevance of condition (6.23) in guaranteeing the existence of positive solutions to (6.24). It is therefore interesting to ﬁnd suﬃcient conditions in order that (6.23) holds. We begin with the following elementary result. Proposition 6.9. Let (M, , ) be a complete manifold, a(x) ∈ C 0 (M ), and L = Δ + a(x). Set v(r) = vol(∂Br ) and denote by 1 a(x) (6.37) a ¯(r) = vol(∂Br ) ∂Br the spherical mean of a(x). Let A(r) ≤ a ¯(r) and suppose that α is a solution of the problem (v(r)α ) + v(r)A(r)α ≥ 0, (6.38) α(0) = α0 > 0, α (0) = 0, and has a ﬁrst zero at r = T > 0. Then λL 1 (M ) < 0.

(6.39)

6.2. Subsolutions and existence

171

Proof. We consider the geodesic ball BT (o) and deﬁne ϕ(x) = α(r(x)). Then, using the co-area formula (1.87) and (6.38) and integrating by parts we compute |∇ϕ|2 − a(x)ϕ2 = |∇ϕ|2 − a ¯(r)ϕ2 BT BT ≤ |∇ϕ|2 − A(r)ϕ2

BT T

= 0

[(α )2 v(r) dr − A(r)α2 v(r)] dr

=−

T 0

[(v(r)α ) + A(r)αv(r)]α dr ≤ 0.

By the Rayleigh characterization λL 1 (BT ) ≤ 0 and by domain monotonicity we get (6.39). Recalling that the solution of an ordinary diﬀerential equation is said to be oscillating if is has an inﬁnite number of zeroes, we observe that, in order to show that a solution α of (6.38) has a ﬁrst zero at T > 0, any oscillation criterion would do. For instance: Proposition 6.10. Let α be any solution of (6.38) with equality on [0, +∞) and assume that v1 ∈ L1 (+∞). Then α is oscillating provided !

+∞

lim inf A(r) v(r) r→+∞

r

ds v(s)

"2 >

1 . 4

(6.40)

Proof. We perform the standard change of variables

+∞

t = K(r) = r

ds v(s)

−1 .

(6.41)

Then, K : (0, +∞) → (0, +∞) is strictly increasing and setting γ(t) = tα K −1 (t) a direct computation shows that γ satisﬁes γ + p(t)γ = 0 with p(t) =

v 2 (K −1 (t)A(K −1 (t)) . t4

(6.42)

172

Chapter 6. Existence

An application of Cauchy’s theorem, together with the deﬁnition of K and (6.40) shows that +∞ 1 p(s) ds > . lim inf t t→+∞ 4 t It follows from the Hille-Nehari oscillation theorem, see [Swa68] Chapter 2, that α is oscillating. Remark. We note that in the above oscillating result A(r) has necessarily to be positive in a neighborhood of +∞, but neither here nor in Proposition 6.9 is A(r) required to have a deﬁnite sign. Suppose now we have an estimate of the type Δr ≤ (m − 1)

h (r), h

h > 0 on (0, +∞)

which would typically follow from an appropriate lower bound for the Ricci curvature. Consider a solution of hm−1 (r)β + hm−1 (r)A(r)β = 0, β (0) = 0, β(0) = β0 > 0 and let T be the ﬁrst zero of β. We set ψ(x) = β(r(x)); then, if A(r) ≤ a ¯(r), ¯(r)β ≥ β + Δrβ + A(r)β. Δψ + a ¯(r(x))ψ = β + Δrβ + a Therefore, if β ≤ 0 (which is necessarily the case if A(r) ≥ 0 satisﬁes Δψ + a ¯(r(x))ψ ≥ 0. Since ψ is radial, using again the co-area formula, |∇ψ|2 − a(x)ψ 2 = |∇ψ|2 − a ¯ψ 2 = − BT

BT

on [0, T )), then ψ

ψ(Δψ + a ¯ψ) ≤ 0 BT

and we conclude that λL 1 (BT ) ≤ 0. Note that in this case v(r) ≤ Chm−1 (r). We conclude this short account by quoting some new results in oscillation theory obtained in [BMR09] and [MRV], which we refer to for a complete treatment. With the notation of Proposition 6.9, let v(r) = vol(∂Br ) and let a ¯(r) be the

6.2. Subsolutions and existence

173

spherical mean of the potential a. Suppose that there exists an oscillating solution z ∈ Liploc of the problem (v(t)z (t)) + a ¯(t)v(t)z(t) = 0 on (0, +∞), (6.43) z (t) = O(1) as t 0+ , z(0+ ) = z0 > 0, and let 0 < t2 < t3 be two consecutive zeros, that is, z(t2 ) = z(t3 ) = 0 and z = 0 in (t2 , t3 ). Similarly to what we did in Proposition 6.9, we consider the function ϕz : M → R deﬁned as z(r(x)) r(x) ∈ [t2 , t3 ] ϕz (x) = 0 r(x) ∈ [0, +∞) \ [t2 , t3 ]. Then, L

λ1

#

¯t ≤ B t3 \ B 2

¯t Bt 3 \ B 2

# t3 =−

t2

2

|∇ϕz | − # ¯t Bt 3 \ B 2

#

¯t Bt3 \B 2 2 ϕz

aϕ2z

# t3 =

t2

¯vz z (vz ) + a = 0. # t3 vz 2 t2

#t v(z )2 − t23 a ¯vz 2 # t3 vz 2 t2

Therefore, by the domain monotonicity of eigenvalues, the oscillation of the solution z implies λL 1 (M \ BR ) < 0,

for all R ≥ 0.

(6.44)

Recall now that the index of L is deﬁned as the number of negative eigenvalues of −L. Hence condition (6.44), together with a result by D. Fisher-Colbrie [FC85] shows that L has inﬁnite index, that is indL (M ) = sup {indL (Ω)} = +∞. Ω⊂⊂M

In fact the conclusion follows without appealing to [FC85] by noting that taking pairs of consecutive zeros of z the corresponding functions ϕz constructed above are L2 -orthogonal and this is enough to conclude that indL (M ) = +∞. Now ﬁx a constant B ≥ 0 and deﬁne, for every t1 , t2 ∈ [0, +∞], V (t1 , t2 ) = e2B

t2 t1

ds v(s)

.

(6.45)

Then we have the following ([MRV], Theorem 14) Theorem 6.11. Let V and B as in deﬁnition (6.45) and suppose that the spherical mean a ¯(r) of a(x) satisﬁes a ¯(r)v 2 (r) ≥ −B 2 for all r > 0.

174

Chapter 6. Existence

i) If there exist 0 < a < b such that 2B a(x)dx > (b,+∞) 2B V V(b,+∞)−1 Bb \Ba

if v −1 ∈ / L1 (+∞), −1 if v ∈ L1 (+∞),

then λL 1 (M ) < 0. ii) L is unstable at inﬁnity, i.e., λL 1 (M \ BR ) < 0 for every R > 0, provided either v ∈ L1 (+∞) and, for some R > 0, / ∞ ds > 1, lim sup a(x)dx v(s) t→∞ Bt \BR t or v ∈ / L1 (+∞) and lim

t→∞

/

a(x)dx

sup

t≤q1
> 2B.

Bq2 \Bq1

In particular, in the assumption ii) L has inﬁnite index. Theorem 6.11 is a consequence of Theorems 6.12 and 6.14 below which are proved in [MRV] (see also Proposition 1.2 and Theorem 1.4 in [BMR09]). Theorem 6.12. Let v(t) and a ¯(t) ∈ L∞ loc ([0, +∞)) be such that v(t) ≥ 0,

1 ∈ L∞ loc ((0, +∞)), v(t)

1 ∈ / L1 (0+ ), v

and a ¯(t) ≥ −

lim v(t) = 0

t→0+

B2 v(t)2

(6.46)

(6.47)

for some real constant B ≥ 0. Let z(t) ∈ Liploc ([0, +∞)) be a solution of problem (6.43). If z(t) = 0 for all t ∈ (0, +∞), then for every 0 ≤ a < b, b 2B if v1 ∈ / L1 (+∞), a ¯(s)v(s)ds ≤ V (b,+∞) 1 2B V (b,+∞)−1 if v ∈ L1 (+∞). a In order to prove Theorem 6.12 let us recall the following Riccati comparison result ([MRV], Lemma 18; compare also with Lemma 1.10 in Chapter 1): Lemma 6.13. Let G, v ∈ C 0 ([0, +∞)), v > 0, and qi ∈ AC(T, Ti ), i = 1, 2, be solutions of the Riccati diﬀerential inequalities q1 (t) ≥ G(t) +

1 2 q (t), v(t) 1

q2 (t) ≤ G(t) +

1 2 q (t) v(t) 2

(6.48)

6.2. Subsolutions and existence

175

a.e. in (T, Ti ) satisfying q1 (T ) = q2 (T ).

(6.49)

Then T1 ≤ T2 and q1 (t) ≥ q2 (t) in [T, T1 ). Conversely, if qi ∈ AC(Ti , t¯), i = 1, 2, are solutions of (6.48) a.e. in (Ti , T ) satisfying q1 (T ) = q2 (T ), then T1 ≥ T2 and q1 (t) ≤ q2 (t) in (T1 , T ]. A proof of the lemma is a minor modiﬁcation of that of Corollary 2.2 in [PRS08]. We are now ready for the Proof of Theorem 6.12. Since, by assumption, z(t) > 0 on (0, +∞), the function y(t) = −v(t)

z (t) z(t)

is well deﬁned on (0, +∞) and it is locally Lipschitz, as can be immediately seen by integrating (6.43) once. Furthermore, y satisﬁes y =

y2 +a ¯(t)v(t) v(t)

(6.50)

so that, according to (6.47), y ≥

y2 − B 2 v(t)

on (0, +∞).

(6.51)

First we consider the case v1 ∈ L1 (+∞). Since, by (6.46) we also have v1 ∈ L1 (0+ ), it follows that given any ﬁxed α ∈ (0, +∞) there exists tα > 0 such that tα 1 ds = log α. (6.52) v(s) 2B 1 With the notation introduced in (6.45) deﬁne yα (t) = B

α + V (1, t) α − V (1, t)

and observe that, because of (6.52), i) yα t+ α = −∞; Furthermore yα satisﬁes yα =

ii) yα t− α = +∞. yα2 − B 2 v(t)

(6.53)

(6.54)

(6.55)

where deﬁned. We claim that −B ≤ y(t) ≤ B

on (0, +∞).

(6.56)

176

Chapter 6. Existence

By contradiction assume that there exists T ∈ (0, +∞) such that y(T ) = Y > B, the case Y < −B being similar. Let Y +B V (1, T ) Y −B

(6.57)

yα (T ) = Y = y(T )

(6.58)

α= and observe that

and that yα will thus be deﬁned on [T, T2 ) for some maximal T2 ≤ +∞. Because of (6.51), (6.55) and (6.58) we can apply the Riccati comparison lemma above 2 with the choices q1 = y, q2 = yα , G = − Bv2 and T = T . Noting that y is deﬁned on [T, +∞) and applying the ﬁrst part of the lemma we have that yα is deﬁned on [T, +∞). However, from (6.57) and (6.52) we have tα > T and we obtain a contradiction using (6.54) ii) (in case Y < −B we would have obtained a contradiction via (6.54) i)); thus we have the validity of (6.56). From (6.50) we obtain y ≥ a ¯(t)v(t). (6.59) Fix any 0 ≤ a < b as in the statement of the theorem and integrate (6.59) on [a, b]. Using (6.56) we have

b

a ¯(s)v(s) ds ≤ y(b) − y(a) ≤ 2B.

(6.60)

a

Suppose now only for

1 v

∈ L1 (+∞). In this case, there exists tα > 0 such that (6.52) holds α ∈ (0, V (1, +∞)).

(6.61)

We claim that −B ≤ y(t) ≤ B

V (t, +∞) + 1 V (t, +∞) − 1

on (0, +∞).

(6.62)

As above, we only prove the right-hand side inequality, the left-hand inequality being proved in a similar way. Assume by contradiction that there exists T ∈ (0, +∞) such that V (T, +∞) + 1 y(T ) = Y > B . V (T, +∞) − 1 Note that, since V (1, +∞) > 1, if we let α=

Y +C , Y −C

6.2. Subsolutions and existence

177

then, for C > 0 suﬃciently small, α < V (1, +∞). It follows that the function yα = C

α + V (1, t) α − V (1, t)

satisﬁes the Riccati equation (6.55), yα (T ) = Y and there exists tα > T such that yα (t− α ) = +∞. Applying the Riccati comparison lemma and arguing as above we obtain a contradiction which proves that y(t) satisﬁes the required upper estimate. Integrating (6.59) as before we get

b

a ¯(s)v(s) ds ≤ a

2BV (b, +∞) . V (b, +∞) − 1

(6.63)

The estimate of Theorem 6.12 now follows from (6.60) and (6.63).

The proof of the next result can be obtained by iterating the technique of the proof above; for details we refer to the original proof of Theorem 10 in [MRV]. Theorem 6.14. Let v, a ¯ and z be deﬁned as in Theorem 6.12. Then z is oscillating provided either v1 ∈ L1 (+∞) and &

∞

a ¯(s)v(s)ds

lim sup t→∞

t R

t

ds v(s)

' >1

for some R > 0, or v −1 ∈ / L1 (+∞) and & ' q2 lim sup a ¯(s)v(s)ds > 2B. t→∞

t≤q1
(6.64)

(6.65)

q1

6.2.3 A more general case We continue with the general strategy of applying Theorem 6.5 to guarantee the existence of solutions. In the next result we replace the spectral assumption λL 1 (M ) < 0 with a pointwise condition. It should be pointed out that, as the case of hyperbolic space shows, the two assumptions are related but independent. Theorem 6.15. Let (M, , ) be a complete manifold, a(x), b(x) ∈ C 0,μ (M ), 0 < μ ≤ 1. Assume that a(x) > 0 on M and b(x) ≥ 0. Assume also that the Ricci tensor satisﬁes δ/2 Ric ≥ −(m − 1)H 2 1 + r(x)2 on M, (6.66) for some δ ≥ −2 and H > 0. Let B0 = {x ∈ M : b(x) = 0} and assume that λL 1 (B0 ) > 0

(6.67)

178

Chapter 6. Existence

and θ

i) a(x) ≥ Ar(x)α ;

ii) b(x) ≤ Br(x)β eDr(x) , for r 1, (6.68) for some α > 2δ − 1, 0 < θ < min 1 + α − 2δ , 1 + α2 , and A, B, D > 0, β ∈ R. Then there exists a positive maximal solution u on M of Δu + a(x)u − b(x)uσ = 0, σ > 1.

(6.69)

Remark. Consider the case of the hyperbolic m-dimensional space Hm −H 2 of constant negative curvature −H 2 . In this case the parameter α must satisfy α > −1, so that a behavior of a(x) ≥ 0 of the type Ar(x)−1+ε ≤ a(x) ≤ C(1 + r(x))

−1+2ε

for r(x) 1 and some ε > 0 is admissible. In this case, for a(x) suﬃciently small, λL 1 (M ) ≥ 0, with L = Δ + a(x). This shows that Theorem 6.15 is not contained in Theorem 6.7 when λL 1 (M ) < 0. Remark. The upper bound for θ is sharp in the sense that, in the case of equality, m+2 , α = 0, δ = 0 and existence may fail; see for instance Theorem 4.8 where σ = m−2 2 θ = 1, and D = 2B, β = m−1 + γ, γ > 0. Proof of Theorem 6.15. According to Theorem 6.5 we only need to construct a global subsolution u− ≥ 0, u− ≡ 0. We observe ﬁrst of all that we may assume that a(x) ≥ A(1 + r(x))α

b(x) ≤ B(1 + r(x))β eDr(x)

θ

on M . Because of (6.66), Proposition 1.15 and Theorem 1.11 imply that δ/2 Δr ≤ Hr and r ≥ R. On the other hand, by the asymptotic behavior for some constant H of the metric coeﬃcients in geodesic polar coordinates we have Δr =

m−1 + o(1) as r → 0+, r

and there exists a constant C such that 1 + rδ/2 weakly on M. Δr ≤ C r It follows that, if w(r) is a nonincreasing positive C 2 function with w (0) = 0 satisfying 1 (6.70) + rδ/2 w (r) + A(1 + r)α w − B(1 + r)β wσ ≥ 0, w + C r

6.2. Subsolutions and existence

179

then the function u− (x) = w(r(x)) is the required (weak) subsolution. We look for a function w of the form w(r) = (μ + P (r))ξ with P (r) ≥ 0, P (0) = 0, P (r) ≥ 0 and P (r) = eDr for r ≥ R, where μ > 0 is a constant to be chosen later, D > 0 and θ are the constants in (6.68) and 1 ξ < 1−σ < 0. Letting, for ease of notation θ

1 Hw = w + C( + rδ/2 )w (r) + A(1 + r)α w, r we claim that, for an appropriate choice of μ > 0, we have Hw (r) > 0 and

θ Hw (r) ≥ Crβ eDr w(r)σ

on [0, +∞)

for some C > 0 and r 1,

(6.71)

(6.72)

It follows that if E is a suﬃciently small positive constant, then Ew satisﬁes (6.70). Indeed, with obvious notation, HEw (r) Hw (r) = E 1−σ (Ew(r))σ (w(r))σ and using (6.71) and (6.72) we may choose E small enough that θ HEw (r) ≥ B(1 + r)β eDr σ (Ew(r))

on [0, +∞). We compute w (r) = ξ(μ + P (r))ξ−1 P (r)

w (r)

= ξ(ξ − 1)(μ + P (r))ξ−2 (P (r))2 + ξ(μ + P (r))ξ−2 P (r) and inserting this in the expression of Hw it is easy to see that we may choose μ large enough that Hw > 0 on any chosen interval [0, R]. On the other hand, using the expression of P (r) for r large, a straightforward computation shows that

θ ξ(1−θ) Hw (r) ∼ Arα μ + eDr , σ w(r) 1 so that the conditions ξ < 1−σ < 0 and D, θ > 0 easily imply (6.72). Thus we may ﬁrst ﬁnd R such that (6.72) holds in [R, +∞) with a constant C > 0. Next, if necessary we may increase μ in order that (6.71) holds on [0, R]. It is clear that (6.72) will continue to hold with a possibly smaller C.

180

Chapter 6. Existence

Remark. We note that if we replace the upper bound on the coeﬃcient b(x), then the same procedure works with diﬀerent choices of w(r). For instance, if we assume that b(x) ≤ B(1 + r)β on M, then the same arguments show that one may ﬁnd a subsolution of the form u− (x) = w(r(x)) with w(r) = (μ + r2 )−γ provided β + α ≤ 2γ(σ − 1). Clearly the solution u supplied by the theorem satisﬁes the lower bound u(x) ≥ C(μ + r2 )−γ . In applications to the Yamabe problem this is relevant in connection to the completeness of the conformal metric.

6.3

Global sub- and supersolutions

We now give another version of Theorem 6.15, where it is no longer assumed that b is nonnegative. This will require the explicit construction of a global supersolution, and the procedure will prove to be rather delicate. The weakening of the assumptions on b is also reﬂected in the stronger curvature conditions and an upper bound imposed on the coeﬃcient a. Theorem 6.16. Let (M, , ) be a complete manifold of dimension m ≥ 4 satisfying Ric ≥ −(m − 1)H 2 for some constant H > 0. Let a(x), b(x) ∈ C 0,μ (M ), 0 < μ ≤ 1, σ ≥ suppose that, for some R0 > 0 we have

(6.73) m+2 m−2

and

2

a(x) ≤

(m − 1) H 2 m−2 σ−1

b(x) > 0

on M \ BR0 ,

on M \ BR0 ,

Δ+ m−2 4 (σ−1)(a(x))

λ1

(BR0 ) > 0,

a(x) ≥ A(1 + r(x))α θ

b(x) ≤ Cr(x)β eDr(x)

(6.74) (6.75) (6.76)

on M,

(6.77)

for r(x) 1

(6.78)

and for some constants A, C, D > 0, β ∈ R, −1 < α ≤ 0, θ < 1 + α. Then, there exists η > 0 such that, if b(x) ≥ −η on M , the equation Δu + a(x)u − b(x)uσ = 0 admits a positive solution u on M .

(6.79)

6.3. Global sub- and supersolutions

181

Proof. As usual, it suﬃces to construct sub- and supersolutions v− and v+ in Liploc such that 0 < v− ≤ v+ . The construction of the subsolution is done as in the proof of Theorem 6.15. Note that in this respect allowing b to take negative values only helps matters. We recall that the function v− constructed in Theorem 6.15 is radial and satisﬁes lim v− (r) = 0. r→+∞

Recalling that for every E < 1, Ev− is still a subsolution shows that the condition v + ≥ v− can always be satisﬁed whenever v+ > 0 on M

and

lim v+ (r(x)) = +∞.

r+∞

We therefore concentrate on the construction of the supersolution. We ﬁrst m+2 . Let a(x) = (m−2)(σ−1) a(x) and reduce the analysis to the case where σ = m−2 4 b = (m−2)(σ−1) b(x). Then a simple computation shows that, if u+ is a positive 4 supersolution of m+2 Δu + a(x)u − b(x)u m−2 = 0 (6.80) on M , then

4

v+ = u+(m−2)(σ−1) is a positive supersolution of (6.79) provided σ ≥ (6.76), we have Δ+ a(x) λ1 (BR0 ) > 0. Furthermore,

(6.81) m+2 m−2 .

Note that, because of (6.82)

b(x) > 0

(6.83) on M \ BR0 where B is the appropriate and if b(x) ≥ −B on M for some B > 0, then b ≥ −B multiple of B. We now proceed to construct a supersolution of (6.80) in BR0 and in M \ BR0 , and then glue them together to form a global supersolution. We divide the argument, which is adapted from the proof of Theorem 0.1 in [RRV97], into several steps. Step 1. Construction of a supersolution of (6.80) inside BR0 . If there exists x0 ∈ BR0 such that a(x0 ) > 0,

(6.84)

it follows by results of J. Escobar (see [Esc87], Theorems 3.2 and 4.2), that there exists a solution u > 0 of the problem m+2 Δu + a(x)u + u m−2 = 0 on BR0 , (6.85) u=0 on ∂BR0 , u > 0 on BR0 .

182

Chapter 6. Existence

On the other hand, if a ≤ 0 on BR0 , then, using the fact that the λΔ 1 (BR0 ) >, we a1 can choose a small enough constant a1 > 0 such that λΔ+ (BR0 ), and apply the 1 above argument to obtain a solution u > 0 of problem (6.85) with a1 instead of a(x). Then m+2 m+2 Δu + a(x)u + u m−2 ≤ Δu + a1 (x)u + u m−2 = 0 and we conclude that (6.85) always admits a positive supersolution u vanishing on ∂BR0 . Step 2. Construction of a radial supersolution on M \ BR0 . Using the Ricci curvature assumption (6.73) and the Laplacian comparison theorem we have Δr ≤ (m − 1)H coth(Hr) on M. (6.86) Moreover, it follows from (6.74) that a≤6 a=

(m − 1)2 2 H 4

on M \ BR0 .

(6.87)

Finally, we choose a positive, decreasing function 6b(t) on [0, +∞) of the form 6b(t) = e−P (t)

(6.88)

with P ∈ C 2 ([0, +∞)) satisfying P (0) = P (0) = 0, P ≥ 0, P ≥ 0, P (t) → +∞ as t → +∞ and such that

b ≥ 6b(r(x))

(6.89)

on M \ BR0 .

(6.90)

We look for a supersolution of (6.80) on M \ BR0 of the form v(x) = β(r(x)) with β : [0, +∞) → R, where β solves the problem m+2 β + (m − 1) coth(Hr)β + 6 aβ − 6b(r)β m−2 ≤ 0, (6.91) β (0) = 0, β ≥ 0, β(r) > 0. Furthermore, we require lim inf β(r) > 0.

(6.92)

r→+∞

Indeed, if β solves (6.91), it follows from (6.86), (6.87) and (6.88) that v is positive and satisﬁes m+2 m+2 Δv + a(x)v − b(x)v m−2 = β + Δrβ + a(x)β − b(x)β m−2

≤ β + (m − 1)H coth(Hr)β + 6 aβ − 6b(r)β m−2 ≤ 0 m+2

on M \ BR0 and lim inf v(x) > 0. r→+∞

(6.93)

6.3. Global sub- and supersolutions

183

The previous computation makes it clear that the requirement β ≥ 0 in (6.91) is vital for our approach to work. As we shall see this will be a consequence of the deﬁnition (6.88) of 6b . To solve (6.91) we choose an increasing divergent sequence {Tn } ⊂ (0, +∞). For a given n we look for a positive solution on [0, Tn ] of m+2 a(r)β − 6b(r)β m−2 = 0, β + (m − 1) coth(Hr)β + 6 (6.94) β (0) = 0, β(Tn ) = i ∈ N. To motivate the argument that follows assume that w > 0 is a solution of the problem m+2 (6.95) Δw + a ˇw − ˇbw m−2 = 0 on an open set Ω of a manifold (N, g), and let ϕ > 0 be a C 2 solution of Δϕ + a ˇ ϕ + λ1 ϕ = 0

on Ω.

(6.96)

If we deﬁne the conformally related metric 4

g = ϕ m−2 g,

(6.97)

and use the transformation law (3.81) and equations (6.95), (6.96), an elementary computation shows that the function

satisﬁes

u = ϕ−1 w,

(6.98)

m+2 4 − λ1 ϕ− m−2 u − ˇbu m−2 = 0 on Ω, Δu

(6.99)

denotes the Laplace-Beltrami operator of g. Hence, if u is a positive where Δ solution of (6.99), then w = ϕu is a positive solution of (6.95). Note now that (6.94) is precisely the expression of the equation ΔHm

u+6 au − 6bu m−2 = 0 m+2

−H 2

2 on the m-dimensional hyperbolic space Hm −H 2 of constant curvature −H when u(x) = β(r(x)) is radial. We also observe that the ﬁrst eigenvalue of the Dirichlet Laplacian on the ball BR in Hm −H 2 satisﬁes Δ Hm

λ1

−H 2

(BR ) ≥

(m − 1)2 2 H [coth(HR)]2 . 4

Indeed, ΔHm

−H 2

r = (m − 1)H coth(Hr)

(6.100)

184

Chapter 6. Existence

and a standard application of the divergence theorem shows that if φ is a smooth function with compact support in BR , then 2 (m − 1)H coth(HR) φ ≤ φ2 ΔHm 2 r −H

= −2

φ∇φ, ∇r ≤ 2

φ

2

1/2 |∇φ|

and the claim follows. It follows that for some λ1 > 0 we have a solution ϕ of aϕ + λ1 ϕ = 0 on BTn+1 , ΔH m 2 + 6 −H

ϕ=0

on ∂BTn+1 , ϕ > 0 on BTn+1

2

1/2 ,

(6.101)

and the radial symmetry of the problem implies that the eigenfunction ϕ is radial. The above discussion shows that in order to ﬁnd a solution of (6.94) we may let (N, g) be the hyperbolic space Hm −H 2 and look for a radial solution of

m+2 4 − λ1 ϕ− m−2 Δu u − 6bu m−2 = 0 on BTn , u > 0, u ≡ ϕ(Ti n ) on ∂BTn .

(6.102)

Since λ1 > 0, the functional associated to (6.102) is strictly convex and unbounded at inﬁnity. Thus problem (6.102) admits a radial solution u. It follows that problem (6.94) admits a radial solution βn,i > 0 on [0, Tn ]. In order to determine the sign of βn,i we proceed as follows: we consider the function v− (r) = Λe

m−2 4 P (r)

, Λ=

m−1 H 2

m−2 2 .

Using (6.88) and the deﬁnition of 6 a we compute / 2 m+2 4 (m − 1) m−2 H 2 − Λ m−2 ≡ 0. 6 av− − 6b(r)v− = v− 4 With the aid of (6.89) we see that v− solves ⎧ m+2 ⎨ m ΔH 2 v− + 6 av− − 6b(r)v−m−2 ≥ 0 on BTn , (in fact on Hm −H 2 ) −H ⎩v− (x) < i on ∂BTn provided i is suﬃciently large. On the other hand, having set z(x) = βn,i (r(x)),

(6.103)

(6.104)

6.3. Global sub- and supersolutions

185

because of (6.94) we have m+2 ΔHm 2 z + 6 az − 6b(r)z m−2 = 0

on BTn ,

z(x) = i

on ∂BTn .

−H

(6.105)

Applying Proposition 6.1, βn,i (r) ≥ v− (r)

on [0, Tn ].

(6.106)

We ﬁx r0 ∈ [0, Tn ] and we consider the function m+2 y(t) = 6b(r0 )t m−2 − 6 at

on (0, +∞).

We have

4 m + 26 1 a ≥ y(t). b(r0 )t m−2 − 6 m−2 t We let t0 = v− (r0 ). Then, y(t0 ) = 0 because of (6.103); thus y (t0 ) ≥ 0. On the other hand 4 m+2 6 m−2 −1 > 0 y (t) = 4 2 b(r0 )t (m − 2)

y (t) =

and we deduce that

y (t) ≥ 0 ∀ t ≥ t0 .

Thus, the function y is nondecreasing on [t0 , +∞). By (6.106) βn,i (r0 ) ≥ v− (r0 ) = t0 It follows that

and y(t0 ) = 0.

m+2 6b(r0 )βn,i (r0 ) m−2 −6 aβn,i (r0 ) ≥ 0.

Since r0 ∈ [0, Tn ] was chosen arbitrarily, it follows that + (m − 1)H coth(Hr)β ≥ 0 on [0, Tn ], βn,i βn,i (0) = 0. Integration of the above immediately yields (r) ≥ 0 βn,i

on [0, Tn ].

(6.107)

Now let i → ∞. Using Proposition 6.1 we see that the sequence of solutions {βn,i } of (6.94) is nondecreasing. Furthermore, it is uniformly bounded on compact subsets of [0, Tn ) because of Lemma 4.2. Thus, βn,i converges as i → +∞ to a solution βn of the problem m+2 βn + (m − 1)H coth(Hr)βn + 6 aβn − 6b(r)βnm−2 = 0 on [0, Tn ), βn (0) = 0, βn (r) ≥ 0, βn (r) > 0, , βn (r) → +∞ as r → Tn−

186

Chapter 6. Existence

and, moreover, βn (r) ≥ v− (r).

(6.108)

A second application of Proposition 6.1 shows that the sequence {βn } is decreasing and therefore it converges to a solution β of m+2 aβ − 6b(r)β m−2 = 0 on [0, +∞), β + (m − 1)H coth(Hr)β + 6 (6.109) β (0) = 0, β ≥ 0, β(r) > 0. Furthermore, because of (6.108), which is independent of n, β(r) ≥ v− (r) → +∞ as r → +∞, so that (6.92) is certainly satisﬁed. Thus v(x) = β(r(x)) satisﬁes (6.93) and m+2 Δv + a(x)v − b(x)v m−2 ≤ 0 on M \ BR0 , (6.110) v(x) > 0, so is the required radial supersolution on M \ BR0 . Step 3. Gluing the supersolutions. 1,2 , for σ = First, given w ∈ C 0 (M ) ∩ Wloc

m+2 m−2

to simplify the writing we deﬁne

bw = w−σ [Δw + a(x)w]

(6.111)

in the weak sense. Note that for E > 0 constant bEw = E 1−σ bw .

(6.112) 1,2 Hloc

such that Next, we let u + be a positive function in C 0 (M ) ∩ u(x) on BR0 −ε , u + (x) = v(x) on M \ BR0

(6.113)

for some ε > 0 suﬃciently small and where u(x) has been deﬁned in (6.85) while v(x) has been deﬁned in (6.110). We then set u+ (x) = E u+ (x).

(6.114)

Having chosen ε > 0 suﬃciently small we can suppose by (6.75) that b(x) > 0 on M \ BR0 −ε . Next we note that w is a supersolution of (6.80) if and only if bw (x) ≤ b(x) on M . Using (6.112) and (6.113) we have: i) On M \ BR0 , bu+ (x) = E 1−σ bu+ (x) = E 1−σ bv (x) ≤ E 1−σb(x) ≤ b(x); therefore u is a supersolution of (6.80) on M \ BR0 ∀ E ≥ 1, since b(x) > 0 on M \ BR0 .

6.3. Global sub- and supersolutions

187

ii) On BR0 + \ B R0 −ε , bu+ (x) = E 1−σ bu+ (x) ≤ E 1−σ

sup B R0 \BR0 −ε

bu+ (x) ≤

inf

B R0 \BR0 −ε

b(x) ≤ b(x),

provided E ≥ E0 ≥ 1 is suﬃciently large. This is possible since b(x) > 0 on B R0 \ BR0 −ε . iii) On BR0 −ε , since u is a supersolution of (6.85) we have bu+ (x) = E 1−σ bu+ (x) = E 1−σ bu (x) =≤ −E 1−σ . To obtain a supersolution on BR0 −ε we need −E 1−σ ≤ b(x).

(6.115)

Thus u+ is a subsolution if and only if b ≥ −E 1−σ where E is determined in ii) that is b(x) ≥ −η where η = cm E 1−σ . If this condition is satisﬁed, we have therefore found the required positive supersolution u+ to (6.79) with the property that lim r(x)→+∞

u+ (x) = +∞.

Remark. It should be noted that the upper bound on b(x) aﬀects only the behavior of the subsolution, and, as observed after the proof of Theorem 6.15, diﬀerent upper bounds for b(x) give rise to subsolutions with diﬀerent asymptotic behaviour, and therefore to diﬀerent asymptotic estimates for the solution. See the remark after the proof of Theorem 6.18 More generally, the positivity of a(x) together with (6.77) and (6.78) may by replaced by any other set guaranteeing the existence of a subsolution of (6.79). For instance we can assume that Δ+a(x)

λ1

(M ) < 0

to construct a subsolution v− on a suﬃciently large ball BR as in the proof of Theorem 6.7. In this case v− satisﬁes σ ≥ 0, v− > 0 on BR , Δv− + a(x)v− − b+ (x)v− v− ≡ 0 on ∂BR .

Therefore u− (x) =

Dv− (x) 0

on BR , on M \ BR

yields the desired subsolution for D ∈ (0, 1] suﬃciently small.

188

6.4

Chapter 6. Existence

The case of the Yamabe problem

It is clear that we can specialize the conclusions of the previous theorems to the case of the Yamabe equation m+2

cm Δu − S(x)u + K(x)u m−2 = 0 −1 K(x). We will state such a simply replacing a(x) = −c−1 m S(x) and b(x) = −c result explicitly only in the case of Theorem 6.16, leaving the other cases to the interested reader. It should also be pointed out that, in application of the theorems of previous sections to the Yamabe equation, it is often possible to replace the condition that S(x) be everywhere negative with the assumption that it is nonpositive and strictly negative oﬀ a compact set. This is based on the following proposition (compare with Proposition 3.10).

Proposition 6.17. Let Ω0 ⊂⊂ Ω1 ⊂ (M, , ) be relatively compact domains with smooth boundaries. Assume that the scalar curvature S(x) veriﬁes S(x) ≤ c2 h(x)

on M \ Ω0

(6.116)

for some c > 0, h ∈ C 0 (M ) nonpositive, h(x) < 0 on Ω1 \ Ω0 . Then there exists η > 0 such that, if S(x) ≤ η on M, (6.117) then there exists a complete conformal metric , homothetic to , on M \ Ω1 , whose scalar curvature S(x) satisﬁes S(x) ≤ c2 h(x)

on M,

(6.118)

for some c > 0. Proof. We elaborate some ideas of [AM88]. It is clear that, if , is a metric conformal to , and homothetic to , on M \ Ω1 , then (6.116) implies (6.118) c > 0. Therefore, up to modifying c, it is enough to on M \ Ω1 for some suitable show that , can be chosen so that S(x) < 0 on Ω1 . We ﬁx a constant δ > 0 and solve the Dirichlet problem cm Δψ = δ on Ω1 , (6.119) ψ=0 on ∂Ω1 . Next, we choose a domain Ω2 such that Ω0 ⊂⊂ Ω2 ⊂⊂ Ω1 and a smooth cut-oﬀ function ξ : M → [0, 1] such that ξ ≡ 1 on Ω0

and

ξ ≡ 0 on M \ Ω2 .

6.4. The case of the Yamabe problem

189

We deﬁne ϕ = ξψ on M ; now we choose a positive constant ϕ0 so that ϕ(x) + ϕ0 > 0 on M and S(x)(ϕ(x) + ϕ0 ) − cm Δϕ ≤ −

(6.120)

δ 2

on Ω1

(6.121)

(note that, in order to achieve (6.121), we use h(x) < 0 on Ω1 \ Ω0 and S(x) ≤ η). Finally, we set 4 , = (ϕ(x) + ϕ0 ) m−2 , . Then S(x) is given by − S(x) = (ϕ(x) + ϕ0 ) m−2 [S(x)(ϕ(x) + ϕ0 ) − cm Δϕ(x)]. m+2

Now (6.120) and (6.121) imply S(x) < 0 on Ω1 .

We now give the announced version of Theorem 6.16 for the Yamabe equation. It is a variation of Theorem 0.1 in [RRV97]. Theorem 6.18. Let (M, , ) be a complete manifold of dimension m ≥ 3, scalar curvature S(x), sectional curvature SectM and let K(x) ∈ C ∞ (M ). Let R > 0 and suppose that the ball BR centered at o does not intersect the cut locus of o. Suppose that i) Ric ≥ −(m − 1)H 2 2

ii) SectM ≤ −A

on M, (6.122)

on BR , 2

iii) S(x) ≤ −m(m − 1)C (1 + r(x)]

α

on M,

for some 0 ≤ A, C ≤ H and −1 < α ≤ 0 and 0 < A ≤ B such that 1 2 A2 [coth(AR)]2 . H < 1+ m(m − 2) Assume also that K(x) ≥ −C1 r(x)β eDr(x)

θ

(6.123)

(6.124)

on M \ BR for some constants C1 , D > 0, β ∈ R, and θ < 1 + α. Then, there exists η > 0 such that, if K(x) ≤ η on M, the metric , can be pointwise conformally deformed to a new metric of scalar curvature K(x). Proof. We only need to verify that the assumptions of Theorem 6.16 are satisﬁed −1 K(x). First of all, the bound on the Ricci with a(x) = −c−1 m S(x) and b(x) = −c curvature implies s(x) ≥ −m(m − 1)H 2 ,

190

Chapter 6. Existence

so that, recalling that cm = 4 m−1 m−2 , −c−1 m s(x) ≤

m(m − 2) 2 (m − 1)2 2 H ≤ H , 4 4

(6.125)

and condition (6.74) in the statement of Theorem 6.16 is automatically satisﬁed. Observe next that the sectional curvature condition and the fact that the ball BR does not intersect the cut locus of o implies, by the Laplacian comparison theorem, that Δr ≥ (m − 1)A coth(Ar) on BR . (6.126) Arguing as in the proof of Theorem 6.16, this in turn implies that the ﬁrst eigenvalue of the Dirichlet Laplacian on BR satisﬁes λΔ 1 (BR ) ≥

2

(m − 1) 2 A [coth(AR)]2 , 4

whence, using (6.125), Δ−c−1 m S(x)

λ1

(BR ) ≥

(m − 1)2 2 m(m − 2) 2 A [coth(AR)]2 − H >0 4 4

by condition (6.123).

Remark. The conclusion of the theorem in fact continues to hold even if equality holds in (6.123). This requires a diﬀerent argument for the construction of the supersolution inside the ball BR . We refer to [RRV97] for the details. Note also that the upper bound (6.122) iii) for S(x) and the lower bound (6.78) for K(x) are used to construct the subsolution and determine a lower estimate for the solution u. As noted after the proof of Theorem 6.15, replacing (6.78) with K(x) ≥ −C1 r(x)β r(x) 1, with β < α allows us to produce a subsolution, and therefore a solution which satisfy the estimate u(x) ≥ Cr(x)γ , r(x) 1, with γ =

m−2 4 (β

+ α). In particular if β + α = 2, then u(x) ≥ Cr−

m−2 2

,

and we may guarantee that the conformal metric 4 , = u m−2 ,

is complete.

6.5. Appendix: the Monotone Iteration Scheme

6.5

191

Appendix: the Monotone Iteration Scheme

In this section we describe the monotone iteration scheme, due to H. Amann (see [Ama76]), that has been one of the main tools to obtain our existence results. Our presentation follows closely that of D. H. Sattinger, [Sat73]. Let Ω be an open domain, and let f (x, s) be a continuous function on Ω × R, and consider the diﬀerential equation Δu + f (x, u) = 0,

in Ω.

(6.127)

1,2 We say that a function u+ ∈ Wloc (Ω) ∩ L∞ loc (Ω) is a supersolution of (6.127) if it satisﬁes the diﬀerential inequality

Δu+ + f (x, u+ ) ≤ 0,

weakly in Ω,

that is, for every test compactly supported function ρ ≥ 0 in W 1,2 (Ω), − ∇u+ , ∇ρ + f (x, u+ )ρ ≤ 0. A function u− is a subsolution if it satisﬁes the reverse diﬀerential inequality. 1,2 Similarly, if Ω is bounded, and g is a continuous function on ∂Ω, u+ ∈ Wloc (Ω) ∩ C(Ω) is a supersolution of the boundary value problem Δu + f (x, u) = 0 in Ω, (6.128) u=g on ∂Ω, if

Δu+ + f (x, u+ ) ≤ 0 in Ω, u+ ≥ g on ∂Ω.

The deﬁnition of a subsolution is obtained reversing the inequalities. Theorem 6.19. Let Ω be a relatively compact open domain with smooth boundary older function such that s → f (x, s) ∂Ω and let f : Ω × R → R be a locally H¨ is locally Lipschitz with respect to s uniformly with respect to x. Suppose that ϕ and ψ ∈ C 0,1 (Ω) are respectively a subsolution and a supersolution of (6.128) satisfying ϕ ≤ ψ on Ω and ϕ ≤ g ≤ ψ on ∂Ω. Suppose moreover that g extends to a C 2,α (Ω) function, which we denote with the same letter. Then the boundary value problem (6.128) has a solution u ∈ C 2,α (Ω) satisfying ϕ ≤ u ≤ ψ. Proof. Since f (x, s) is locally Lipschitz in s uniformly with respect to x in Ω, there exists λ > 0 such that s → f (x, s) + λs = F (x, s)

192

Chapter 6. Existence

is monotone increasing for every x in Ω and every s satisfying minΩ ϕ ≤ s ≤ maxΩ ψ. For every function w ∈ C α (Ω) we let v = T w ∈ C 2,α (Ω) be the solution of the boundary value problem (Δ − λ)v = −[f (x, w) + λw] in Ω, (6.129) v=g on ∂Ω, which exists and is unique by standard elliptic theory, (see, e.g., [GT01] Theorem 6.14). We claim that the operator T is monotone, that is, if w1 ≤ w2 on Ω, then T w1 ≤ T w2 . Indeed, by the monotonicity of the function F (x, s) the diﬀerence v = T w2 − T w1 satisﬁes (Δ − λ) v = − F (x, w1 ) − F (x, w2 )] ≤ 0 in Ω, v = 0 on ∂Ω, and therefore by the minimum principle, v ≥ 0 in Ω, i.e., v2 = T w2 ≥ v1 = T w1 , as claimed. + − − Now we set u− 1 = T ϕ, u1 = T ψ and for every k ≥ 1, uk+1 = T uk and + + uk+1 = T uk . Recalling that ϕ and ψ are respectively sub- and supersolutions of the boundary value problem (6.128) and arguing as above, the maximum principle and an induction argument show that − − + + + ϕ ≤ u− 1 ≤ u2 ≤ · · · ≤ uk ≤ · · · ≤ uk ≤ u2 ≤ u1 ≤ ψ. + − + Thus, as k tends to inﬁnity u− k → u and uk → u . − + 2,α We claim that u and u are in C (Ω) and solve the boundary value (6.128). Indeed, the solution v of (6.129) satisﬁes Lp Sobolev estimates up to the boundary of the form ||v||W 2,p ≤ C ||g||W 2,p + ||F (x, w)||Lp ,

(see, e.g., [ADN59], Theorem 15.2) and it follows easily that for every p, the 2,p to a solution (u± ) of the diﬀerential equation sequences {u± k } converge in W Δu = f (x, u). By Morrey’s embedding lemma, see [GT01], Theorem 7.26, for every p > n the space W 1,p embeds continuously into C β (Ω) and therefore, by choosing p large enough, the above convergence takes place in C α (Ω). Finally, by [ADN59], Theorem 7.3, the solution v of (6.129) satisﬁes Schauder’s estimates up to the boundary ||v||C 2,α ≤ C( ||g||C 2,α + ||F (x, w)||C α , 2,α and therefore the sequences u± (Ω) to solutions u± of (6.128), as k converge in C required.

6.5. Appendix: the Monotone Iteration Scheme

193

We conclude this section with the following result that extends to the situation at hand the well-known fact that the maximum of two subharmonic functions is subharmonic. Theorem 6.20. Let f be a continuous function on Ω × R and suppose that ui ∈ 1,2 L∞ loc (Ω) ∩ Wloc (Ω), i = 1, 2 are subsolutions of (6.127) on Ω. Then = max{u1 , u2 } is also a subsolution. Similarly if vi are supersolutions of (6.127), then so is v = min{v1 , v2 }. Proof. The proof is adapted from [Le98]. We only show that the maximum of subsolutions is a subsolution. The argument in the case of supersolutions is similar. 1,2 Set u = max{u1 , u2 } and note that u ∈ L∞ loc (Ω) ∩ Wloc (Ω) and ∇u =

∇u1 ∇u2

if u1 ≥ u2 , if u1 ≤ u2 .

By assumption, for every 0 ≤ ϕ ∈ C0∞ (Ω) we have ∇ui , ∇ϕ − f (x, ui ) ≤ 0, i = 1, 2. Let D ⊂ Ω be a relatively compact open set with smooth boundary such that supp ϕ ⊂ D, and set D1 = {x ∈ D : u1 (x) > u2 (x)}, Do = {x ∈ D : u1 (x) = u2 (x)} and D2 = {x ∈ D : u1 (x) < u2 (x)}. Let also γ : R → R be such that γ ≥ 0, γ(t) = 0 if t ≤ 0 and = 1 is t ≥ 1, and let γn (t) = γ(nt), so that γn (t) = 0 if t ≤ 0, = 1 if t ≥ 1/n and 0 ≤ γn (t) ≤ nM 1,2 with M = sup γ . Let w = u2 − u1 . Since w ∈ Wloc (Ω) there exists wn ∈ C0∞ (Ω) 1,2 such that wn → w in W (D). By passing to a subsequence, if necessary, we may also assume that wn → w a.e. in D and ||wn − w||W 1,2 (D) ≤ n12 . Finally, set ϕ1 = (1−γn (wn ))ϕ and ϕ2 = γn (wn )ϕ, so that 0 ≤ ϕi ∈ C0∞ (D). We claim that for a.e. x ∈ D2 there exists n0 = n0 (x) such that wn (x) > 1/n for every n ≥ n0 . Indeed, since w > 0 on D2 , there exists n1 = n1 (x) such that w(x) > 2/n1 . Moreover, wn → w a.e., for a.e. x there exists n0 = n0 (x) ≥ n1 such that for every n ≥ n0 |wn (x) − w(x)| ≤ 1/n1 , and therefore, for n ≥ n0 ≥ n1 , wn (x) ≥ w(x) − |wn (x) − w(x)| ≥

2 1 1 1 − = ≥ . n1 n1 n1 n

Thus, for a.e. x and every n ≥ n0 we have γn (wn (x)) = 1 and we conclude that γn (wn ) → 1 a.e. in D2 . Similarly, since w(x) < 0 in D1 , we see that γn (wn (x)) → 0 a.e. in D1 . Since 0 ≤ ϕi ∈ C0∞ (Ω), and ui are subsolutions we have ∇ui , ∇ϕi − f (x, ui ) ≤ 0,

194

Chapter 6. Existence

whence, adding the two inequalities, and using the deﬁnition of ϕi , a computation yields

∇u1 , (1 − γn (wn ))∇ϕ − γn (wn )ϕ∇wn + ∇u2 , γn (wn )∇ϕ + γn (wn )ϕ∇wn − f (x, u1 )(1 − γn (wn )∇ϕ + f (x, u2 )γn (wn )∇ϕ = ∇u2 − ∇u1 , γn (wn )∇ϕ + ∇u2 − ∇u1 , γn (wn )ϕ∇wn + [f (xu1 ) − f (x, u2 )]γn (wn )ϕ + ∇u1 , ∇ϕ − f (x, u1 )ϕ

0≥

(6.130)

= I1 (n) + I2 (n) + I3 (n) + I4 . Observe now that, since ∇ui ∈ L2 (D), ∇ϕ ∈ L∞ (D), ∇u1 = ∇u2 on Do , 0 ≤ γn (wn ) ≤ 1, and γn (wn )tends to 1 a.e. on D2 and to 0 a.e. on D1 , , letting n → ∞, by dominated convergence we have I1 (n) =

D1 ∪D2

∇u2 − ∇u1 , γn (wn ))∇ϕ →

∇u2 − ∇u1 , ∇ϕ. D2

Similarly, since f (x, ui ) is bounded on D, f (x, u1 ) = f (x, u2 ) on Do and ϕ is compactly supported I4 (n) =

[−f (x, u2 )+f (x, u1 )]γn (wn ))ϕ →

D1 ∪D2

[−f (x, u2 )+f (x, u1 )]γn (wn ))ϕ. D2

To deal with I2 note that I2 = ≥

∇u2 −

∇u1 , ∇wn γn (wn )ϕ

=

∇w, ∇wn − ∇wγn (wn )ϕ ≥ −

∇w, ∇w + (∇wn − ∇w)γn (wn )ϕ |∇w||∇wn − ∇w|γn (wn )ϕ.

Since γn (wn ) ≤ M n,

and

||wn − w||W 1,2 (D) ≤

1 , n2

the integral on the rightmost side is bounded above by M n||ϕ||∞ ||∇ϕ||L2 (D) n12 = C n and we deduce that lim inf I2 (n) ≥ 0. n

6.5. Appendix: the Monotone Iteration Scheme

195

Thus, letting n → +∞ in (6.130) we obtain 0 ≥ lim I1 (n) + lim inf I2 (n) + lim I3 (n) + I4 n n n ≥ ∇u2 − ∇u1 , ∇ϕ + [−f (x, u2 ) + f (x, u1 )ϕ] + ∇u1 , ∇ϕ − f (x, u1 )ϕ D2 D = ∇u2 , ∇ϕ − f (x, u2 )ϕ + ∇u1 , ∇ϕ − f (x, u1 )ϕ. D2

D\D2

Recalling that u = max{u1 , u2 } =

u2 u1

on D2 , and that ∇u = on D \ D2 ,

∇u2 ∇u1

on D2 , on D \ D2 ,

the last inequality amounts to 0≥

∇u, ∇ϕ − f (x, u)ϕ,

as required to show that u is a subsolution of (6.127).

Chapter 7

Some special cases In this ﬁnal chapter we present some special cases where one can use particular techniques to relax assumptions. Typically this happens in Euclidean and hyperbolic spaces, and more generally in the case of models, or manifolds with some special symmetry. For the models, the main advantage is to have a precise expression for Δr which is available, on general manifolds, only in the form of an upper or a lower bound under certain assumptions on the curvature and the cut locus. We provide here reﬁned techniques and a few results that, as a side product, show the degree of sharpness of the general theory and of the methods that we have developed dealing with generic complete Riemannian manifolds. As a ﬁnal remark we underline that, here and there, in this chapter we explicitly link our geometric point of view to more familiar Euclidean procedures and tools (for instance, the Rellich-Pohozaev formula), as shown below in detail.

7.1

A nonexistence result

We begin by showing that under appropriate assumptions on a(x) and b(x) there are no positive solutions u on M of Δu + a(x)u − b(x)uσ ≥ 0. To prove this fact we shall use a technique that does not require the determination of a priori estimates on u. The idea is to perform a sort of radialization and then apply ODE methods. However, as we will explain later, this has some geometrical heavy side eﬀects. We need a collection of preliminary technical results; the next lemma is motivated by the work of K.S. Cheng and J. T. Lin, [CL87]. Lemma 7.1. Let 0 < R0 ≤ R1 ≤ R2 ≤ +∞, σ > 1 and assume that the functions 0 l, ψ, b ∈ C ([R0 , +∞)) satisfy i) l(t) > 0 on [R0 , +∞); ii) ψ(t) > 0 on [R0 , +∞); (7.1) iii) tb(t) ∈ L1 (+∞); iv) tb(t) > 0 and nonincreasing on [R1 , +∞).

P. Mastrolia et al., Yamabe-type Equations on Complete, Noncompact Manifolds, Progress in Mathematics 302, DOI 10.1007/978-3-0348-0376-2_7, © Springer Basel 2012

197

198

Chapter 7. Some special cases

Suppose that lim inf t→+∞

l(t)ψ(t)σ > 0. tb(t)

(7.2)

If α is a positive solution of &

s

α(s) ≥ ψ(s) c1 + c2 Ro

' t σ l(t)α(t) dt 1− s

(7.3)

on [R0 , R2 ) for some constant c1 , c2 > 0, then R2 < +∞. Proof. We argue by contradiction and let R2 = +∞. We set β(s) =

α(s) ψ(s)

so that (7.3) implies that β satisﬁes & ' s t σ σ l(t)ψ(t) β(t) dt 1− β(s) ≥ c1 + c2 ∀ s ≥ R0 . s Ro Because of (7.2), there exist c3 > 0 and R3 ≥ R0 such that s t tb(t)β(t)σ dt ∀ s ≥ R ≥ R3 . 1− β(s) ≥ c1 + c3 s R

(7.4)

We choose R4 > max {R1 , R3 } and we deﬁne

t

η(t) =

vb(v) dv + R42 b(R4 ).

(7.5)

R4

We observe that η(t) ≥ 0 and

i) η (t) = t b(t) > 0;

ii) η(t) → +∞ as t → +∞

(7.6)

because of assumptions (7.1) iii), iv). We let R ≥ R4 and we consider η : [R, s] → [ν = η(R), ξ = η(s)]. We indicate with t(η) its inverse. We set γ(η) = β(t(η)) and we perform a change of variable in (7.4) to obtain

ξ

γ(ξ) ≥ c1 + c3

! 1−

ν

" t(η) γ(η)σ dη. t(ξ)

(7.7)

7.1. A nonexistence result

199

Next, we observe that for η suﬃciently large, say η ≥ η0 , η −1 t(η) is nondecreasing. Indeed, it is enough to show that ηt (η) − t(η) ≥ 0,

η ≥ η0 .

This, in turn, is equivalent to showing that

t

vb(v) dv + R42 b(R4 ) − t2 b(t) ≥ 0

(7.8)

R4

for t suﬃciently large. But (7.1) iv) and our choice of R4 imply (7.8) for t ≥ R4 . From (7.7) we deduce

ξ

γ(ξ) ≥ c1 + c3 ν

η γ(η)σ dη 1− ξ

(7.9)

for all ξ > ν ≥ η(R4 ). With the aid of (7.6) ii) and (7.1) we then have: ∀ ν > η(R4 ) and ∀ ξ ∈ [ν, 2ν], ξ 1 γ(ξ) ≥ c1 + c3 (ξ − η)γ(η)σ dη. (7.10) 2ν ν To complete the proof we deﬁne g(ξ) = c1 + c3

1 2ν

ξ

(ξ − η)γ(η)σ dη

(7.11)

ν

and observe that g satisﬁes i) g(ν) = c1 , ii) g (ν) = 0 #ξ c3 iii) g (ξ) = 2ν γ(η)σ dη, iv) g (ξ) = ν

c3 σ 2ν γ(ξ)

≥

c3 σ 2ν g(ξ)

(7.12)

where in (7.12) iv) we have been using (7.10) and (7.11). From (7.12) we get

[g (ξ)]

2

= 2g (ξ)g (ξ) ≥

c3 c3 g (ξ)g(ξ)σ = g(ξ)σ+1 . ν (σ + 1)ν

(7.13)

Therefore, integrating (7.13) over [ν, ξ], with the aid of (7.12) ii) we deduce !

c3 g (ξ) ≥ (σ + 1)ν

" 12

g(ξ)σ+1 − g(ν)σ+1

12

.

(7.14)

Integrating (7.14) again over [ν, ξ] we have

ξ ν

!

g (x) dx [g(x)σ+1 − g(ν)σ+1 ]

1/2

c3 ≥ (σ + 1)ν

"1/2 (ξ − ν).

(7.15)

200

Chapter 7. Some special cases

Next, we perform the change of variable u=

g(x) g(ν)

and we rewrite (7.15) in the form 1

g(ξ) c1

9 √

du uσ+1 − 1

≥

cσ−1 c3 1 (ξ − ν). (σ + 1)ν

(7.16)

− 1 But σ > 1; hence uσ+1 − 1 2 ∈ L1 ([1, +∞)). On the other hand, choosing ξ = 2ν in (7.16), we immediately obtain a contradiction by taking ν suﬃciently large. To state the next result we introduce some notation. Given σ > 1 and a nonnegative f ∈ C 0 (M ), we set 1 ¯ f (r) = f, r ∈ [0, +∞); (7.17) vol(∂Br ) ∂Br '1−σ & 1 1 f 1−σ , r ∈ [0, +∞) (7.18) f¯σ (r) = vol(∂Br ) ∂Br respectively for the spherical mean (as in the previous chapters) and the weighted spherical mean of f , with the convention that f¯σ (r) = 0 in case the integral in (7.18) is inﬁnite. Furthermore Br = Br (o) for some ﬁxed origin o ∈ M as before. We prove Lemma 7.2. Let (M, , ) be a complete manifold of dimension m ≥ 2 and assume Δr ≥ (m − 1)z(r)

(7.19)

for some z(r) ∈ C 0 ([0, +∞)) in the weak sense on M . Let a(x), b(x) ∈ C 0 (M ) satisfy a(x) ≤ p(r(x)) on M \{o} (7.20) for some p(t) ∈ C 0 ((0, +∞)), b(x) ≥ 0

on M.

(7.21)

Let u ∈ C 2 (M ) be a positive solution of the diﬀerential inequality Δu + a(x)u − b(x)uσ ≥ 0, σ > 1,

on M.

(7.22)

Assume that, for any ﬁxed 0 < δ < s, there exists a function ϕ ∈ C 2 ([δ, s]) with the following properties: ϕ + (m − 1)z(t)ϕ + p(t)ϕ ≥ 0 on [δ, s],

(7.23)

7.1. A nonexistence result

201

i) ϕ (s) =

1 ; vol(∂Bs )

−

ii) ϕ (t) ≥ 0 on [δ, s]

(7.24)

ϕ(s) = 0

(7.25)

ϕ(δ) ≤C ϕ (δ)

(7.26)

for some constant C > 0 independent of δ and s. Then, up to choosing δ suﬃciently small (and consequently ϕ), there exists a constant C1 > 0, independent of δ and s, such that for all R ∈ [δ, s) u ¯(s) satisﬁes s ϕ(t) vol(∂Bt ) ¯bσ (t) u ¯(t)σ dt. (7.27) u ¯(s) ≥ C1 ϕ (δ)δ m−1 − R

Proof. We ﬁx 0 < δ < s, ϕ and R ∈ [δ, s]. We consider the function ϕ(r(x)) on M ; because of (7.19), (7.20) and (7.24) ii) we have Δϕ(r)(x) ≥ −a(x)ϕ(r(x))

(7.28)

¯s \Bδ . The second Green’s identity, (7.24) i), (7.25) together in the weak sense on B with the deﬁnition (7.17) of u ¯ give u ¯(s) = uΔϕ − ϕΔu + uΔϕ − ϕΔu + u ∇ϕ, ∇r − ϕ ∇u, ∇r . Bs \BR

BR \Bδ

∂Bδ

(7.29) Observe that (7.22), (7.28), (7.21) and positivity of u yield uΔϕ − ϕΔu ≥ 0.

(7.30)

BR \Bδ

Similarly

Bs \BR

uΔϕ − ϕΔu ≥

ϕ(r)b(x)uσ .

(7.31)

Bs \BR

Next, a straightforward application of H¨ older’s inequality shows b(x)uσ ≥ ¯bσ (t)¯ u(t)σ vol(∂Bt )

(7.32)

∂Bt

where ¯bσ (t) has been deﬁned in (7.18). Putting together (7.31), (7.32) and using the co-area formula (1.87), we ﬁnally get s uΔϕ − ϕΔu ≥ − ϕ(t)¯bσ (t)¯ u(t)σ vol(∂Bt ) dt. (7.33) Bs \BR

R

We now take care of the boundary term in (7.29). On the one hand, u ∇ϕ, ∇r − ϕ ∇u, ∇r = ϕ (δ)¯ u(δ) vol(∂bt ) − ϕ(δ) Δu. ∂Bδ

Bδ

(7.34)

202

Chapter 7. Some special cases

On the other hand, vol(∂Bδ ) # δ m−1 ; vol(Bδ ) # δ m as δ ↓ 0+ .

(7.35)

Using positivity of u, (7.35), and (7.34) it is easy to see that, up to choosing 0 < δ ≤ δ0 suﬃciently small, there exists a constant C1 > 0 independent of δ and s such that u ∇ϕ, ∇r − ϕ ∇u, ∇r ≥ C1 ϕ (δ)δ m−1 . (7.36) ∂Bδ

Estimate (7.27) now follows from (7.29), (7.30), (7.33) and (7.36). Remark. In case u is a radial solution of u + (m − 1)z(r)u + a(r)u − b(r)uσ ≥ 0, σ > 1 u (0) = 0, u(0) = u0 > 0 with z(r) =

h (r) h(r) ,

on [0, +∞),

(M, h) a model (see Chapter 4), (7.27) becomes

u(s) ≥ C1 ϕ (δ)δ m−1 −

s

ϕ(t)h(t)m−1 b(t)u(t)σ dt.

(7.37)

R

We apply the above results to produce a nonexistence theorem for complete Riemannian manifolds with sectional curvature bounded above by a negative constant −B 2 , B > 0. The case B = 0 can be dealt with similarly and later, in the special case of Euclidean space, we shall apply this technique to diﬀerential inequalities of a certain type. Theorem 7.3. Let (M, , ) be a complete manifold of dimension m ≥ 2 with a pole o ∈ M and suppose (7.38) Riem ≤ −B 2 for some constant B > 0. Let a(x), b(x) ∈ C 0 (M ) satisfy a(x) ≤ AB 2 coth(Br(x)) on M, with A ≤ b(x) ≥ 0

(m − 1)2 , 4

on M.

Suppose that for some constant σ > 1, ⎧ b(t) vol(∂Bt )σ−1 ⎪ ⎪ C ⎪ ⎨ tσ−1 e m−1 2 B(σ−1)t ¯bσ (t) ≥ tb(t) vol(∂Bt )σ−1 ⎪ ⎪ C ⎪ (m−1)2 ⎩ B m−1 −A∗ (σ−1)t 2 + 4 e

(7.39) (7.40)

in case A =

(m−1)2 , 4

in case A <

(m−1)2 4

(7.41)

for t 1, some constant C > 0 and with A∗ = max {0, A},

(7.42)

7.1. A nonexistence result

203

where i) tb(t) ∈ L1 (+∞),

ii) tb(t) > 0 and nonincreasing at inﬁnity.

(7.43)

Then the diﬀerential inequality Δu + a(x)u − b(x)uσ ≥ 0

(7.44)

has no positive C 2 -solution on M . Remark. In the proof of Theorem 7.3 we shall make use of the Hessian comparison theorem (see Chapter 1) which in the above assumptions yields Δr ≥ (m − 1)B coth Br

(7.45)

pointwise on M \{o}. Remark. We would like to comment on the case of hyperbolic space Hm −B 2 and Yamabe equation m+2

cm Δu + m(m − 1)B 2 u + k(x)u m−2 = 0, where cm = 4 m−1 m−2 . Here, vol(∂Bt ) ∼ C1 e(m−1)Bt

as t → +∞

for some C1 > 0; furthermore, a(x) =

m(m − 2) 2 B , 4

b(x) = −

1 k(x). cm

Thus, a(x) ≤ AB 2 coth(Br(x)),

with A =

m(m − 2) (m − 1)2 < . 4 4

It follows that the requirement (7.41) on ¯bσ (t) becomes ¯bσ (t) ≥ Ctb(t)e2Bt , t 1, C > 0. Now admissible choices of b(t) are b(t) =

1 , t2

b(t) =

t2

1 , log t

b(t) =

t2

1 , log t(log log t)

Thus, for instance, the above becomes 2Bt ¯bσ (t) ≥ e , t

t 1, C > 0.

. . . , t 1.

204

Chapter 7. Some special cases

This is implied by b(x) ≥ C

e2Br(x) r(x)

for r(x) 1, C > 0.

(7.46)

This is expressed, in terms of k(x), as k(x) ≤ −C

e2Br(x) r(x)

for r(x) 1, C > 0.

(7.47)

Inequality (7.47) relaxes the second of (4.86) in the case of m-dimensional hyperbolic space Hm −B 2 . However, assumption (7.38) is deﬁnitely a stronger requirement than (4.84) of Theorem 4.8. Note that (7.47) is independent of the dimension of Hm −B 2 . Proof of Theorem 7.3. We reason by contradiction and we assume that (7.44) has a positive C 2 -solution u on M . Note that, by the ﬁrst remark after the statement of the theorem, (7.45) holds on M . Next, we begin by considering the case A = (m−1)2 . We ﬁx 0 < δ ≤ R < s and deﬁne 4 ϕ(t) =

m−1 t−s e 2 B(s−t) vol(∂Bs )

on [δ, s].

Then ϕ satisﬁes (7.23) of Lemma 7.2 with z(t) = B coth Bt. (7.24), (7.25) and (7.26) are veriﬁed up to choosing in (7.26) C = m−1 2 B. From Lemma 7.2 it follows that u ¯(s) satisﬁes (7.27), that is, for some constant C1 > 0 independent of δ and s, s ϕ(t) vol(∂Bt )¯bσ (t)¯ u(t)σ dt (7.48) u ¯(s) ≥ C1 ϕ (s)δ m−1 − R

for all R ∈ [δ, s). On the other hand, ϕ (δ) =

e

m−1 2 B(s−δ)

vol(∂Bs )

m−1 B 2

1 + (s − δ)

m−1

≥ C2

s e 2 Bs vol(∂Bs )

with C2 > 0 independent of s and δ ∈ (0, δ0 ] for some δ0 > 0 ﬁxed. Furthermore, −ϕ(t) =

m−1 t − m−1 Bt s e 2 Bs 1− e 2 . vol(∂Bs ) s

Therefore we choose m−1

s e 2 Bs , ψ(s) = vol(∂Bs ) to obtain

l(t) = e−

& u ¯(s) ≥ ψ(s) C3 + C4

s R

m−1 2 Bt

vol(∂Bt )¯bσ (t)

t 1 − l(t)¯ u(t)σ dt s

'

7.1. A nonexistence result

205

that is, (7.3) of Lemma 7.1. Next we observe that ψ(t) > 0, l(t) > 0 for t suﬃciently large (so that we shall choose R suﬃciently large in the above). Furthermore assumption (7.41) implies that (7.2) is satisﬁed. From Lemma 7.1 it follows that u ¯(s) is deﬁned on a ﬁnite interval; contradiction. 2 The case A < (m−1) is treated similarly choosing on [δ, s], 0 < δ < s, the 4 function ϕ as follows: eαB(s−t) 1 − e−αB(s−t) ϕ(t) = − α vol(∂Bs ) with m−1 α= + 2

:

(m − 1)2 − A∗ . 4

Remark. The proof above shows how (7.41) is sensitive to the coeﬃcient A of (7.39). One can also explore the dependence of (7.41) on the possible decay of the right-hand side of (7.39) at inﬁnity. This is a very delicate question 1 we 0 to which (m−1)2 . shall return later. Note that, given (7.39), (7.41) is sharp for A ∈ 0, 4 We shall now apply the above result in a special geometrical setting. Consider the complete manifold Sm−p−1 × Hp+1 with the product metric (recall that Sm is the standard sphere of dimension m and that Hm is the standard hyperbolic space of dimension m: see also Chapter 2); a computation shows that its scalar curvature S(x) is S(x) = (m − 1)(m − 2 − 2p).

(7.49)

Therefore, S(x) is positive for zero for negative for

m−2 , 2 m−2 p= , 2 m−2 < p ≤ m − 2. 2 0≤p<

(7.50) (7.51) (7.52)

Let (θ, x) denote the variables in Sm−p−1 × Hp+1 ; we want to study the Yamabe equation m+2 (7.53) cm Δu − (m − 1)(m − 2 − 2p)u + k(θ, x)u m−2 = 0, u > 0 on Sm−p−1 × Hp+1 . We restrict our attention to the special case k(θ, x) = k(x). Given w = w(θ, x) deﬁne 1 w(x) 6 = w(θ, x) dθ, (7.54) ωm−p−1 Sm−p−1

206

Chapter 7. Some special cases

where dθ is the volume element on Sm−p−1 , can and ωm−p−1 = vol Sm−p−1 . Note that, for all σ > 1, using H¨older’s inequality we have w(x) 6 σ≤

1 ;σ (x). w ωm−p−1

(7.55)

The following result is straightforward. Lemma 7.4. Let u be a positive solution of (7.53) on Sm−p−1 × Hp+1 If k(x) ≤ 0, then u 6 satisﬁes m+2

6 − (m − 1)(m − 2 − 2p)6 u + ωm−p−1 k(x)6 u m−2 ≥ 0. cm ΔHp+1 u

(7.56)

Proof. We observe, as one can easily check, that Δ = ΔSm−p−1 + ΔHp+1 . Therefore, using Stokes’ theorem we have ; = ΔS m−p−1 u + Δ Δu Hp+1 u =

1 ωm−p−1

Sm−p−1

ΔSm−p−1 u(θ, x) dθ

6. + Δ Hp+1 u = ΔHp+1 u(θ, x) = ΔHp+1 u Hence, m+2 ; = (m − 1)(m − 2 − 2p)6 cm ΔHp+1 u u − k(x)u m−2 6 = cm Δu m+2 = (m − 1)(m − 2 − 2p)6 u − k(x)u m−2

and using k(x) ≤ 0 and (7.55) we obtain (7.56).

We are now ready to prove the following Theorem 7.5. Consider the complete manifold Sm−p−1 × Hp+1 with the product < p ≤ m − 2. Then its metric cannot be conformally deformed to metric for m−2 2 a metric of scalar curvature k(θ, x) = k(x) ∈ C ∞ (Hp+1 ) satisfying k(x) ≤ 0 and k(x) ≤ −C

e2

2p−m+2 m−2 r(x)

r(x)

for r(x) 1,

(7.57)

where r(x) is the hyperbolic distance from a ﬁxed origin in Hp+1 and C > 0 constant. Proof. Simply apply Theorem 7.3 to the diﬀerential inequality (7.56) reasoning by contradiction. Note the special case p =

m−2 2

k(x) ≤ −

in which (7.57) reduces to C r(x)

for r(x) 1.

7.1. A nonexistence result

7.1.1

207

A Rellich-Pohozaev formula

The aim of this section is to provide a Pohozaev formula, a tool well-known to analysts in Euclidean space, which is used in order to obtain nonexistence results. The formula can be readily generalized from Rm to a general Riemannian manifold M , but its eﬀectiveness in this wider context depends on the possibility of choosing a conformal vector ﬁeld X on M appropriately related to the problem at hand. For Euclidean space, one has the canonical simple choice X = r∇r, where r(x) is the distance function from the origin o. Similar choices are naturally available for spaces of constant sectional curvature or for models in the sense of Greene and Wu, [GW79]. However, for general Riemannian manifolds the mere existence of a globally deﬁned conformal vector ﬁeld X implies subtle geometrical restrictions, and the behavior of X and its divergence (or, alternatively, the Lie derivative of the metric in the direction of X) is diﬃcult to understand. This is the reason why in this section we shall limit ourselves to special cases, and, in the next section, to hyperbolic space, and more speciﬁcally to its Poincar´e disc B1 (0) realization. What follows can be considered as a geometric motivation of the Pohozaev formula in the particular case of the Yamabe equation in Rm : indeed, this special case relates the formula to the Kazdan-Warner obstruction obtained by integration of (2.17) (see Chapter 2, Theorem 2.3; see also the discussion in Section 5 of [PRS03a]). Let (Rm , , ) be the Euclidean space with its ﬂat canonical metric. For the sake of simplicity let us suppose m ≥ 3. We consider the conformally related metric 4 , = u m−2 , and, as in the previous chapters, we denote by T and T the trace-free Ricci tensors of , and , respectively. Then clearly T ≡ 0 and equation (2.26) yields " ! ! " 2m −2 1 1 −1 2 Δu , − Hess(u) + u du ⊗ du − |∇u| , . (7.58) T = 2u m m−2 m Let now X = r∇r. Since Lr∇r , = 2 , , X is conformal with respect to , , and therefore also with respect to , . Recalling (2.17) we have +W + = m − 2 X(S). div 2m , to the Applying the divergence theorem on Br with respect to the metric above equation, and expressing the result in terms of the original metric we obtain 2m m−2 r ∇r, ∇S u m−2 dvolm = ru2 T(∇r, ∇r)dvolm−1 , (7.59) 2m Br ∂Br where we have used the fact that the outward unit normal to ∂Br with respect to , is u−2/(m−2) ∇r, that the m- and (m − 1)-dimensional volume elements are

208

Chapter 7. Some special cases

+ m−1 = u2(m−1)/(m−2) dvolm−1 respec+ m = u2m/(m−2) dvolm and dvol given by dvol + is deﬁned by + , Y = T(X, Y ). Now, tively, and that, according to (2.14), W W from the expression of T and X = r∇r, setting ∂u for ∇r, ∇u we obtain ∂r

ru T(∇r, ∇r) = 2ru 2

!

5 . " 2 2m ∂u 1 1 2 Δu − Hess(u)(∇r, ∇r) + r − |∇u| . m m−2 ∂r m

We insert this formula into (7.59) and observe that, using the expression for Δ in spherical coordinates, Hess(u)(∇r, ∇r) =

m − 1 ∂u ∂2u = Δu − − Δ∂Br u. ∂r2 r ∂r

Now we apply the divergence theorem on ∂Br and use |∇u|2 =

∂u ∂r

2

+ |∇∂Br u|2∂Br ,

to deal with the term Δ∂Br u, and ﬁnally use m+2 m−2 cm Δu = −Su ,

cm = 4

m−1 m−2

to obtain, after some simpliﬁcations,

2m 2m m−2 m−2 r∇r, ∇Su dvolm = rSu dvolm−1 Br ∂Br 5 %. $ 2 ∂u 1 ∂u 2 + mcm − |∇u| u dvolm−1 . + r 2 ∂r m−2 ∂r ∂Br

(7.60)

The above formula is in the typical form of a Rellich-Pohozaev identity. We are going to describe a simpler and more general way to obtain it. Let u ∈ C 2 (M ) be a solution of the equation Δu = h(x)f (u) (7.61) on a complete Riemannian manifold (M, , ) with h ∈ C 0 (M ) and f ∈ C 0 (R). Let u F (u) = f (s) ds, 0

and deﬁne the vector ﬁeld 1 W = h(x)F (u) + |∇u|2 X − X, ∇u∇u − αu∇u, 2

(7.62)

7.1. A nonexistence result

209

where X is a vector ﬁeld on M and α ∈ R. Using the identities ∇|∇u|2 , X = 2 Hess(u)(∇u, X), 1 ∇u∇u, X = Hess(u)(∇u, X) + LX , (∇u, ∇u), 2 we compute div W = h(x)(F (u) div X − αuf (u)) 1 1 div X − α |∇u|2 + F (u)∇h, X − LX , (∇u, ∇u). + 2 2 If we assume that X is a conformal vector ﬁeld, the above inequality becomes div W = h(x) F (u) div X − αuf (u) m−2 + div X − α |∇u|2 + F (u)∇h, X. 2m Integrating over the annulus Br \ Bro , 0 ≤ r0 < r, and using the divergence theorem we obtain r ∂u 1 ∂u 2 − αu div W = h(x)F (u) + |∇u| X, ∇r − X, ∇u . 2 ∂r ∂r Br ∂Bt ro

In particular, if (M, , ) is R with its canonical metric, and we choose X = r∇r, m+2 m−2 f (u) = u m−2 , h(x) = c−1 m S(x) and α = 2 , the above formula yields m

m−2 2m cm

∇ru m−2 r∇S, 2m

Br

0m−2

= ∂Br

2m cm

rSu

2m m−2

∂u 2 m − 2 ∂u 1r r u − |∇u|2 + r + , 2 ∂r 2 ∂r

(7.63)

ro

and, letting ro = 0, we recover (7.60). As an application of this identity, we consider the prescribed scalar curvature equation on Sm , m+2

cm ΔSm u − m(m − 1)u + K(x)u m−2 = 0. We use (the inverse of) stereographic projection to pull back the standard metric of Sm onto Rm . The pull-back metric is conformal to Euclidean metric with conformality factor 4(1 + |x|2 )−2 . Thus, the original conformal change of metric on Sm may be viewed as a conformal change of metric on Rm and we are led to investigating the equation m+2

cm Δu + b(x)u m−2 = 0

(7.64)

210

Chapter 7. Some special cases

on Rm , where b(x) is bounded and admits a ﬁnite limit at inﬁnity, and to look for positive solution u ∈ C ∞ (Rm ) such that u(x) ∼ |x|2−m

as |x| → +∞.

(7.65)

We claim that if b is radial, somewhere positive and monotone increasing where positive, then there are no positive radial solutions of (7.65) satisfying (7.65). Indeed, assume by contradiction that such a solution exists, and identify radial functions on Rm with functions on [0, +∞). Since b(x) is bounded, and u(r) ∼ r2−m as r → +∞, it follows from (7.64) that u (r) = O(r1−m ) as r → +∞. To prove this, note that since u is a radial solution of (7.64) it satisﬁes the ode m−1 m+2 m−1 r u = −c−1 b(r)u m−2 m r so that, integrating between ε and r, we obtain r m+2 m−1 m−1 −1 u (r) = ε u (ε) − cm b(t)tm−1 u(t) m−2 dt. r ε 2−m

Since b(t) is bounded, and u(t) ∼ t

, we have

m+2 b(t)tm−1 u(t) m−2 = O t−3 ,

so that the integral on the right-hand side is absolutely convergent at inﬁnity, and we deduce that r

m+2 |b(t)|tm−1 u(t) m−2 dt ≤ Er1−m as r → +∞ |u (r)| ≤ r1−m εm−1 |u (ε)|+c−1 m ε

as required. Next, by the assumption on b, there exists ro ≥ 0 such that b(r) ≤ 0 for 0 ≤ r ≤ ro and b(r) > 0, b ≥ 0, b ≡ 0 for r > ro .

(7.66)

We apply equation (7.63) with b(r) instead of S and with the value of ro speciﬁed above, and let r → +∞. Since b ≥ 0, by monotone convergence, the integral over Br \Bro converges to the integral over Rm \Bro , while, by the asymptotic behavior of u and u the boundary integral over ∂Br tends to zero, and we obtain 2m ro m−2 m−2 2 u (ro ) + u(ro )u (ro ) = − rb (r)u m−2 . (7.67) 2 2 2m c m m R \Bro Bro Now, if ro = 0, the left-hand side of the above equation vanishes, while the righthand side does not, and we immediately deduce the required contradiction. Otherwise, b ≤ 0 on Bro and it follows from (7.64) that Δu ≥ 0

on Bro ,

so that, by the maximum principle maxBro u = u(ro ) and u (ro ) ≥ 0, and, again, (7.67) yields a contradiction. We have therefore proved the following result due to W. Chen and C. Li ([CL95], Theorem 3)

7.1. A nonexistence result

211

Theorem 7.6. Let K be a nonconstant rotationally symmetric function on Sm , m ≥ 3,, and assume that K is somewhere positive, and monotone increasing in the region where it is positive. Then the equation m+2

cm Δu − m(m − 1)u + Ku m−2 = 0 has no positive smooth rotationally symmetric solutions on Sm .

7.1.2

A nonexistence result for hyperbolic space

As already mentioned in the introduction, in this section we will work in the Poincar´e model for hyperbolic space. The main reason is that the Rellich-Pohozaev formula has a particularly nice expression in this setting, and allows us to give a more transparent proof of our main nonexistence result. We leave it to the interested reader to translate the result in the standard model Hm = Rm with metric in polar coordinates on Hm \ {0} = (0, +∞) × Sm−1 given by , Hm = dr2 + sinh2 dθ2 , which has been used throughout the book. We recall that the Poincar´e model of Hm is (Bm , gH ), where Bm is the unit ball of Rm and gH is the metric conformally related to the canonical Euclidean metric , given by gH =

4 , , (1 − |x|2 )2

(7.68)

where, using standard notation, |x| = distRm (x, 0). We state the following problem: Given K ∈ C ∞ (Bm ), m ≥ 3, does there exist a positive function v on Bm such that 4

(i) gv = v m−2 gH has scalar curvature K and (ii) gv is a complete Riemannian metric on Bm ? Setting

5 u(x) = v(x)

. m−2 4

4 (1 − |x|2 )

2

,

we have that the above problem admits an aﬃrmative answer if and only if the equation m+2 cm Δu + K(x)u m−2 = 0, (7.69)

212

Chapter 7. Some special cases

where cm = 4 m−1 m−2 (as in the previous chapters) and Δ is the Euclidean Laplacian, admits a positive solution u such that the conformal metric 4

gu = u m−2 , is a complete Riemannian metric on Bm . We note that the requirement of completeness implies that u must diverge at the boundary of Bm , while equation (7.69) of course means that K is the scalar curvature of gu . We also recall that, letting r = |x| , then the Riemannian distance ρ from the origin of Bm with respect to the hyperbolic metric (7.68) is given by (see [RRV94b]) 1+r , (7.70) ρ = log 1−r while

ρ

eρ − 1 . (7.71) 2 eρ + 1 For the developments below, we will use the Rellich-Pohozaev-type formula which is obtained in the following way. Fix xo ∈ Bm , let X = x − xo and deﬁne a vector ﬁeld by the formula ' & 2m m−2 cm 2 m−2 |∇u| − Ku (x−xo )−cm x−xo , ∇u∇u+cm αu∇u. (7.72) W = 2 2m r = tanh

=

Using the fact that Lx−xo , = , , a computation similar to the one carried out in the previous section shows that 0 1 2m m−2 m−2 m−2 div W = cm K− − α |∇u|2 + α − x − xo , ∇K u m−2 , 2 2 m (7.73) and we obtain a Rellich-Pohozaev formula integrating and using the divergence theorem. We are now ready to state (see [RRV94b], Theorem 1.2) Theorem 7.7. Let K ∈ C ∞ (Bm ) and suppose that lim sup(1 − |x|2 )K(x) < 0. |x|→1−

Assume also that either (i) K ≤ 0 in Bm , or (ii) there exists α ≤ m−2 and xo ∈ Bm such that 2 m−2 m−2 K(x) − x − xo , ∇K(x) ≥ 0 on Bm , α− 2 2m with strict inequality in at least one point.

(7.74)

7.1. A nonexistence result

213

Then equation (7.69) does not admit any positive solutions on Bm . Note that if we assume that u is nonnegative and satisﬁes (7.69), then, by the strong maximum principle (see, [GT01], p. 35), either u > 0 or u ≡ 0. We split the proof into several steps. We note that steps 1 to 3 are devoted to showing that the solution u(x) must tend to zero in a speciﬁc way as the point x tends to the boundary of Bm . This fact could be obtained as an application of Theorem 4.4 in Chapter 4. However we follow the alternative method presented below to describe a technique of common use in Analysis. Proof. We assume that (7.69) admits a positive solution u and derive a contradiction. Step 1. Assumption (7.74) implies that u is bounded. To prove this fact we ﬁrst establish the following Proposition 7.8. Let BR ⊂ Rm be the ball of radius R centered at 0, let σ > 1, and let f be a positive function in C 0 (B R ). Then there exists a constant C = C(σ, m) > 0 such that, if u ∈ C 2 (BR ) satisﬁes the diﬀerential inequality Δu ≥ f (x)uσ on BR , then

1

u(o) ≤ C(R2 min f )− σ−1 .

(7.75) (7.76)

BR

Proof. Set A = minB R f , so that u satisﬁes Δu ≥ Auσ . For 0 < R1 < R deﬁne where

λ=

(7.77) 2

v(x) = λ(R12 − |x|2 )− σ−1 , B 2 R A 1

1 σ−1

B = max

) 2m σ+1 * . ,4 σ − 1 (σ − 1)2

A straightforward computation shows that Δv ≤ Av σ on BR1 ,

(7.78)

and v(x) → +∞ as |x| → R1 . Thus the function w = u − v attains an absolute maximum at a point xo ∈ BR1 , and satisﬁes the inequality Δw ≥ A(uσ − v σ ) = h(x)w, where h ∈ C 0 (BR1 ) is the nonnegative function deﬁned by # u(x) σ−1 σA t dt if u(x) = v(x), v(x) h(x) = u(x)−v(x) if u(x) = v(x). σAu(x)σ−1

(7.79)

214

Chapter 7. Some special cases

If w(xo ) > 0, then h(xo ) > 0, and Δw(xo ) ≥ h(xo )w(xo ) > 0, which is impossible since w attains a maximum at xo . Thus for every x ∈ BR1 , w(x) ≤ w(xo ) ≤ 0, that is u(x) ≤ v(x) in BR1 , and in particular, u(0) ≤ v(0) =

B A

1 σ−1

2 − σ−1

R1

− 1 1 = B σ−1 AR12 σ−1 ,

and letting R1 → R we obtain the desired estimate.

To prove the claim in Step 1, let u be a positive solution of (7.69). We ﬁx γ in (0, 1) and applying Proposition 7.8 in the ball B γ (x) = {y ∈ Rm : |y − x| ≤ γ(1 − |x|)}, we deduce that & − m−2 4 u(x) ≤ C (1 − |x|)2 max (−K(z)) . γ B (x)

Using (7.74) we deduce that u is bounded above as |x| → 1− and, it is therefore bounded. Step 2. We show that lim|x|→1− u(x) = 0. By (7.74) there exists constants 0 ≤ η < 1 and c > 0 such that c K(x) ≤ − (1 − |x|)−2 2

(7.80)

on the set Annη,1 = B1 \ B η = {x ∈ Bm : η < |x| < 1}. Let δ ∈ (η, 1) and let vδ be the maximal solution of the problem m+2 on Annη,δ , cm Δv = 2c (1 − |x|)−2 v m−2 lim|x|→η v(x) = lim|x|→δ v(x) = +∞, which exists by Lemma 6.4 in Chapter 6, and is radially symmetric since it must coincide with the maximal radial solution to (7.1.2). Since vδ (x) diverges as x tends to the boundary of Annη,δ , by usual comparison arguments, u < vδ on this set. Moreover, as δ → 1− , vδ decreases and converges to a radially symmetric function v which satisﬁes m+2 c cm Δv = (1 − |x|)−2 v m−2 on Annη,1 , 2

7.1. A nonexistence result

215

and 0 ≤ u(x) ≤ v(|x|). Therefore, it suﬃces to show that lim v(|x|) = 0.

|x|→1−

To this end, we ﬁrst note that, arguing as in Step 1, we can show that v is bounded in a neighborhood of ∂Bm . Next, since v is radial, it satisﬁes the ode cm (rm−1 v ) =

m+2 c m−1 (1 − r)−2 v m−2 , r 2

for η < r < 1. It follows that the function rm−1 v (r) is increasing in η < r < 1, hence it has constant sign in a left neighborhood of 1. Thus v itself tends to a nonnegative limit β as r → 1− . Assume by contradiction that β > 0. Then there exists η < τ < 1 and a constant θ > 0 such that (rm−1 v ) ≥ θrm−1 (1 − r)−2 , and integrating twice we deduce that v(r) → +∞ as r → 1− contradicting the fact that v is bounded. The contradiction shows that β = 0 and therefore limr→1− u(x) = 0. Step 3. There exists a constant M > 0 such that u(x) ≤ M (1 − |x|).

(7.81)

Indeed, the function ψ(x) = M (1 − |x|) is a supersolution of (7.69) in Annη,1 and, by taking M large enough we can arrange that u(x) ≤ ψ(x) on ∂Annη,1 . Thus, the usual comparison principle shows that u ≤ ψ on Annη,1 , and by increasing M if necessary, the inequality holds on Bm . Step 4. Suppose that K ≤ 0 on Bm . Then u is a nonnegative subharmonic function which vanishes on ∂Bm , and by the maximum principle u ≡ 0. Step 5. Using the Pohozaev-type formula (7.73), we prove that, if K satisﬁes condition (ii) in the statement of the theorem, then (7.69) has no positive solution on Bm .

216

Chapter 7. Some special cases m

Suppose ﬁrst that u ∈ C 1 (B ). We ﬁx ε > 0, integrate (7.73) over B1−ε and let ε → 1− . Using the fact that u = 0 on ∂Bm−1 , so that ∇u = ∂u ∂r ∇r, we obtain cm m−2 2 |∇u, ∇r| x − xo , ∇u = cm − − α |∇u|2 2 ∂Bm 2 Bm " ! 2m m−2 m−2 K− x − xo , ∇K u m−2 . α− + 2 2m Bm Since the left-hand side of the equality is nonpositive, and α ≤ m−2 we conclude 2 that " ! 2m m−2 m−2 (7.82) K− x − xo , ∇K u m−2 ≤ 0. α− 2 2m m B On the other hand, condition (ii) in the statement of the theorem and u > 0 imply that the left-hand side is strictly positive, and the required contradiction follows. However, in general we do not know that u is C 1 up to the boundary, and we need to justify that one may take the limit as ε → 0 to obtain (7.82). This can be done by the following approximation technique. Write K = K + − K − , and, for a ﬁxed δ > 0, let 1 Kδ− (x) = min{ , K − (x)}, δ and let v = vδ be the solution of the problem m+2 m+2 cm Δv − Kδ− v m−2 = −K + u m−2 v = 0 on ∂Bm .

(7.83)

in Bm ,

(7.84)

m+2

By the asymptotic behavior of K, the function K + u m−2 is Lipschitz and compactly supported in Bm , and since Kδ− is Lipschitz and bounded on Bm , the existence of v is guaranteed by a minimization argument, and by standard regularity, v is C 2 m in B . Moreover, if δ1 > δ2 , then Kδ−1 ≤ Kδ−2 , and by comparison, v δ2 ≤ v δ1

in Bm .

(7.85)

Recalling the equations satisﬁed by vδ and u and noting that Kδ− ≤ K − , we deduce that the function w = u − v satisﬁes m+2

m+2

cm Δw ≥ Kδ− (u m−2 − v m−2 ) = h(x)Kδ− w, m+2 . Since Kδ− ≥ 0 in Bm , and u = v = 0 with h(x) ≥ 0 deﬁned by (7.79) and σ = m−2 m on ∂B , by the maximum principle w ≥ 0 and therefore u ≤ vδ for every δ > 0. Thus as δ → 0, vδ converges to a function vo which vanishes on ∂Bm and satisﬁes vo ≥ u. Now, by Step 3, vδ is a solution of (7.84) and is uniformly bounded on Bm . Moreover, if r < 1, Kδ− = K on Br for δ small enough, so that, by standard

7.1. A nonexistence result

217

elliptic estimates (see, e,g., [Aub98], Theorem 4.40), we deduce that vδ is uniformly bounded in C 1 (Br ) and therefore it converges to vo uniformly on Br . On the other hand, again by Step 3, vδ satisﬁes the estimate (7.81) and therefore {|∇r, ∇vδ |} is bounded and decreasing, and it follows that vδ tends to zero as x → ∂Bm uniformly in δ. Combining these two facts we conclude easily that vδ → vo uniformly on Bm . Integrating equation (7.74) against a test function and letting δ → 0 shows that vo belongs to W 1,2 (Bm ) ∩ C 0 (Bm ) and it is a weak solution of m+2

m+2

cm Δvo = K − vom−2 + K + u m−2 . Thus w = vo − u is a nonnegative weak solution of m+2 m+2 m−2 − m−2 −u Δw = K vo = h(x)w, where h ≥ 0 is deﬁned in (7.79). Since w = 0 on ∂Bm , by comparison w ≤ 0 and therefore w ≡ 0, that is vo = u. Next we ﬁx δ > 0 and we consider the vector ﬁeld & ! "' 2m m+2 m−2 cm − m−2 2 + m−2 −K u vδ Wδ = Kδ vδ |∇vδ | + (x − xo ) 2 2m − cm x − xo , ∇vδ ∇vδ + cm αvδ ∇vδ . Writing v instead of vδ , a computation similar to that leading to (7.73) shows that 2m m+2 m−2 m−2 − α |∇v|2 + x − xo , ∇Kδ− vδm−2 − ∇K + u m−2 vδ divWδ = cm 2 2m 2m m+2 m−2 Kδ− vδm−2 − K + u m−2 vδ − L(v), (7.86) + α− 2 where L(v) = L(vδ ) =

m+2 4 m + 2 + m−2 m + 2 + m−2 + x − xo , ∇u K vu K vu 2 m−2 m+2 m + 2 m−2 + x − xo , ∇K + . vu 2m

Integrating (7.86) over Bm , noting that v is C 1 on Bm and vanishes on ∂Bm , and using the divergence theorem we obtain cm − x − xo , ∇r∇v, ∇r2 2 ∂Bm m+2 2m m−2 m−2 − α |∇v|2 + x − xo , ∇Kδ− v m−2 − ∇K + u m−2 v = cm 2 2m Bm

m+2 2m m−2 ∇Kδ− v m−2 − ∇K + u m−2 v − L(v). −α + 2 Bm Bm

218

Chapter 7. Some special cases

Since K + has compact support in Bm , as δ → 0, the integral L(vδ ) converges to L(u). Moreover, L(u) =

m+2 2m m + 2 + m−2 m + 2 + m−2 + x − xo , ∇u K u K u 2 m−2 2m 2m m + 2 m−2 m+2 + x − xo , ∇K + = u div K + u m−2 (x − xo ) , 2m 2m

so that L(u) = 0 and L(vδ ) = o(δ) as δ → 0. Summing up, recalling that α ≤ m−2 2 , we have obtained that 0 m+2 2m m − 2 + m−2 K vu α− − Kδ− v m−2 o(δ) + 2 Bm 1 m+2 2m m−2 − x − xo , vu m−2 ∇K + − v m−2 ∇Kδ− , < 0 2m

(7.87)

with o(δ) → 0 as δ → 0. Next, we rewrite the integrand in the above formula as 0 1 2m m−2 m−2 )(K + − Kδ− ) − x − xo , ∇(K + − Kδ− ) v m−2 (α − 2 2m 0 1 m+2 2m m−2 + m−2 )K − x − xo , ∇K + vu m−2 − v m−2 , + (α − 2 2m and note that since K + is compactly supported in Bm and vδ → u uniformly, the integral of the second summand in tends to zero as δ → 0. Hence, (7.87) yields 0 1 2m m−2 m−2 (α − lim sup )(K + − Kδ− ) − x − xo , ∇(K + − Kδ− ) v m−2 ≤ 0. 2 2m δ→0 Bm (7.88) On the other hand, let Gδ = {x : K(x) < −1/δ}, so that Gδ is open and K = −Kδ− = −1/δ on Gδ . It follows that ∇K = −∇Kδ− = 0 on Gδ , and, using again − − α ≤ m−2 2 , the fact that Kδ = K in the set where K ≥ −1/δ and using condition (ii) in the statement we deduce that

α−

m−2 m − 2 + (K − Kδ− )(x) − x − xo , ∇(K + − Kδ− )(x) ≥ 0 2 2m

(7.89)

almost everywhere on Bm , so that the limit in (7.88) exists and is equal to zero. Finally, since the integrand in (7.88) is nonnegative, and, as δ → 0, tends to 1 2m m−2 m − 2 + (K − K − )(x) − x − xo , ∇(K + − K − )(x) u m−2 α− 2 2m almost everywhere in Bm , we conclude from Fatou’s lemma that " ! 2m m − 2 m−2 α− K− x − xo , ∇K u m−2 ≤ 0, 2 2m m B and, using (ii) and u > 0 a contradiction is reached as above.

(7.90)

7.1. A nonexistence result

7.1.3

219

An integral obstruction

The aim of this subsection is to give a necessary condition for the existence of a metric conformally related to the Poincar´e metric (7.68) on Bm with assigned scalar curvature K(x); we recall that, as observed before, this problem consists in ﬁnding a positive solution of equation (7.69). The nature of the obstruction that we are going to describe is manifestly diﬀerent from that of the Kazdan-Warner condition (see Chapter 2) which, basically, succesfully applies only in the compact setting. However, its generalization to a generic complete manifold is still missing. We have the following (see [RRV94b], Proposition 1.1) Proposition 7.9. Suppose that K ∈ L1 (Bm ). If equation (7.69) admits a positive solution u such that lim u(x) = +∞, (7.91) − |x|→1

then the following inequalities hold: B $

m

1

r1−m

K(x) < 0; %

(7.92)

K(x) dr < 0

(7.93)

|x|
s

for all s ∈ [0, 1). m+2 . For 1 > ε > 0 we consider a Proof. Step 1. First we prove (7.92). Let γ = m−2 family of nonnegative, nondecreasing functions ρε (r), r ∈ [0, 1), such that 0 if 0 ≤ r ≤ 2ε , ρε (r) = (7.94) r − 2ε if ε ≤ r < 1.

We multiply equation (7.69) by ρε cm Δu ρε

1 uγ

1 uγ

to obtain

+ Kuγ ρε

1 uγ

= 0.

(7.95)

The function ρε u1γ (x) is compactly supported in Bm , because of (7.91); thus we can apply the ﬁrst Green’s formula in (7.95) to obtain - , 1 1 γ ∇u, ∇ ρε γ + (7.96) Ku ρε γ = 0. −cm u u Bm Bm An easy computation shows that the ﬁrst integral in (7.96) is equal to 1 2 u−γ−1 ρε γ |∇u| . cm γ u Bm

220

Chapter 7. Some special cases

Since ρε > 0 if u < ε1γ , we deduce that, for ε > 0 small enough, 1 1 2 Kuγ ρε γ = −γ u−γ−1 ρε γ |∇u| < 0. u u Bm Bm

(7.97)

We let ε → 0; since 0 < ρεr(r) < 1, by Lebesgue’s theorem the left-hand side term in (7.97) admits a limit, which is K(x) ≤ 0. (7.98) Bm

To be more precise, we can arrange to have that the function ε → ρε (r) is nondecreasing for all r ≥ 0; in this way, we conclude that the right-hand side in (7.97) decreases to its limit as ε tends to 0. Thus we ﬁnd 2 K(x) = −γ u−α−1 |∇u| < 0. (7.99) Bm

Bm

Step 2. We prove now (7.93). First we observe that 2 div u−α ∇u = u−α Δu − αu−α−1 |∇u| .

(7.100)

Next, we rewrite equation (7.69) as −cm u−α Δu = K.

(7.101)

We use (7.100) in (7.101) and integrate over |x| < r to obtain 1 2 K(x) = −γ u−γ−1 |∇u| − u−α ∇u, ν , cm |x|
(7.102)

where ν is the exterior unit normal vector to Sr = rSm−1 . Using polar coordinates (r, σ) ∈ (0, 1) × Sm−1 , we rewrite the last integral in (7.102) as ∂ −γ+1 rm−1 −α u ∇u, ν = (r, σ) dσ, (7.103) u 1 − γ ∂r |x|=r Sm−1 where dσ is the canonical measure on Sm−1 . We also observe that u−γ+1 (1, σ) dσ = 0

(7.104)

Sm−1

by (7.91). We multiply both sides of (7.102) by cm r1−m and integrate over the interval (s, 1) using (7.103) and (7.104); we deduce 5 $ % % $ 1

s

1

r1−m

|x|
K(x) dr = − cm γ

r1−m

s

+

1 γ−1

from which (7.93) follows immediately.

|x|
u−γ−1 |∇u|

u1−γ (s, σ) dσ

2

dr

" (7.105)

Sm−1

7.2. Special symmetries and existence

7.2

221

Special symmetries and existence

In this section we prove an existence result (see Theorem 7.12 below) where we allow the coeﬃcient a(x) to be nonpositive everywhere on M . However, this is obtained by assuming the strong curvature assumption (7.116). A geometrical application of this result is contained in Corollary 7.13. In what follows we shall use the following version of Gronwall inequality. Lemma 7.10. Let ϕ and ψ be nonnegative, locally integrable functions on [0, +∞) and let η be a solution of the diﬀerential inequality η (r) ≤ ϕ(r)η(r) + ψ(r)

on [0, +∞).

(7.106)

Then, ∀ r ∈ [0, +∞), η(r) ≤ η(0)e

r 0

ϕ(t) dt

r

+

ψ(t)e

r t

ϕ(ξ) dξ

dt.

(7.107)

0

Proof. First of all we observe that, ∀ s ∈ [0, +∞),

η(r)e−

r s

ϕ(t) dt

=e

s r

ϕ(t) dt

{η (r) − ϕ(r)η(r)},

so that, using (7.106), we deduce

η(r)e−

r s

ϕ(t) dt

≤ ψ(r)e−

r s

ϕ(t) dt

.

Integrating this inequality on [0, r] gives η(r)e−

r s

ϕ(t) dt

≤ η(0)e

s 0

ϕ(t) dt

r

+

ψ(t)e−

t s

ϕ(ξ) dξ

dt

0

and therefore the validity of (7.107).

A second ingredient we shall use is the following Proposition 7.11. Let Mh be an m-dimensional model and suppose that α2 ≤ h (r) Cr h(r)

(7.108)

> 0, α > −2. Let σ > 1; then, given ε > 0, there for r 1 and some constants C exists δ > 0 such that for β0 < δ the initial value problem β + (m − 1) hh (r)β = g(r)β σ , (7.109) β (0) = 0, β(0) = β0

222

Chapter 7. Some special cases

with g ∈ C 0 ([0, +∞)) satisfying

(i) g > 0 (ii) g(r) ≤

α r 1− 2

C (log r)1+ε

on [0, +∞), for r 1

(7.110)

has a positive, (bounded) C 2 -solution β on [0, +∞). Proof. First, we produce a subsolution of (7.109) on [0, +∞). We follow an idea contained in the proof of Proposition 3.2 of [RRS95]. Fix R > 1 and deﬁne g ∈ C 0 ([0, +∞)) by setting ) * ⎧ ε(log R)σε h (r) 1 1+ε ⎨ r(log (m − 1) for r ≥ R, − − 1+ε r) h(r) r r log r ) * g(r) = σε h (R) ⎩ ε(log R) 1 1+ε for 0 ≤ r < R. R(log R)1+ε (m − 1) h(R) − R − R log R Next, set γ(r) =

1 1 − (log R)ε (log r)ε

on [R, +∞).

A direct computation shows that γ + (m − 1)

h 1 (r)γ = g(r) h (log R)σε

on [R, +∞). Furthermore, γ is increasing on [0, +∞) and lim γ(r) =

r→+∞

It follows that γ(r)σ ≤

1 (log R)εσ

1 . (log R)ε

on [R, +∞)

and therefore γ satisﬁes γ + (m − 1)

h (r)γ ≥ g(r)γ(r)σ h

(7.111)

on [R, +∞). However, from (7.108), using also α > −2 we have that, up to choosing R suﬃciently large, for some absolute constant T > 0, g(r) ≥

T ε(log R)σε α r1− 2 (log r)1+ε

∀ r ∈ [R, +∞) and we can also suppose that R is large enough so that T ε(log R)σε > C

7.2. Special symmetries and existence

223

where C is the constant appearing in (7.110) (ii). It follows that g(r) ≥ g(r)

on [R, +∞).

(7.112)

h (r)γ ≥ g(r)γ(r)σ . h

(7.113)

Then, (7.111) and (7.112) imply γ + (m − 1)

Next we choose δ > 0 so small that the solution of (7.109) with 0 < β0 < δ is deﬁned on a maximal interval [0, R1 ) with R + 1 < R1 ≤ +∞ and in such a way that, having set R0 = R + 1, β(R0 ) < γ(R0 ), β (R0 ) < γ (R0 ).

(7.114)

If R1 = +∞ we have nothing to prove; thus let R1 < +∞. On [R0 , R1 ) set w = β − γ so that, by (7.109) and (7.113), w satisﬁes w + (m − 1)

h (r)w − g(r)(β σ − γ σ ) ≤ 0. h

(7.115)

Let r¯ ∈ [R0 , R1 ) be chosen; then on [R0 , r¯] we have the validity of (7.115). Deﬁne z(r) on [R0 , r¯] as # γ(r) σ−1 σ t dt if β(r) = γ(r), β(r) z(r) = γ(r)−β(r) if β(r) = γ(r). σβ(r)σ−1 Then z(r) ∈ C 0 ([R0 , r¯]), z(r) ≥ 0 and (7.115), (7.114) can be now written as w + (m − 1) hh (r)w − c(r)w ≤ 0, w(R0 ) < 0, w (R0 ) < 0, with c(r) = g(r)z(r) ≥ 0. Therefore, by the maximum principle, w attains its negative minimum either at r¯ or at R0 . The latter case cannot occur because it would imply w (R0 ) ≥ 0; hence the negative minimum is achieved at r¯, that is, β(¯ r) < γ(¯ r). Since r¯ ∈ [R0 , R1 ) was chosen arbitrarily, we have that β(r) < γ(r)

on [R0 , R1 )

contradicting the fact that β(r) → +∞ as r → R1 since [0, R1 ) was its maximal interval of deﬁnition. We are now ready to prove Theorem 7.12. Let (M, , ) be a complete Riemannian manifold with a pole o satisfying Riem(M, , ) ≤ −B 2 , (7.116)

224

Chapter 7. Some special cases

for some constant B > 0. Let a(x), b(x) ∈ C 0,μ (M ) for some 0 < μ ≤ 1. Assume that b(x) is nonnegative and strictly positive outside some compact set and that B0 = {x ∈ M : b(x) = 0} a(r), 6b(r) ∈ C 0 ([0, +∞)) be such that satisﬁes λL 1 (B0 ) > 0 with L = Δ + a(x). Let 6 6 a(r) ≤ 0 6b(r) ≤ C e

on [0, +∞), σ−1 B(m−1)

r 0

(7.117)

a(ξ) dξ

(7.118)

r(log r)1+ε

for r 1 and some σ > 1, ε > 0, and suppose that a(x) ≥ 6 a(r(x)), 6 b(x) ≤ b(r(x))

(7.119)

on M . Then, the equation Δu + a(x)u − b(x)uσ = 0 admits a positive maximal C 2 -solution on M . Proof. We ﬁrst consider the ODE problem a(r)α = 0 on [0, +∞), (i) α + (m − 1)B coth(Br)α + 6 (ii) α (0) = 0, α(0) = α0 ∈ (0, 1].

(7.120)

Standard Picard iteration (see for instance [CL55]) guarantees the existence of a solution α on [0, +∞). Then α solves the integrated equation t r 1−m m−1 α(r) = α0 − (sinh Bt) (sinh Bs) 6 a(s)α(s) ds (7.121) 0

0

on [0, +∞). Thus, using (7.117), we have r 1−m B(cosh Bt)(sinh Bt) α(r) = α0 + 0

≤ α0 +

r

B(cosh Bt)(sinh Bt) 0

t 0 t

m−1

(sinh Bs) (−6 a(s))α(s) ds B cosh Bs 0 /r m−1 (sinh Bs) (−6 a(s))α(s) ds B cosh Bs

1−m

t 1 2−m (sinh Bt) 2−m 0 r 1 + tanh Bt(−6 a(t))α(t) dt B(m − 2) 0 r 1 6 a(t)α(t) dt, ≤ α0 − B(m − 2) 0 = α0 +

m−1

(sinh Bs) (−6 a(s))α(s) ds B cosh Bt

0

7.2. Special symmetries and existence

225

that is, α(r) ≤ α0 −

1 B(m − 2)

Setting 1 η(r) = − B(m − 2)

r

6 a(t)α(t) dt.

0

(7.122)

r

6 a(t)α(t) dt,

0

from (7.122) we deduce η (r) ≤

α0 1 (−6 a(r))η(r) + (−6 a(r)). B(m − 2) B(m − 2)

Furthermore, η(0) = 0. Applying Lemma 7.10 and (7.122) we obtain r r 1 α0 (−6 a(t))e t B(m−2) (− a(ξ)) dξ dt. α(r) ≤ α0 + η(r) ≤ α0 + 0 B(m − 2) It follows that

α(r) ≤ α0 − α0

r 0

d tr e dt

that is,

− a(ξ) B(m−2)

1

dξ

α(r) ≤ α0 e− B(m−2)

dt = α0 e

r 0

a(t) dt

r

− a(ξ) 0 B(m−2)

dξ

,

.

(7.123)

Note that from (7.120) and (7.117) we also deduce α ≥ 0. We now deﬁne g(r) = 6b(r)α0σ−1 e− B(m−2) σ−1

r 0

a(ξ) dξ

>0

(7.124)

and observe that, because of (7.124) and (7.118), g(r) ≤

C r(log r)1+ε

(7.125)

for r 1 and some constant C > 0. Applying Proposition 7.11 we deduce the existence of a positive solution β of the ODE problem β + (m − 1)B coth(Br)β − g(r)β σ = 0 on [0, +∞), (7.126) β (0) = 0, β(0) = β0 > 0. Note that β ≥ 0 since g(r) ≥ 0. Next we deﬁne t r ) * 1−m m−1 6 b(s)γ(s)σ − 6 γ(r) = γ0 + (sinh Bt) (sinh Bs) a(s)γ(s) ds. 0

0

Choosing γ0 ∈ (0, α0 β0 ) we claim that γ(r) is deﬁned on all of [0, +∞). Observe that if γ(r) is bounded above on each interval of the type [0, T ], then we are done. ¯ as Towards this aim it is suﬃcient to show that deﬁning R ¯ = sup {r0 ∈ (0, +∞) : γ(r) ≤ α(r)β(r) R

on [0, r0 ]}

226

Chapter 7. Some special cases

¯ = +∞. We suppose the converse, that is, R ¯ < +∞. Then, using our we have R choice of γ0 and (7.123), t R¯ ) * 1−m m−1 6 ¯ b(s)γ(s)σ − 6 (sinh Bt) (sinh Bs) a(s)γ(s) ds γ(R) = γ0 + 0

0

(7.127) < α0 β0 R¯ t 1−m m−1 + (sinh Bt) (sinh Bs) {g(s)α(s)β(s)σ − 6 a(s)α(s)β(s)} ds. 0

0

Since α , β ≥ 0 we have aαβ, ΔHm 2 αβ = αΔHm 2 β + βΔHm 2 α + 2α β ≥ gαβ σ − 6 −B

−B

−B

in other words,

m−1 m−1 (sinh Br) (αβ) ≥ (sinh Br) {g(r)αβ σ − 6 a(r)αβ}. ¯ and using α (0) = β (0) = 0, α(0) = α0 , β(0) = β0 we get Integrating over [0, R] R¯ 1−m (sinh Bt) α(R)β(R) ≥ α0 β0 + 0 (7.128) t m−1 σ × (sinh Bs) {g(s)α(s)β(s) − 6 a(s)(r)α(s)β(s)} ds. 0

Since α0 ∈ (0, 1], comparing (7.127) and (7.128) we obtain ¯ < α(R)β( ¯ R) ¯ γ R ¯ It follows that γ is deﬁned on [0, +∞) and it contradicting the deﬁnition of R. satisﬁes γ + (m − 1)B coth(Br)γ + 6 a(r)γ − 6b(r)γ σ = 0 on [0, +∞), (7.129) γ (0) = 0, γ(0) = γ0 > 0. Furthermore, since 6 a(r) ≤ 0 and 6b(r) ≥ 0 we have γ (r) ≥ 0

on [0, +∞).

Deﬁning u− (x) = γ(r(x))

on M,

assumptions (7.116), (7.119), γ ≥ 0 and the Hessian comparison theorem (see Chapter 1) show that u− is a positive subsolution of Δu + a(x)u − b(x)uσ = 0. Applying Theorem 6.5 we obtain the desired conclusion.

7.3. The case of Euclidean space and further results

227

Applying Theorem 7.12 we have that the equation m+2

cm ΔHp+1 v − (m − 1)(m − 2 − 2p)v + K(x)v m−2 = 0 with 0 ≤ p < on Hp+1 and

m−2 2

(7.130)

has a smooth positive solution v on Hp+1 whenever K(x) ≤ 0 K(r) ≥ −C

e−

m−2−2p m−1 r(x)

r(log r)

(7.131)

1+ε

for r(x) 1 and some constants C, ε > 0. Letting (θ, x) be the generic point in Sm−p−1 × Hp+1 we deﬁne u(θ, x) = v(x). Then

m+2

cm Δu − (m − 1)(m − 2 − 2p)u + K(x)u m−2 = 0 on Sm−p−1 × Hp+1 . We have thus proved Corollary 7.13. Let the generic point of Sm−p−1 × Hp+1 be denoted by (θ, x) and 6 x) = K(x) with K(x) ≤ 0 on Sm−p−1 × Hp+1 and satisfying (7.131). Let let K(θ, m−p−1 × Hp+1 can be conformally 0 ≤ p < m−2 2 ; then, the product metric on S 6 deformed to a new metric with scalar curvature K(θ, x).

7.3

The case of Euclidean space and further results

We observe that the techniques introduced above can be used to produce positive supersolutions of Δu − a(x)u = 0 on M with a controlled behaviour at inﬁnity; this of course requires again the strong geometric assumption (7.38). However, we shall later use this technique on Rm to produce interesting results from the point of view of the study of the qualitative behaviour of positive solutions of Yamabe type equations.

7.3.1

A linear comparison result

We begin with the following comparison result. Lemma 7.14. Let Mg be a model and let a(r) ∈ C 0 ([0, +∞) be nonnegative. Let u = α ◦ r be a positive, radial, C 2 -solution on Mg of Δu + a(r)u ≤ 0

(7.132)

and let v = β ◦ r be a positive radial C 2 -subsolution of (7.132) on M \BR (o) for some R > 0 such that β (R) > 0. Then, there exists a constant C > 0 such that u(x) ≤ Cv(x)

on M \BR (o).

(7.133)

228

Chapter 7. Some special cases

Proof. First of all we observe that existence of a solution u > 0 of (7.132) implies by a result of Fisher-Colbrie and Schoen, [FCS80], that the operator L = Δ + a(x) satisﬁes λL (7.134) 1 (M ) ≥ 0. Next, (7.132) yields (r) α + a(r)α ≤ 0 on [0, +∞), α + (m − 1) gg(r) α (0) = 0

(7.135)

so that nonnegativity of a(r), positivity of α and (7.135) imply α (r) ≤ 0. Having made this observation, we choose ξ > 0 to satisfy ξβ (R) > α (R)

and

ξβ(R) > α(R).

(7.136)

¯ > R and let ϕ be a positive radial solution of the Dirichlet eigenvalue Next, we ﬁx R problem for L on BR+1 . Because of (7.134) ϕ satisﬁes ¯ (r) ϕ + (m − 1) gg(r) ϕ + a(r)ϕ ≤ 0, (7.137) ¯ ϕ (0) = 0, ϕ(R + 1) = 0. In particular

ϕ (t) ≤ 0

¯ + 1). on [0, R

(7.138)

¯ we consider the function On the interval [R, R] w(t) = α(t) − ξβ(t); because of our choice ξ it follows that i) w (R) < 0;

ii) w(R) < 0.

(7.139)

On the other hand, w satisﬁes Δw + a(r)w ≤ 0

on BR¯ \BR .

(7.140)

But, on BR¯ \BR , ϕ is positive and solves Δϕ + a(r)ϕ ≤ 0. By the generalized maximum principle (see [PW67], page 73) we have, either w = cϕ ¯ for some negative constant c or w ϕ has its negative absolute minimum at R or R. However, in this latter case, we must have w (R) ≥ 0; (7.141) ϕ

7.3. The case of Euclidean space and further results

229

but from (7.138) and (7.139), w (R)ϕ(R) − ϕ (R)w(R) < 0 ¯ < 0. Since R ¯ > R was contradicting (7.141). We therefore conclude that w(R) chosen arbitrarily we have proved u ≤ ξw

on M \BR .

This shows the validity of (7.133).

7.3.2

Back to Corollary 5.8

We give the next result in order to comment again on Corollary 5.8. We begin with the following observation which is contained in the proof of Theorem 7.3. Theorem 7.15. Let (M, , ) be a complete, connected, simply connected manifold satisfying Riem ≤ −B 2 for some constant B > 0. Let u be a nonnegative C 2 -solution of Δu + a(x)u ≥ 0 on BR with u(0) > 0; suppose a(x) ≤ AB 2 coth(Br(x))

on M, with A ≤

Then

(m − 1)2 . 4

m−1

r e 2 Br u ¯(r) ≥ C vol(∂Br )

(7.142)

for some constant C > 0 independent of r. A similar result holds when B = 0 but we shall analyze this and the next proposition later on in more details. Proposition 7.16. Let (M, , ) be a complete manifold of dimension m ≥ 2, connected and simply connected. Suppose that Riem ≤ −B 2 ,

B>0

and for a(x) ∈ C 0 (M ), a(x) ≤

(m − 1)2 2 B coth(Br(x)) 4

on M.

Then, there exists a positive C 2 -supersolution w of Δw + a(x)w ≤ 0

(7.143)

230

Chapter 7. Some special cases

satisfying C −1 r(x)e−

m−1 2 Br(x)

≤ w(x) ≤ Cr(x)e−

m−1 2 Br(x)

(7.144)

for some constant C > 0 and r(x) 1. Proof. We choose 6 a(r) ∈ C 0 ([0, +∞) to satisfy i) 6 a(r) > 0 on [0, +∞); and, for some ﬁxed ε > 0, 6 a(r)

= ≤

(m−1)2 2 B 4 (m−1)2 2 B 4

ii) a(x) ≤ 6 a(r(x)) on M

coth Br

on [ε, +∞),

coth Br

on [0, ε).

(7.145)

(7.146)

Next, we consider hyperbolic space Hm , ds2 with constant sectional curvature −B 2 . On Hm \{o} = (0, +∞) × Sm−1 we represent the metric in the form ds2 = dt2 +

1 sinh2 (Bt) dθ2 B

(7.147)

where dθ2 is the canonical metric on Sm−1 and t(x) = dist(Hm ,ds2 ) (x, o). On Hm the equation a(r)u = 0 (7.148) Δds2 u + 6 has a positive radial solution. Indeed, by (7.147) a radial solution of (7.148) is of the form u = α ◦ t with α a solution of α + (m − 1)B coth(Bt)α + 6 a(t)α = 0 on [0, +∞), (7.149) α (0) = 0, α(0) = α0 , and it is well known that (7.149) admits a solution. Having chosen α0 > 0 so that u(0) = α0 > 0, Theorem 7.15 implies that u is positive on Hm . We ﬁx b > 0 and consider − m−1 v(t(x)) = [t(x) − b][sinh Bt(x)] 2 on (b, +∞). It is immediate to verify that v is a positive radial subsolution of (7.148) on Hm \B¯b . Furthermore v (b + ε) > 0 for some ε > 0 suﬃciently small. Then, according to Lemma 7.14 and Theorem 7.15, u(x) = α(t(x)) satisﬁes C −1 t(x)e−

m−1 2 Bt(x)

≤ u(x) ≤ Ct(x)e−

m−1 2 Bt(x)

(7.150)

for some constant C > 0 and t(x) 1. We deﬁne, on (M, , ), w(x) = α(r(x));

(7.151)

7.3. The case of Euclidean space and further results

231

it is then clear that w satisﬁes (7.144). Furthermore, from (7.145) and (7.149) we have (7.152) α (t) ≤ 0 on [0, +∞). By the Hessian comparison theorem, see (7.45), we then deduce that w satisﬁes (7.143) and it is C 2 and positive. We go back to Corollary 5.8. In the assumptions of Proposition 7.16 we see that two positive solutions u and v of Δu + a(x)u − b(x)uσ = 0, σ > 1 b(x) ≥ 0, b(x) ≡ 0, coincide provided

m−1 u(x) − v(x) = o r(x)e− 2 Br(x)

on M,

as r(x) → +∞

(7.153)

or, in other words, (u(x) − v(x))

e

m−1 2 Br(x)

r(x)

→0

as r(x) → +∞.

(7.154)

This may seem to be a stronger requirement than (5.81), but this is not the case, because there (m − 1)2 m(m − 2) < a(x) ≡ 4 4 and it is in this latter case that we have produced w in Proposition 7.16. We can also use Proposition 7.16 to explicitate assumption (3.35) in the nonexistence result of Theorem 3.3. However, we shall now focus on the case of Rm .

7.3.3

The Euclidean space

To deal with the case of Rm we need a number of reﬁned results contained in [BRS98]. Lemma 7.17. Let a(t), b(t) ∈ C 0 ([0, +∞)) satisfy b(t) ≥ 0

(7.155)

and

A2 m−2 , m ≥ 3. with A ≤ t2 2 Let α be a solution of the problem α + (m − 1) 1t α + a(t)α − b(t)α|α|σ−1 ≥ 0, α(0) = α0 > 0, α (0) = 0 a(t) ≤

(7.156)

(7.157)

232

Chapter 7. Some special cases

on [0, T ), T ≤ +∞, σ > 1. Then α is positive; moreover, for all δ < T suﬃciently small and for all t ∈ [δ, T ) we have ⎧ δ m−2 2 ⎪ ⎨ m−2 if A = m−2 log δt 4 α0 t 2 ,

√ √ (7.158) α(t) ≥ 1 2 2 2 2 m−2− (m−2) −4A −4A ⎪ √ (m−2) ⎩ 14 m−2+ α0 δt 2 if A < m−2 2 . 2 2 (m−2) −4A

Remark. In particular, any solution of the problem β + m−1 t β + a(t)β ≥ 0, β(0) = β0 > 0, β (0) = 0

(7.159)

with a(t) as in the statement of Lemma 7.17, is deﬁned and positive on [0, +∞) and satisﬁes the estimate from below given in (7.158). Proof. We follow the lines outlined above for the general case of a Riemannian manifold with curvature bounded above. Let [0, T) be the maximal subinterval of [0, T ) on which α is positive, and ﬁx 0 < δ < s ≤ T. We are going to show that T = T by applying the following version of Green’s second identity, (7.160) sm−1 [ϕ (s)α(s) − ϕ(s)α (s)] = δ m−1 [ϕ (δ)α(δ) − ϕ(δ)α (t)] s m−1 [ϕ (t)α(t) − ϕ(t)α (t)] + (m − 1)tm−2 [ϕ (t)α(t) − ϕ(t)α (t)] dt t + δ

to the functions α and ϕ, where ϕ = ϕs is the solution of the problem A2 ϕ + m−1 t ϕ + t2 ϕ = 0 on [δ, s], ϕ(s) = 0, ϕ (s) = s−(m−1) .

(7.161)

By (7.161), ϕ is negative and monotonically increasing in (δ, s), and it is easy to check that ⎧ 2−m 2−m if A = m−2 s 2 t 2 log st ⎪ 2 , ⎪ ⎪

√ ⎪ 1 ⎨ (m−2)2 −4A2 1 2 2−m− ϕ(t) = − √(m−2)2 −4A2 t

⎪ √ ⎪ 1 √(m−2)2 −4A2 2 2 ⎪ ⎪ ⎩× s 2 2−m+ (m−2) −4A 1 − st if A < m−2 2 . A straightforward computation shows that α(t)ϕ (t) − α (t)ϕ(t) ≥ −

m−1 [α(t)ϕ (t) − α (t)ϕ(t)] − ϕ(t)b(t)α(t)|α(t)|σ−1 , t

so that, substituting in Green’s identity above yields s tm−1 ϕ(t)b(t)α(t)|α(t)|σ−1 dt+δ m−1 [ϕ (δ)α(δ) − ϕ(δ)α (δ)]. (7.162) α(s) ≥ − δ

7.3. The case of Euclidean space and further results

233

Since s ≤ T, the above integral is negative so that α(s) ≥ δ m−1 [ϕ (δ)α(δ) − ϕ(δ)α (δ)]. s 2−m m m−2 2 δ− 2 1 + For A = m−2 2 , using ϕ (δ) = s 2 log δ , we can estimate the right-hand side of (7.162) by & ' s

m m−2 1 δ 2 α(δ) − |α (δ)| m−2 log , 2δ δ s 2 and since α(0) = α0 > 0 and α (0) = 0, the quantity in braces is greater than m−2 4δ α0 > 0 for δ > 0 suﬃciently small. We therefore conclude that the ﬁrst inequality of (7.158) holds in [δ, T]. Similarly, if A < m−2 2 , one veriﬁes that the second inequality of (7.158) holds in [δ, T]; together with the deﬁnition of T, this implies that T = T, and the proof is complete. The next lemma deals with the case where a(t) decays at inﬁnity faster than 2 the critical decay At2 . Lemma 7.18. Let a(t), b(t) ∈ C 0 ([0, +∞)). Assume that b(t) ≥ 0, and that there exist constants A > 0 and ε > 0 such that & ' (m − 2)2 A2 a(t) ≤ min for t ≥ 0. , 4t2 t2+ε If α is a solution of (7.159) deﬁned on [0, +∞), then there is a constant C > 0 depending only upon α(0) and a(t), such that α(t) ≥ C

for t ∈ [0, +∞).

(7.163)

Proof. The idea of the proof is the same as that of Lemma 7.17 and consists of applying Green’s identity (7.160) with suitable test functions ϕ having the properties speciﬁed in the next Claim. For all s suﬃciently large, there exists a piecewise C 2 -function ϕs with the following properties: (i) ϕs (t) +

m−1 ϕs (t) + a(t)ϕs (t) ≥ 0 t

(ii) ϕs < 0

in (0, s), ϕs (s) = 0

and

on (0, s];

lim sm−1 ϕs (s) = 1;

s→+∞

(iii) there exist constants c1 , c2 independent of s such that, for δ small enough, |ϕs (δ)| ≤ c1

1 log δ

− m−2 2 and

ϕs (δ)

≥ c2

1 −m log δ 2. δ

234

Chapter 7. Some special cases

Postponing for the moment the proof of the claim (after Proposition 7.19), we ﬁnish the proof of the lemma. Arguing as in the proof of Lemma 7.17, we have s m−1 ϕs (s)α(s) ≥ − tm−1 ϕs (t)b(t)α(t)|α(t)|σ−1 dt (7.164) s δ

+ 2m−1 [ϕs (δ)α(δ) − ϕs (δ)α (δ)] for δ small enough. Since sm−1 ϕs (s) → 1 and ϕs (t) < 0, this yields α(s) ≥

1 m−1 s [ϕs (δ)α(δ) − ϕs (δ)α (δ)]. 2

Using property (iii) of the claim, we see that the right-hand side is bounded from below by ! " α(δ) c1 c2 m 1 δ 2 log − |α (δ)| , 2 δ δ c2

which is positive, provided δ > 0 is small enough.

Using Lemma 7.18 we now proceed similarly to what we did in the proof of Proposition 7.16 to establish Proposition 7.19. Let a(t) ∈ C 0 ([0, +∞)), a(t) ≥ 0 and satisfying, for m ≥ 3, a(t) ≤

i)

A2 t2

ii) min

)

(m−2)2 A2 4t2 , t2+ε

*

for 0 ≤ A ≤

m−2 2 ,

for some ε > 0, A > 0.

(7.165)

Then the equation Δu + a(|x|)u = 0 has a positive radial solution u ∈ C 2 (Rm ) satisfying, for some constant C > 0 and |x| 1, in case (7.165) i),

C −1 (log |x|)|x|−

m−2 2

≤ u(x) ≤ C(log |x|)|x|−

m−2 2

C −1 |x|−γ ≤ u(x) ≤ C|x|−γ where γ =

1 2

m−2−

if A = if 0 ≤

m−2 2 , A < m−2 2

(7.166)

( (m − 2)2 − 4A2 ,

while in case (7.165) ii), C −1 ≤ u(x) ≤ C. We now go back to Lemma 7.18 to provide a

(7.167)

7.3. The case of Euclidean space and further results

235

Proof of the claim. We divide the argument into three steps. Step 1. We begin by considering the diﬀerential equation ϕ +

m−1 A2 ϕ + 2+ε = 0. t t

(7.168)

is not an integer, the general Assuming, without loss of generality, that ν = m−2 ε solution is given in terms of the Bessel function of the ﬁrst kind by m−2 m−2 2A − ε 2A − ε ϕ(t) = C1 t− 2 Jν t 2 + C2 t− 2 J−ν t 2 , ε ε see [Wat66] page 97. Given s > 0 we deﬁne ϕ1,s (t) =

Γ(ν)Γ(−ν) − m−2 − m−2 (7.169) s 2 t 2 ! ε " 2A − ε 2A − ε 2A − ε 2A − ε 2 2 2 2 × −J−ν s t s t Jν + Jν J−ν ε ε ε ε

so that ϕ1,s is a solution of (7.168) satisfying ϕ1,s (s) = 0.

4

2 λ cos x − λπ Since Jλ (x) behaves like πx 2 − 4 as x → +∞, the function ϕ1,s is not of deﬁnite sign in (0, s); however, we claim that there exists p1 independent of s such that, for all s > p1 , ϕ1,s (t) < 0 in [p1 , s). Indeed, from the power series representation of Bessel functions Jλ (x) =

+∞ x λ

2

0

k

(−1)k (x/2) , Γ(k + 1)Γ(k + λ + 1)

it follows easily that Jν (x) (resp. Γ(1 − ν)J−ν (x)) is positive and increasing (resp. decreasing) in an interval (0, x+ ) (resp. (0, x− )). By rewriting the right-hand side of (7.169) as 2A − ε 2A − ε Γ(ν) 2 2 t s Γ(1 − ν)J−ν − m−2 m−2 Jν ε ε εt 2 s 2 5 2A − ε 2A − ε . Jν ε s 2 Γ(1 − ν)J−ν ε t 2 , × 1 − 2A ε − 2ε Jν ε t− 2 Γ(1 − ν)J−ν 2A ε s we may therefore conclude that there exists p1 > 0, which depends only upon x+ and x− , such that ϕ1,s (t) is negative for p1 ≤ t < s. We will also need to know the asymptotic behavior of ϕ1,s (t) and ϕ1,s (t) for s and t large. Using the formula x λ Jλ (x) =

2

Γ(1 + λ)

+ o(xλ )

as x → 0,

236

Chapter 7. Some special cases

which can be read oﬀ from the power series representation above, and the recurrence formulas for the derivative of Jλ ,

xλ Jλ (x)

= xλ Jλ−1 (x),

x−λ Jλ (x)

= x−λ Jλ+1 (x),

a lengthy but straightforward computation shows that s2−m − t2−m 2−m + s2−m (ηt + ηs ), + t m−2 εA2 −(m−1) s2−m t−ε−1 + t1−m (ηs + ηt ) ϕ1,s (t) = t + (m − 2)(m − 2 − ε) ϕ1,s =

+ s2−m t−ε−1 (ηs + ηt ), where limt→+∞ ηt = 0 and lims→+∞ ηs = 0. This immediately gives lim sm−1 ϕ1,s (s) = 1

s→+∞

which implies that there exists p2 ≥ p1 , independent of s, such that |ϕ1,s (t)| ≤

2m − 1 2−m t , 2(m − 2)2

4m − 9 1−m t ≤ ϕ1,s (t) ≤ 2t1−m 4(m − 2)

(7.170)

for all t, p2 ≤ t ≤ s. Step 2. Fix p ∈ (p2 , s) and let ϕ2 (t) = ϕ2,p,s (t) be the solution of the diﬀerential equation m − 1 (m − 2)2 ϕ + ϕ + ϕ = 0 on (0, +∞) t 4t2 satisfying ϕ2 (p) = ϕ1,s (p), ϕ2 (p) = ϕ1,s (p). It is easy to verify that ϕ2 is given by the formula ! p " m−2 2−m 2−m m−2 ϕ1,s (p) + pϕ1,s (p)t 2 log . ϕ2 (t) = p 2 ϕ1,s (p)t 2 − 2 t According to (7.170) above, we have m−2 m − 3 2−m ϕ1,s (p) + pϕ1,s (p) > p , 2 2(m − 2) which implies that ϕ2 (t) < 0 A2 t2+ε

for 0 < t ≤ p.

Step 3. Let t0 be such that a(t) ≤ for t ≥ t0 and let p = max {t0 , p2 }. Deﬁne ϕ2,p,s (t) for 0 < t ≤ p, ϕs (t) = ϕ1,s (t) for t > p.

7.3. The case of Euclidean space and further results

237

By steps 1 and 2 above, ϕs is a piecewise C 2 -solution of m−1 ϕ (t) + a(t)ϕ(t) ≥ 0 t

ϕ (t) +

which is negative in (0, s], vanishes for t = s and satisﬁes lim sm−1 ϕs (s) = 1.

s→+∞

To conclude the proof of our claim, it remains to be shown that ϕs satisﬁes condition (iii), that is, there are constants c1 and c2 independent of s such that |ϕs (δ)| ≤ c1

1 log δ

− m−2 2 and

ϕs (δ)

≥ c2

1 −m log δ 2 δ

for δ small enough. Indeed, if δ < p the deﬁnition of ϕ2 and (7.170) imply |ϕs (δ)| ≤ p

m−2 2 +1

ϕ1,s (p)δ

m−2 2

log

2−m p ≤ 2p 2 (log p + 1) log δ

1 2−m δ 2 . δ

On the other hand, a simple computation gives ! m−2 m−2 m−2 ϕs (δ) = p 2 − ϕ1,s (p) + ϕ1,s (p) + pϕ1,s (p) 2 2 p " m m−2 δ− 2 × 1+ log 2 δ p m m − 2 m−2 p 2 mϕ1,s (p) + pϕ1,s (p) log δ− 2 , ≥ 2 δ whence again by (7.170) ϕs (δ) ≥

p m m − 3 2−m p 2 log δ− 2 4 δ

and the proof of the claim is completed.

As a ﬁrst consequence of Proposition 7.19 we give the next uniqueness result for ground states (see Chapter 3 for the deﬁnition) of a Yamabe type equation in Rm . This is exactly the subtle case for uniqueness and in what follows we shall apply Corollary 5.8. Theorem 7.20. Let m ≥ 3 and a(x), b(x) ∈ C 0 (Rm ), b(x) ≥ 0, b(x) ≡ 0, with ⎧ ⎨i) A22 for 0 ≤ A ≤ m−2 |x| 2 , * ) a(x) ≤ (7.171) 2 (m−2)2 ⎩ii) min 4|x|2 , |x|A2+ε for some ε > 0, A > 0.

238

Chapter 7. Some special cases

Let u(x) and v(x) be positive solutions of Δu + a(x)u − b(x)uσ = 0

on Rm .

Assume, as |x| → +∞, in case (7.171) i), ⎧

m−2 ⎪ ⎨u(x) − v(x) = o (log |x|)|x|− 2

√ − 1 m−2− (m−2)2 −4A2 ⎪ ⎩u(x) − v(x) = o |x| 2

if A =

m−2 2 ,

if 0 ≤ A <

m−2 2 ;

in case (7.171) ii), u(x) − v(x) → 0. Then, u(x) ≡ v(x)

on Rm .

Remark. As already observed, this case is particularly signiﬁcant when u(x) and v(x) are ground states of equation Δu + a(x)u − b(x)uσ = 0 on Rm . As an immediate application of Proposition 7.19 we consider the nonexistence result given in Theorem 3.3. We then have Theorem 7.21. Let a(x), b(x) ∈ C 0 (Rm ) and assume b(x) ≥ 0. Let H ≥ 1 and A ∈ R be constants such that max {0, A} ≤ H − 1. Let m ≥ 3 and suppose that Ha(x) satisfy (7.171). Then the diﬀerential inequality uΔu + a(x)u2 − b(x)uσ+1 ≥ −A|∇u|2 , σ ≥ 1 has no nonnegative C 2 -solutions u on Rm satisfying supp u ∩ {x ∈ Rm : b(x) > 0} = ∅ and, in case (7.171) i), ⎧& '−1 0 1 β+1 ⎪ H (2−β) ⎪ − m−2 2(β+1) ⎨ # 2 (log |x|)|x| u ∈ L1 (+∞) ∂Br ) * −1 ⎪ # β+1 ⎪ ⎩ |x|−γ H (2−β) u2(β+1) ∈ L1 (+∞) ∂Br

( with γ = 12 m − 2 − (m − 2)2 − 4A2 , while in case (7.171) ii) & u

2(β+1)

'−1

∂Br

for some β > 1 and max 0, A ≤ β ≤ H − 1.

∈ L1 (+∞)

if A =

m−2 2 ,

if 0 ≤ A <

m−2 2 ,

Bibliography [ADN59] S. Agmon, A. Douglis, and L. Nirenberg. Estimates near the boundary for solutions of elliptic partial diﬀerential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math., 12:623–727, 1959. [Ahl38]

L. V. Ahlfors. An extension of Schwarz’s lemma. Trans. Amer. Math. Soc., 43:359–364, 1938.

[AM85]

P. Aviles and R. C. McOwen. Conformal deformations of complete manifolds with negative curvature. J. Diﬀerential Geom., 21(2):269– 281, 1985.

[AM88]

P. Aviles and R. C. McOwen. Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds. J. Differential Geom., 27(2):225–239, 1988.

[Ama76]

H. Amann. Supersolutions, monotone iterations, and stability. J. Differential Equations, 21(2):363–377, 1976.

[Aub76]

´ T. Aubin. Equations diﬀ´erentielles non lin´eaires et probl`eme de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. (9), 55(3):269– 296, 1976.

[Aub98]

T. Aubin. Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998.

[BE87]

J.P. Bourguignon and J.P. Ezin. Scalar curvature functions in a conformal class of metrics and conformal transformations. Trans. Amer. Math. Soc., 301:723–736, 1987.

[Bes08]

A. L. Besse. Einstein manifolds. Classics in Mathematics. SpringerVerlag, Berlin, 2008. Reprint of the 1987 edition.

[Bis77]

R.L. Bishop. Decompositon of cut loci. Proc. Amer. Math. Soc., 65:133– 137, 1977.

[BMR09] B. Bianchini, L. Mari, and M. Rigoli. Spectral radius, index estimates for Schr¨ odinger operators and geometric applications. J. Funct. Anal., 256:1769–1820, 2009.

P. Mastrolia et al., Yamabe-type Equations on Complete, Noncompact Manifolds, Progress in Mathematics 302, DOI 10.1007/978-3-0348-0376-2, © Springer Basel 2012

239

240 [BRS98]

Bibliography L. Brandolini, M. Rigoli, and A. G. Setti. Positive solutions of Yamabe type equations on complete manifolds and applications. J. Funct. Anal., 160:176–222, 1998.

[BVV91] M.-F. Bidaut-V´eron and L. V´eron. Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations. Invent. Math., 106:489–539, 1991. [Cal57]

E. Calabi. An extension of Hopf’s maximum principle with an application to Riemannian geometry. Duke Math. J., 25:45–56, 1957.

[Car88]

´ Cartan. Le¸cons sur la g´eom´etrie des espaces de Riemann. Les Grands E. ´ Classiques Gauthier-Villars. [Gauthier-Villars Great Classics]. Editions Jacques Gabay, Sceaux, 1988. Reprint of the second (1946) edition.

[CGT82] J. Cheeger, M. Gromov, and M. Taylor. Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Diﬀerential Geom., 17(1):15–53, 1982. [Cha84]

I. Chavel. Eigenvalues in Riemannian geometry, volume 115 of Pure and Applied Mathematics. Academic Press Inc., Orlando, FL, 1984.

[Cha06]

I. Chavel. Riemannian geometry, volume 98 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 2006. A modern introduction.

[Che55]

S.S. Chern. An elementary proof of the existence of isothermal parameters on a surface. Proc. Amer. Math. Soc., 6:771–782, 1955.

[CL55]

E. A. Coddington and N. Levinson. Theory of ordinary diﬀerential equations. McGraw-Hill, New York, 1955.

[CL87]

K. S. Cheng and J.-T. Lin. On the elliptic equations Δu = K(x)uσ and Δu = K(x)e2u . Trans. Amer. Math. Soc., 304(2):639–668, 1987.

[CL95]

W. X. Chen and C. Li. A note on the Kazdan-Warner type conditions. J. Diﬀerential Geom., 41(2):259–268, 1995.

[CN92]

K.-S. Cheng and W.-M. Ni. On the structure of the conformal scalar curvature equation on Rn . Indiana Univ. Math. J., 41(1):261–278, 1992.

[Dav95]

E. B. Davies. Spectral theory and diﬀerential operators, volume 42 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995.

[dC92]

M. P. do Carmo. Riemannian geometry. Mathematics: Theory & Applications. Birkh¨ auser Boston Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty.

Bibliography

241

[Eis49]

L. P. Eisenhart. Riemannian Geometry. Princeton University Press, Princeton, N. J., 1949. 2d printing.

[EL78]

J. Eells and L. Lemaire. A report on harmonic maps. Bull. London Math. Soc., 10:1–68, 1978.

[Esc87]

J. Escobar. Positive solutions for some semilinear elliptic equations with critical Sobolev exponents. Comm. Pure. Appl. Math., 40:623– 657, 1987.

[Esc90]

J. Escobar. Uniqueness theorems on conformal deformations of metrics, Sobolev inequalities, and an eigenvalue estimate. Comm. Pure. Appl. Math., 43:857–883, 1990.

[FC85]

D. Fischer-Colbrie. On complete minimal surfaces with ﬁnite Morse index in three-manifolds. Invent. Math., 82(1):121–132, 1985.

[FCS80]

D. Fischer-Colbrie and R. Schoen. The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature. Comm. Pure. Appl. Math., 33:199–211, 1980.

[Fed69]

H. Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.

[For91]

O. Forster. Lectures on Riemann surfaces, volume 81 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. Translated from the 1977 German original by Bruce Gilligan, Reprint of the 1981 English translation.

[GHV72] W. Greub, S. Halperin, and R. Vanstone. Connections, curvature, and cohomology. Vol. I: De Rham cohomology of manifolds and vector bundles. Academic Press, New York, 1972. Pure and Applied Mathematics, Vol. 47. [Gri76]

P. A. Griﬃths. Entire holomorphic mappings in one and several complex variables. Princeton University Press, Princeton, N. J., 1976. The ﬁfth set of Hermann Weyl Lectures, given at the Institute for Advanced Study, Princeton, N. J., October and November 1974, Annals of Mathematics Studies, No. 85.

[Gri99]

A. Grigor yan. Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Amer. Math. Soc. (N.S.), 36(2):135–249, 1999.

[GT01]

D. Gilbarg and N. Trudinger. Elliptic partial diﬀerential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.

242

Bibliography

[GW79]

R. E. Greene and H. Wu. Function theory on manifolds which possess a pole, volume 699 of Lecture Notes in Mathematics. Springer, Berlin, 1979.

[HW53]

P. Hartman and A. Wintner. On the existence of Riemannian manifolds which cannot carry non-constant analytic or harmonic functions in the small. Amer. J. Math., 75:260–276, 1953.

[Ili96]

S. Ilias. In´egalit´es de Sobolev et r´esultats d’isolement pour les applications harmoniques. J. Funct. Anal., 139(1):182–195, 1996.

[Jin88]

Z. R. Jin. A counterexample to the Yamabe problem for complete noncompact manifolds. In Partial diﬀerential equations (Tianjin, 1986), volume 1306 of Lecture Notes in Math., pages 93–101. Springer, Berlin, 1988.

[Kor14]

A. Korn. Zwei Anwendungen der Methode der sukzessiven Ann¨ aherungen. Schwarz-Festschr., pages 215–229, 1914.

[Kui49]

N. Kuiper. On conformally ﬂat spaces in the large. Ann. of Math., 50:916–924, 1949.

[KW74a] J. L. Kazdan and F. W. Warner. Curvature functions for compact 2-manifolds. Ann. of Math. (2), 99:14–47, 1974. [KW74b] J. L. Kazdan and F. W. Warner. Curvature functions for open 2manifolds. Ann. of Math. (2), 99:203–219, 1974. [KW75a] J. L. Kazdan and F. W. Warner. Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures. Ann. of Math. (2), 101:317–331, 1975. [KW75b] J. L. Kazdan and F. W. Warner. Scalar curvature and conformal deformation of Riemannian structure. J. Diﬀerential Geometry, 10:113–134, 1975. [Le98]

V. K. Le. On some equivalent properties of sub- and supersolutions in quasilinear elliptic equations. Hiroshima Math. J., 28:373–380, 1998.

[Lee97]

J. M. Lee. Riemannian manifolds, volume 176 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1997. An introduction to curvature.

[Lee03]

J. M. Lee. Introduction to smooth manifolds, volume 218 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2003.

[Lee11]

J. M. Lee. Introduction to topological manifolds, volume 202 of Graduate Texts in Mathematics. Springer, New York, second edition, 2011.

[Li90]

P. Li. On the structure of complete K¨ ahler manifolds with nonnegative curvature near inﬁnity. Invent. Math., 99(3):579–600, 1990.

Bibliography

243

[Lic16]

L. Lichtenstein. Zur Theorie der konformen Abbildung nichtanalytischer, singularit¨ atenfreier Fl¨ achenst¨ ucke auf ebene Gebiete. Krak. Anz., 1916:192–217, 1916.

[Lic58]

A. Lichnerowicz. Ge´ om´etrie des groupes des transformation. Travaux et Recherches Math´ematiques, III. Dunod, Paris, 1958.

[LP87]

J. M. Lee and T. H. Parker. The Yamabe problem. Bull. Amer. Math. Soc., 17:37–91, 1987.

[LR96]

P. Li and M. Ramachandran. K¨ ahler manifolds with almost nonnegative Ricci curvature. Amer. J. Math., 118(2):341–353, 1996.

[LTY98]

P. Li, L. F. Tam, and D. Yang. On the elliptic equation Δu+ku−Kup = 0 on complete Riemannian manifolds and their geometric applications. Trans. Amer. Math. Soc., 350(3):1045–1078, 1998.

[LY90]

P. Li and S.-T. Yau. Curvature and holomorphic mappings of complete K¨ahler manifolds. Compositio Math., 73(2):125–144, 1990.

[Mil63]

J. Milnor. Morse theory. Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51. Princeton University Press, Princeton, N.J., 1963.

[MR10]

P. Mastrolia and M. Rigoli. Diﬀusion-type operators, Liouville theorems and gradient estimates on complete manifolds. Nonlinear Anal., 72:3767–3785, 2010.

[MRS10] L. Mari, M. Rigoli, and A. G. Setti. Keller-Osserman conditions for diﬀusion-type operators on Riemannian manifolds. J. Funct. Anal., 258:665–712, 2010. [MRV]

P. Mastrolia, M. Rimoldi, and G. Veronelli. Myers-Type Theorems and Some Related Oscillation Results. J. Geom. Anal., pages 1–17. 10.1007/s12220-011-9213-0.

[MS10]

O. Munteanu and N. Sesum. The Poisson equation on complete manifolds with positive spectrum and applications. Adv. Math., 223:198–219, 2010.

[Ni82]

W. M. Ni. On the elliptic equation Δu + K(x)u(n+2)/(n−2) = 0, its generalizations, and applications in geometry. Indiana Univ. Math. J., 31(4):493–529, 1982.

[NST01]

L. Ni, Y. Shi, and L.-F. Tam. Poisson equation, Poincar´e-Lelong equation and curvature decay on complete K¨ahler manifolds. J. Diﬀerential Geom., 57:339–388, 2001.

[Oba62a] M. Obata. Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Japan, 14:333–340, 1962.

244

Bibliography

[Oba62b] M. Obata. Conformal transformations of compact Riemannian manifolds. Illinois J. Math, 6:292–295, 1962. [Pet06a]

P. Petersen. Riemannian geometry, volume 171 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2006.

[Pet06b]

P. Petersen. Riemannian geometry, volume 171 of Graduate Texts in Mathematics. Springer, New York, second edition, 2006.

[PR]

S. Pigola and M. Rimoldi. Characterizations of Model Manifolds by Means of Certain Diﬀerential Systems. Canad. Math. Bull., pages 1– 14. doi:10.4153/CMB-2011-134-0.

[PRS03a] S. Pigola, M. Rigoli, and A. G. Setti. Some applications of integral formulas in Riemannian geometry and PDE’s. Milan J. Math., 71:219– 281, 2003. [PRS03b] S. Pigola, M. Rigoli, and A. G. Setti. Volume growth, “a priori” estimates, and geometric applications. Geom. Funct. Anal., 13(6):1302– 1328, 2003. [PRS05a] S. Pigola, M. Rigoli, and A. G. Setti. A Liouville-type result for quasilinear elliptic equations on complete Riemannian manifolds. J. Funct. Anal., 219:400–432, 2005. [PRS05b] S. Pigola, M. Rigoli, and A. G. Setti. Maximum principles on Riemannian manifolds and applications. Mem. Amer. Math. Soc., 174(822):x+99, 2005. [PRS05c] S. Pigola, M. Rigoli, and A. G. Setti. Vanishing theorems on Riemannian manifolds, and geometric applications. J. Funct. Anal., 229:424– 461, 2005. [PRS07]

S. Pigola, M. Rigoli, and A. G. Setti. Some characterizations of spaceforms. Trans. Amer. Math. Soc., 359(4):1817–1828 (electronic), 2007.

[PRS08]

S. Pigola, M. Rigoli, and A. G. Setti. Vanishing and ﬁniteness result in geometric analysis, volume 266 of Progress in Mathematics. Birkhauser Verlag Ag, 2008.

[PRS10]

S. Pigola, M. Rigoli, and A. G. Setti. Existence and non-existence results for a logistic-type equation on manifolds. Trans. Amer. Math. Soc., 362:1907–1936, 2010.

[PW67]

M. H. Protter and H. F. Weinberger. Maximum principles in diﬀerential equations. Englewood Cliﬀs Prentice-Hall, 1967.

[RRS95]

A. Ratto, M. Rigoli, and A.G. Setti. On the Omori-Yau maximum principle and its application to diﬀerential equations and geometry. J. Funct. Anal., 134:486–510, 1995.

Bibliography

245

[RRV94a] A. Ratto, M. Rigoli, and V´eron. Conformal immersions of complete Riemannian manifolds and extensions of the Schwarz lemma. Duke Math. J., 74(1):223–236, 1994. [RRV94b] A. Ratto, M. Rigoli, and L. V´eron. Scalar curvature and conformal deformations of hyperbolic space. J. Funct. Anal., 121:15–77, 1994. [RRV97]

A. Ratto, M. Rigoli, and L. Veron. Scalar curvature and conformal deformations of noncompact riemannian manifolds. Math. Z., 225:395– 426, 1997.

[RS01]

M. Rigoli and A. G. Setti. Liouville type theorems for φ-subharmonic functions. Rev. Mat. Iberoamericana, 17:471–520, 2001.

[RSV05]

M. Rigoli, M. Salvatori, and M. Vignati. Some remarks on the weak maximum principle. Rev. Mat. Iberoamericana, 21(2):459–481, 2005.

[RZ07]

M. Rigoli and S. Zamperlin. A priori estimates, uniqueness and existence of positive solutions of Yamabe type equations on complete manifolds. J. Funct. Anal., 245:144–176, 2007.

[Sat73]

D. H. Sattinger. Topics in stability and bifurcation theory. Lecture Notes in Mathematics, Vol. 309. Springer-Verlag, Berlin, 1973.

[Sch84]

R. Schoen. Conformal deformation of a Riemannian metric to constant scalar curvature. J. Diﬀerential Geom., 20(2):479–495, 1984.

[Spi79]

M. Spivak. A comprehensive introduction to diﬀerential geometry. Vol. II. Publish or Perish Inc., Wilmington, Del., second edition, 1979.

[Swa68]

C. A. Swanson. Comparison and oscillation theory of linear diﬀerential equations. Academic Press, New York, 1968. Mathematics in Science and Engineering, Vol. 48.

[Swa75]

C. A. Swanson. Picone’s identity. Rend. Mat. (6), 8(2):373–397, 1975. Collection of articles dedicated to Mauro Picone on the occasion of his ninetieth birthday, II.

[SY94]

R. Schoen and S.-T. Yau. Lectures on diﬀerential geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge, 1994.

[Tru68]

N. S. Trudinger. Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Pisa (3), 22:265–274, 1968.

[Wat66]

G.N. Watson. A treatise on the theory of Bessel functions. Cambridge University Press, 1966.

[Yam60]

H. Yamabe. On a deformation of Riemannian structures on compact manifolds. Osaka Math. J., 12:21–37, 1960.

246

Bibliography

[Yau73]

S.T. Yau. Remarks on conformal transformations. J. Diﬀ. Geom., 3:369–381, 1973.

[YN59]

K. Yano and T. Nagano. Einstein spaces admitting a one-parameter group of conformal transformations. Ann. of Math., 69:451–461, 1959.

List of Symbols [X, Y ]

Lie bracket of two vector ﬁelds X and Y , page 12

|Hess(u)|

norm of Hess(u), page 19

|X|

norm of the vector ﬁeld X, page 19

|x|

distance of the point x ∈ Rm from the origin 0, page 211

2

(U )

space of skew-symmetric 2-forms, page 14

C ∞ (M )

set of smooth functions deﬁned on M , page 16

(U, ϕ)

local chart, page 8

Δu

Laplacian of the function u, page 18

δij

suggestive version of the Kronecker symbol, page 8

δij

Kronecker symbol, page 8

γ˙

tangent vector of the curve γ, page 22

x1 , . . . , xm

coordinate functions, page 8

∂u ∂ν

directional derivative of the function u in the direction of ν, page 62

Hess(u)

Hessian of the function u, page 18

H λL 1 (Ω)

ﬁrst eigenvalue of the operator LH on the bounded domain Ω, page 74

H λL 1 (M )

ﬁrst eigenvalue of the operator LH on the Riemannian manifold M , page 74

LX ,

Lie derivative of the metric , in the direction of X, page 13

P. Mastrolia et al., Yamabe-type Equations on Complete, Noncompact Manifolds, Progress in Mathematics 302, DOI 10.1007/978-3-0348-0376-2, © Springer Basel 2012

247

248

List of Symbols

LX ω

Lie derivative of the 1-form ω in the direction of X, page 13

LX f

Lie derivative of the function f in the direction of X, page 13

LX Y

Lie derivative of the vector ﬁeld Y in the direction of X, page 13

Bm

unit ball of Rm , page 71

Bm R (0)

open disk of radius R centered at the origin in To M ≈ Rm , page 54

Hm

standard hyperbolic space of dimension m, page 47

Hm −H 2

hyperbolic space of constant sectional curvature −H 2 , page 75

Sm

standard sphere of dimension m, page 47

Sm +

standard upper hemisphere, page 53

√

Sm k +

upper hemisphere of radius k −1/2 , page 63

Sm k2

sphere of constant sectional curvature k 2 , page 61

X(M )

set of smooth vector ﬁelds on M , page 10

Annη,1

annulus B1 \ B η , page 214

cut(o)

cut locus of the point o, page 21

II

second fundamental tensor, page 33

, ij

(local) components of the metric, page 8

∇ω

covariant derivative of the 1-form ω, page 11

∇f

gradient of the function f , page 11

∇X

covariant derivative of the vector ﬁeld X, page 10

∇Y ω

covariant derivative of ω in the direction of Y , page 11

∇Y X

covariant derivative of X in the direction of Y , page 10

!f !∞

L∞ -norm of the function f , page 153

ωm

volume of the unit sphere in Rm , page 28

ωik

covariant derivative of the coeﬃcient ωi , page 11

List of Symbols

249

Conf(M )

group of conformal diﬀeomorphisms on M , page 146

dim (M )

dimension of the manifold M , page 8

div X

divergence of a vector ﬁeld X, page 10

hess(u)

(1, 1) version of Hess(u), page 20

Id

identity matrix, page 23

Iso(M )

group of isometries of M , page 146

Lip0 (M )

set of Lipschitz functions on M with compact support, page 29

ric

(1, 1) version of Ric, page 20

Riem

Riemann curvature tensor of type (1, 3), page 15

Sect(u ∧ v)

sectional curvature of the plane π spanned by u and v, page 16

supp ϕ

support of the function ϕ, page 77

Tor

torsion tensor, page 12

tr

trace, page 10

vol ∂BR (o)

volume of the boundary of the geodesic ball BR (o), page 28

vol BR (o)

volume of the geodesic ball BR (o), page 28

W

Weyl tensor, page 45

⊗

tensor product, page 10

Bm

closed unit ball of Rm , page 71

Kulkarni-Nomizu product, page 46

∂· ∂r

derivative in the radial direction, page 28

∂BR (o)

boundary of the geodesic ball centered at o with radius R, page 21

(ϕt )∗

push-forward of the ﬂow, page 14

ρ

Riemannian distance from the origin of Bm with respect to the hyperbolic metric, page 212

Ric

Ricci tensor, page 16

250

List of Symbols

Ric(∇r, ∇r) radial Ricci curvature, page 23 (M, , ) i θ

local orthonormal coframe, page 8

{ei }

local orthonormal frame, page 8

sgn(·)

signum function, page 110

sharp map, page 11

√

g

Riemannian manifold with metric , , page 7

square root of the determinant of the metric in polar geodesic coordinates, page 23

Θij

curvature forms, page 14

θji

Levi-Civita connection forms, page 8

ϕt

local ﬂow of a vector ﬁeld, page 14

∧

wedge product, page 8

,

conformally deformed metric, page 37

A

Schouten tensor, page 45

a+

positive part of the function a, page 89

Aft

t-level set of a function f , page 29

BR (o)

geodesic ball centered at o with radius R, page 21

C

Cotton tensor, page 45

C ∞ (U )

set of smooth functions deﬁned on the open set U , page 9

C 0,α (M )

space of locally H¨older continuous functions on M with exponent α, page 165

C0∞ (M )

set of smooth function with compact support on M , page 27

cm

constant appearing in the (geometric) Yamabe equation (m ≥ 3), page 39

dω

exterior diﬀerential of the 1-form ω, page 12

df

exterior diﬀerential of the function f , page 11

List of Symbols

251

dxi

diﬀerential of the coordinate function xi , page 8

expo

exponential map of M at o, page 21

f∗

pull-back via the map f , page 32

fi

local components of the diﬀerential df , page 11

G(x, y)

Green kernel, page 95

H

mean curvature vector ﬁeld, page 34

hα ijk

coeﬃcients of the covariant derivative of II, page 35

hα ij

coeﬃcients of the second fundamental tensor II, page 33

hν

mean curvature in the direction of ν, page 34

Kp (π)

sectional curvature of the 2-plane π, page 16

Krad

radial sectional curvature, page 31

R

curvature tensor of type (0, 4), page 15

r

Euclidean distance from the origin in Rm , page 212

r(x)

Riemannian distance function, page 21

i Rjkt

(local) components of the Riemann curvature tensor of type (1, 3), page 15

Rijkt,l

covariant derivatives of the (local) components of the curvature tensor, page 16

Rijkt

(local) components of the curvature tensor of type (0, 4), page 15

Rij

(local) components of the Ricci tensor, page 16

S

scalar curvature, page 16

T

traceless Ricci tensor, page 18

Tp∗ M

cotangent space at p, page 10

Trs (M )

set of tensor ﬁelds of type (r, s), page 10

Tp M

tangent space at p, page 10

252

List of Symbols

Tij

(local) components of the traceless Ricci tensor, page 18

u∗

supremum of the function u, page 102

u∗

inﬁmum of the function u, page 148

uijkt

fourth derivatives of the function u, page 20

uijk

third derivatives of the function u, page 19

uij

(local) components of Hess(u), page 18

W 1,1 (M )

Sobolev space of functions in L1 (M ) with (weak) gradient in L1 (M ), page 29

W 1,2 (M )

Sobolev space of functions in L2 (M ) with (weak) gradient in L2 (M ), page 143

Xki

covariant derivative of the coeﬃcient X i , page 10

Index L∞ a priori estimate, 149 LH , 74 a priori estimates from above, 113 from below, 106 Bessel function, 235 Bianchi identities ﬁrst, 15 second, 16 Bishop-Gromov comparison theorem, 28 Bochner-Weitzenb¨ock formula, 19 boundary point lemma, 66 Cartan’s lemma, 33 co-area formula, 29 Codazzi equations, 35 Codazzi tensor, 46 comparison result, 140 conformal deformation of the metric, 37 diﬀeomorphism, 146 vector ﬁeld, 41, 207 coordinate functions, 8 cotangent space, 10 Cotton tensor, 45 covariant derivative of a 1-form, 11 of a function, 11 of a generic tensor ﬁeld, 11 of a vector ﬁeld, 10 of the metric, 12 curvature forms, 14 curvature tensor

of type (0, 4), 15 Riemann, 15 cut locus, 21 cut point, 21 ordinary, 21 singular, 21 Darboux coframe, 32 frame, 33 frames along f preserving orientations, 34 de Rham cohomology groups, 47 decomposition of the curvature tensor using the Schouten tensor, 46 diﬀerential of a coordinate function, 8 Dirichlet problem for the operator LH , 74 divergence of a vector ﬁeld, 10 dual orthonormal frame, 8 Einstein manifold, 16 Einstein summation convention, 8 entire subsolution, 166 estimate from above, 112 from below, 105 exponential map, 21 exterior diﬀerential of a 1-form, 12 Fatou’s lemma, 218

P. Mastrolia et al., Yamabe-type Equations on Complete, Noncompact Manifolds, Progress in Mathematics 302, DOI 10.1007/978-3-0348-0376-2, © Springer Basel 2012

253

254 ﬁrst Bianchi identities, 15 ﬁrst eigenvalue negativity, 170 nonnegativity, 75 ﬁrst eigenvalue of the operator LH on M , 74 on bounded domains, 74 ﬁrst structure equations, 8 fundamental theorem of Riemannian geometry, 13 Gauss equations, 35 Gauss lemma, 22 Gram-Schmidt orthonormalization process, 8 Green kernel, 94 Gronwall inequality, 221 ground state, 103 group of conformal diﬀeomorphisms, 146 of isometries, 146 Hessian, 18 (1, 1) version, 20 Hessian comparison theorem, 31 Hopf classiﬁcation theorem, 56 Hopf-Rinow theorem, 124 hypersurface, 34 immersion, 32 isometric, 32 minimal, 34 totally geodesic, 34 totally umbilical, 34 index of a Schr¨ odinger operator, 173 integral curve, 22 interior of a set, 74 isothermic coordinates, 46, 48 K¨ unneth formula, 47 Kazdan-Warner obstruction, 18, 41 Kronecker symbol, 8 Kulkarni-Nomizu product, 46

Index Laplace-Beltrami operator, 18 transformation law, 99 Laplacian, 18 Laplacian comparison theorem, 25 Levi-Civita connection forms, 8 Lie bracket of two vector ﬁelds, 12 Lie derivative geometric meaning, 14 of a 1-form, 13 of a function, 13 of a vector ﬁeld, 13 local chart, 8 local orthonormal coframe, 8 locally conformally ﬂat manifold, 46 lowering indices, 15 Mayer-Vietoris argument, 47 mean curvature in the direction of a unit normal vector ﬁeld, 34 of an immersed hypersurface, 34 vector ﬁeld, 34 metric induced by an immersion, 32 parallelism of the, 12 torsion-free, 12 minimizing geodesic, 22 model manifold, 115 Monotone iteration scheme, 191 monotonicity of the ﬁrst eigenvalue, 74 Morse lemma, 56 negative part, 125 Newton inequality, 60 Newton’s inequality, 23 nonparabolic manifold, 94 normal bundle, 35 Obata type vector ﬁeld, 57 Obata’s theorem, 53 oscillating solution, 171 parabolic manifold, 94

Index parallel translation, 22 Picone’s identity, 74 Poincar´e model of the hyperbolic space, 211 Poisson equation, 94 pole, 29, 202 positive part, 112 positive part of a function, 89 radial Ricci curvature, 23, 26 reference point, 22 Rellich-Pohozaev formula, 207 Rellich-Pohozaev ientity, 208 Riccati diﬀerential inequalities, 25 Ricci equations, 36 Ricci identities, 19 Ricci tensor, 16 (1, 1) version, 20 traceless, 18, 40 Riemann theorem, 46 Riemann-K¨ obe uniformization theorem, 48 Riemannian manifold, 7 metric, 7 Riemannian distance function, 21 Riemannian product of locally conformally ﬂat manifolds, 47 scalar curvature, 16 Schouten tensor, 45 Schur’s theorem, 17 second fundamental tensor, 33 second Green formula, 27 second structure equations, 14 pull-back, 34 sectional curvature, 16 radial, 31 Seifert-Van Kampen theorem, 56 sharp map, 11 smallness in a spectral sense, 159

255 solution maximal, 165, 166 minimal, 166 spectral theory of Schr¨ odinger operators, 158 sphere m-dimensional, 43 spherical mean, 170, 173 stretching factor, 146 subsolution, 111, 191 supersolution, 111, 191 of a boundary value problem, 191 support of a function, 77 symmetries of the Riemann curvature tensor, 15 of the second derivatives, 19 tangent space, 10 tangent vector of a curve, 22 tensor ﬁeld of type (r, s), 10 third derivatives, 19 torsion tensor, 12 trace, 10 transformation law for a local o.n. coframe, 37 for the connection forms, 38 for the curvature forms, 38 for the curvature tensor, 38 for the Hessian, 99 for the Laplace-Beltrami operator, 99 for the Ricci tensor, 38 for the scalar curvature, 39 for the traceless Ricci tensor, 43 umbilical point, 34 Van der Waerden-Bortolotti covariant derivation, 35 vector ﬁeld, 10

256 volume of a geodesic ball, 28 of the boundary of a geodesic ball, 28 weak maximum principle at inﬁnity, 133 weakly distance decreasing, 155 weighted spherical mean, 200 Weyl tensor, 44 Yamabe equation(s), 39 invariant, 47 Yamabe equation on hyperbolic space, 211

Index