A Perspective on Canonical Riemannian Metrics [1st ed.] 9783030571849, 9783030571856

Table of contents :
Front Matter ....Pages i-xix
Basic Concepts of Riemannian Geometry (Giovanni Catino, Paolo Mastrolia)....Pages 1-25
Commutations and Variations (Giovanni Catino, Paolo Mastrolia)....Pages 27-49
The Weyl Tensor (Giovanni Catino, Paolo Mastrolia)....Pages 51-70
Canonical Metrics I: Curvature Conditions (Giovanni Catino, Paolo Mastrolia)....Pages 71-108
Canonical Metrics II: Critical Metrics of Riemannian Functionals (Giovanni Catino, Paolo Mastrolia)....Pages 109-134
Bochner-Weitzenböock Formulas and Applications (Giovanni Catino, Paolo Mastrolia)....Pages 135-158
Ricci Solitons: Selected Results (Giovanni Catino, Paolo Mastrolia)....Pages 159-211
Existence Results of Canonical Metrics on Four-Manifolds (Giovanni Catino, Paolo Mastrolia)....Pages 213-229
Back Matter ....Pages 231-247

Citation preview

Progress in Mathematics 336

Giovanni Catino Paolo Mastrolia

A Perspective on Canonical Riemannian Metrics Ferran Sunyer i Balaguer Award Winning monograph

Progress in Mathematics Volume 336

Series Editors Antoine Chambert-Loir, Université Paris-Diderot, Paris, France Jiang-Hua Lu, The University of Hong Kong, Hong Kong SAR, China Michael Ruzhansky, Ghent University, Belgium and Queen Mary University of London, UK Yuri Tschinkel, Courant Institute of Mathematical Sciences, New York, USA

More information about this series at http://www.springer.com/series/4848

Giovanni Catino • Paolo Mastrolia

A Perspective on Canonical Riemannian Metrics

Giovanni Catino Dipartimento di Matematica Politecnico di Milano Milano, Italy

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

ISSN 0743-1643 ISSN 2296-505X (electronic) Progress in Mathematics ISBN 978-3-030-57184-9 ISBN 978-3-030-57185-6 (eBook) https://doi.org/10.1007/978-3-030-57185-6 Mathematics Subject Classification (2010): 53C25, 53C20, 53C21 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed 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. 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. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Ferran Sunyer i Balaguer (1912–1967) was a selftaught Catalan mathematician who, in spite of a serious physical disability, was very active in research in classical mathematical analysis, an area in which he acquired international recognition. His heirs created the Fundaci´o Ferran Sunyer i Balaguer inside the Institut d’Estudis Catalans to honor the memory of Ferran Sunyer i Balaguer and to promote mathematical research. Each year, the Fundaci´o Ferran Sunyer i Balaguer and the Institut d’Estudis Catalans award an international research prize for a mathematical monograph of expository nature. The prize-winning monographs are published in this series. Details about the prize and the Fundaci´o Ferran Sunyer i Balaguer can be found at http://ffsb.espais.iec.cat/en This book has been awarded the Ferran Sunyer i Balaguer 2020 prize. The members of the scientific commitee of the 2020 prize were: Antoine Chambert-Loir Universit´e Paris-Diderot (Paris 7) Tere M-Seara Universitat Polit´ecnica de Catalunya Joan Porti Universitat Aut`onoma de Barcelona Michael Ruzhansky Ghent University and Queen Mary University of London Kristian Seip Norwegian University of Science and Technology

Ferran Sunyer i Balaguer Prize winners since 2009: 2009

Timothy D. Browning Quantitative Arithmetic of Projective Varieties, PM 277

2010

Carlo Mantegazza Lecture Notes on Mean Curvature Flow, PM 290

2011

Jayce Getz and Mark Goresky Hilbert Modular Forms with Coefficients in Intersection Homology and Quadratic Base Change, PM 298

2012

Angel Cano, Juan Pablo Navarrete and Jos´e Seade Complex Kleinian Groups, PM 303

2013

Xavier Tolsa Analytic capacity, the Cauchy transform, and non-homogeneous Calder´ on–Zygmund theory, PM 307

2014

Veronique Fischer and Michael Ruzhansky Quantization on Nilpotent Lie Groups, Open Access, PM 314

2015

The scientific committee decided not to award the prize

2016

Vladimir Turaev and Alexis Virelizier Monoidal Categories and Topological Field Theory, PM 322

2017

Antoine Chambert-Loir, Johannses Nicaise and Julien Sebag Motivic Integration, PM 325

2018

Michael Ruzhansky and Durvudkhan Suragan Hardy Inequalities on Homogeneous Groups, PM 327

2019

The scientific committee decided not to award the prize

Contents Introduction 1

2

3

xi

Basic Concepts of Riemannian Geometry 1.1 Moving frames: Levi-Civita connection, structure equations and curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Levi-Civita connection, first structure equation and covariant derivatives . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Second structure equation and the Riemann tensor . . . . . 1.1.3 Ricci, scalar and sectional curvatures . . . . . . . . . . . . . 1.2 General frame: everything with Christoffel symbols . . . . . . . . . 1.2.1 Levi-Civita connection and covariant derivatives revisited . 1.2.2 Riemann and Ricci curvatures revisited . . . . . . . . . . . 1.2.3 Covariant derivatives revisited, Hessian and Laplacian . . . 1.3 Back to moving frames: the decomposition of the curvature tensor 1.4 Other curvatures: A, B, C . . . . . . . . . . . . . . . . . . . . . . .

1 6 10 14 14 15 16 19 22

Commutations and Variations 2.1 Commutation formulas . . . . . 2.2 Variations of curvatures tensors 2.2.1 General variations . . . 2.2.2 Conformal deformations 2.2.3 Aubin’s deformation . .

. . . . . . . . . . . . . . . . . . and other geometric quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Weyl Tensor 3.1 General properties . . . . . . . . . . . . . . . . . 3.1.1 Some formulas . . . . . . . . . . . . . . . 3.1.2 Some inequalities . . . . . . . . . . . . . . 3.1.3 The Weyl–Schouten and Aubin theorems 3.2 The case of dimension four . . . . . . . . . . . . 3.2.1 Self-dual and anti-self-dual parts of W . . 3.2.2 The Derdzinski basis . . . . . . . . . . . . 3.2.3 Special formulas . . . . . . . . . . . . . .

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27 27 33 33 39 45

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51 51 51 53 57 57 58 58 59

viii

Contents 3.2.4 3.2.5

4

Integral identities and topology . . . . . . . . . . . . . . . . A Kato inequality . . . . . . . . . . . . . . . . . . . . . . .

Curvature Conditions 4.1 Old and new canonical metrics: algebraic and analytic conditions 4.1.1 Ricci solitons . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Canonical metrics revisited: equivalent conditions . . . . . . . . . 4.3 The rigid classes: SFf , LSf , LSEf and PRf . . . . . . . . . . . . 4.4 The class HCf . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Rigidity and characterization results . . . . . . . . . . . . 4.4.2 Two examples . . . . . . . . . . . . . . . . . . . . . . . . 4.5 The class Yf : a possible generalization of the Yamabe problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 An obstruction . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 An existence result in Euclidean space . . . . . . . . . . . 4.6 Non-gradient canonical metrics . . . . . . . . . . . . . . . . . . . 4.6.1 The class SFX . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 The classes LSX and LSEX . . . . . . . . . . . . . . . . . 4.6.3 The class HCX . . . . . . . . . . . . . . . . . . . . . . . . 4.6.4 The class YX . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Final remarks and open problems . . . . . . . . . . . . . . . . . .

5 Critical Metrics of Riemannian Functionals 5.1 The Einstein–Hilbert functional . . . . . . . . . 5.2 Quadratic curvature functionals . . . . . . . . . 5.2.1 A basis . . . . . . . . . . . . . . . . . . 5.2.2 Remarks on two special cases . . . . . . 5.3 Some rigidity results for quadratic functionals . 5.3.1 The Euler-Lagrange equations . . . . . 5.3.2 Proofs of Theorem 5.4 and Theorem 5.5 5.3.3 Proof of Theorem 5.7 . . . . . . . . . . 5.3.4 Proof of Theorem 5.8 . . . . . . . . . . 5.3.5 The Euler-Lagrange equation for Ft,s . .

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6 Bochner-Weitzenb¨ock Formulas and Applications 6.1 A general Bochner-Weitzenb¨ock formula . . . . . . . . . 6.1.1 General dimension . . . . . . . . . . . . . . . . . 6.1.2 Dimension four and an integral identity . . . . . 6.2 Some applications . . . . . . . . . . . . . . . . . . . . . 6.2.1 Harmonic Weyl and Einstein manifolds: the first Bochner-Weitzenb¨ock formula . . . . . . . . . . . 6.2.2 Rigidity results for Einstein manifolds . . . . . . 6.2.3 A general result on four-dimensional manifolds .

65 67

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71 71 76 77 81 83 83 87

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89 89 91 92 102 104 104 105 107 107

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109 109 112 112 113 115 118 120 126 128 131

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135 135 135 136 137

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Contents 6.3

7

8

ix

Higher-order Bochner-Weitzenb¨ock formulas on Einstein manifolds 6.3.1 Rough formulas . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 The second Bochner-Weitzenb¨ock formula . . . . . . . . . . 6.3.3 A rigidity result for Einstein manifolds . . . . . . . . . . . .

145 145 149 150

Ricci Solitons: Selected Results 7.1 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Fundamental formulas for Ricci solitons . . . . . . . . . . . 7.1.2 The tensor D and the integrability conditions . . . . . . . . 7.2 Rigidity I: pointwise conditions . . . . . . . . . . . . . . . . . . . . 7.2.1 Compact shrinkers with strictly positive sectional curvature 7.2.2 Further results in the non-necessarily gradient case . . . . . 7.3 Rigidity II: integral conditions . . . . . . . . . . . . . . . . . . . . . 7.3.1 The compact case: integral pinching conditions . . . . . . . 7.3.2 The non-compact case: L1 conditions and integral curvature decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Rigidity III: vanishing conditions on the Weyl tensor . . . . . . . . 7.4.1 A key integral formula . . . . . . . . . . . . . . . . . . . . . 7.4.2 Proof of the results . . . . . . . . . . . . . . . . . . . . . . . 7.5 Rigidity IV: Weyl scalars . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Weyl scalars on a Ricci soliton . . . . . . . . . . . . . . . . 7.5.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . .

159 161 161 165 167 167 172 174 174

Existence Results of Canonical Metrics on Four-Manifolds 8.1 A new variational problem: Weak harmonic Weyl metrics 8.2 The Euler-Lagrange equation . . . . . . . . . . . . . . . . 8.2.1 Critical metrics . . . . . . . . . . . . . . . . . . . . 8.2.2 The PDE for the conformal factor . . . . . . . . . 8.3 Existence of minimizers . . . . . . . . . . . . . . . . . . . 8.3.1 Proof of Theorem 8.2 . . . . . . . . . . . . . . . . 8.3.2 Some preliminary results . . . . . . . . . . . . . . . 8.3.3 Existence . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Uniqueness . . . . . . . . . . . . . . . . . . . . . . 8.3.5 Further results . . . . . . . . . . . . . . . . . . . .

213 213 217 217 219 219 219 220 221 225 228

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178 191 194 196 200 202 205 208

List of Symbols

231

Bibliography

233

Index

243

Introduction “It is geometers’ dream to find a canonical metric gbest on a given smooth manifold M so that all topology of M will be captured by geometry” (M. Gromov)1

It is a little awkward to write an introduction for a book concerning canonical (“best”) Riemannian metrics, when perfect examples are already available in the literature: for instance, it is impossible not to think about the celebrated Besse’s book ”Einstein manifolds” ([11]), whose presentation has certainly inspired us and many other researchers. The idea of writing a monograph devoted to special Riemannian structures came to us some years ago; however, an exhaustive treatment of the subject would require a series of books, and this would go far beyond our present scope. To make the work sufficiently far-reaching, and at the same time focused on a selection of special topics, we chose to build this book around our past and present research, always having in mind the fundamental contributions made by many experts in the field. Thus, this monograph does not claim to be a comprehensive treatment of the matter, nor it is a standard textbook with a systematic approach to Riemannian geometric problems; however, it contains all the ingredients needed to make it self-contained. Getting to the heart of the topic, let (M, g) be a n-dimensional, n ≥ 3, smooth Riemannian manifold with a given metric g. It is well known that the geometry of (M, g) is encoded in its Riemann curvature tensor Riem: since Riem is a quite involved 4-tensor depending on g (and on the choice of a “compatible” connection ∇, which for us will always be the Levi-Civita connection associated to g), in Riemannian Geometry it is natural to define and study some canonical metrics, in a suitable sense, in order to “capture” the topological properties of the manifold (M, g). To do this, there are, at least, two possible very general points of view, that we can call the “CC” one, where CC stands for Curvature Conditions (which can be algebraic or analytic), and the “CM” one, where CM means Critical Metrics of suitable Riemannian functionals. 1 M.

Gromov, Spaces and questions, GAFA, Special Volume (2000), 118–161.

xii

Introduction

The CC point of view In the CC-algebraic case, one typically imposes the constancy of Riem, or of its algebraic traces, namely the Ricci curvature Ric and the scalar curvature S: thus we have that (M, g) belongs to • SF if Riem =

λ 2(n−1)

g ? g (Space Forms);

• E if Ric = λ g (Einstein manifolds);

• Y if S = nλ (Yamabe metrics, with a slight abuse of terminology), for some λ ∈ R. Here, and in the rest of the book, ? denotes the standard KulkarniNomizu product of symmetric 2-tensors (see Chapter 1 for the precise definitions and notations). Clearly, these three classes of Riemannian manifolds are related by SF ⊂ E ⊂ Y and it is well known that, in dimension n = 3, SF = E. On the other hand, from the CC-analytic point of view, the idea is to simplify the curvature by subjecting it to some differential conditions; for instance, in a quite natural way, this is possible by considering curvature tensors that belong to the kernel of a first-order linear differential operator. Some well-known and studied conditions of this type can be prescribed by saying that (M, g) belongs to • LS if ∇(Riem) = 0 (locally symmetric metrics); • PR if ∇(Ric) = 0 (metrics with parallel Ricci curvature); • HC if div(Riem) = 0 (harmonic curvature metric).

SF

⊂

LS ∪ LSE

⊂ PR ∪ ⊂ E

(1)

∩

⊂

Starting from Bianchi identities, we can redefine the class Y of Yamabe metrics using the condition div(Ric) = 0 or, equivalently, ∇S = 0, where div denotes the divergence operator. Obviously, SF ⊂ LS ⊂ PR and, again by Bianchi identities, PR ⊂ HC, E ⊂ HC ⊂ Y. Thus, we have the inclusions

⊂

HC ⊂ Y

where, by definition, LSE := LS ∩ E (locally symmetric Einstein metrics). We note that, in dimension n ≥ 4, all the inclusions are strict. In principle, one could also consider “higher-order” differential conditions, such as ∇k Riem = 0 or ∇k Ric = 0, k ≥ 2, but these give rise again to the classes LS and PR, respectively, by the results in [130, 142]. However, one can consider other higher-order analytic curvature conditions in order to generalize locally symmetric metrics, leading, for instance, to the class of semi-symmetric spaces introduced by Cartan in [39]. As far as the classification of “classical” canonical metrics is concerned, it is well known that the class SF is the most rigid, since, up to quotients, it contains

Introduction

xiii

only Sn , Rn and Hn with their standard metrics. Locally symmetric spaces LS were classified by Cartan ([38]), while, in view of the de Rham decomposition theorem ([11]), PR metrics are locally Riemannian products of Einstein metrics. On the other hand, given any compact manifold M , there always exists a Riemannian metric g such that (M, g) ∈ Y (see e.g. [111]). The remaining classes are more flexible: in particular, E and HC, in the last decades, have been studied by many researchers, also for their connections with Physics, specifically General Relativity and Yang-Mills Theory. In fact, those metrics arise naturally as solutions of the Euler-Lagrange equations of some variational problems; more precisely, in dimension n ≥ 3, the class E of Einstein metrics coincides with the set of critical points of the Einstein-Hilbert functional S(g) (see below), while the HC equation arises in studying, in a given Riemannian vector bundle π : E → M , critical metric connections ∇ for the Yang-Mills functional Z 1 YM(∇) := |Rm∇ |2 dVg , 2 M where Rm∇ is the curvature of the connection ∇. Yang-Mills connections are characterized by d∗ Rm∇ = 0, where d∗ is the formal adjoint (with respect to the standard volume form dVg ) of the exterior differential d acting on E-valued differential forms on (M, g) (see e.g. [67]). Note that d∗ becomes the ordinary divergence operator div when E = T M and ∇ is the Levi-Civita connection of g. In view of the Bianchi identity dRm∇ = 0, and this means that the curvature of any Yang-Mills connection is harmonic with respect to the standard Hodge Laplacian ∆H := dd∗ + d∗ d, acting on 2-forms. The CM point of view A critical metric of a Riemannian functional is, by definition, a solution of the associated Euler-Lagrange equations, which can be obtained by taking the variation of the functional in question (see [11] and [13]). This solution is tensorial, when we consider general variations of the metric, or scalar, when we consider variations in the conformal class of a reference metric g0 . There are, clearly, many geometrically meaningful functionals, but perhaps the most famous is the Einstein-Hilbert action Z n−2 S(g) := Volg (M )− n Sg dVg , M

where Volg (M ) is the volume of (M, g). It is not difficult to show (see Chapter 5) that all stationary points of S(g) are Einstein metrics, i.e., they belong to E. While the existence of Einstein metrics as critical points of S(g) is not guaranteed, a constrained version of the problem always admits a solution: more precisely, Yamabe, Aubin, Trudinger, and Schoen (see [111]) showed that the Yamabe invariant Y(M, [g]) := inf S(e g) g e∈[g]

xiv

Introduction

is always attained in the conformal class [g]. Moreover, every critical point of the normalized functional in the conformal class has constant scalar curvature (see Chapter 5 for details). Other important examples are Riemannian functionals that are quadratic in the curvature: in [10], Berger commenced the study of these types of Riemannian functionals (see [11, Chapter 4] and [140] for surveys). A basis for the space of quadratic curvature functionals is given by Z Z Z 2 2 W = |W | dV, r = | Ric | dV, S2 = S 2 dV, and, from the decomposition of the Riemann tensor, one has  Z Z  4 2 2 2 2 2 R = | Riem | dV = |W | + | Ric | − S dV . n−2 (n − 1)(n − 2) All such functionals, which also arise naturally as total actions in certain gravitational fields theories in physics, have been studied in depth in the last years by many authors (for instance, Calabi, Tian, Gursky, Viaclovsky, LeBrun, Anderson, Carron and others: see Chapter 5 for references). It is meaningful, then, to define classes of “best” metrics as critical points of Riemannian functionals. How to extend the CC approach The canonical metrics related to the CC point of view can be thought of as solutions of PDEs of the form F[g] = 0, where F is a differential operator acting on the metric g. In recent years many mathematicians have focused their research on more general structures, considering particular conditions that involve the curvature of a metric and a potential, that is, a smooth function defined on the underlying manifold (metric measure spaces, conformal invariants, Einstein-type manifols, dilaton fields, etc.) In this setting, it is natural to study solutions (g, f ), with f ∈ C ∞ (M ), of the equation F[g, f ] = 0, where F is again a differential operator, now acting on the metric g and on the potential f . A particularly important example arised from the pioneering works of Hamilton [91] and Perelman [131] towards the solution of the Poincar´e conjecture in dimension three: indeed, their seminal papers generated a flourishing activity in the research of self-similar solutions, or Ricci solitons, of the Ricci flow. From the static point of view, these structures are characterized by the condition Ricf := Ric + Hess(f ) = λ g , where Ricf is the Bakry-Emery Ricci tensor, f ∈ C ∞ (M ) is called the potential, λ ∈ R and Hess is the Hessian. In this case, we say that (M, g, f ) ∈ Ef (gradient Ricci solitons); such a soliton is called shrinking, steady or expanding according to whether λ > 0, λ = 0 or λ < 0, respectively. It is apparent that this is a reasonable generalization of the Einstein condition which, interpreted as a global prescription

Introduction

xv

SF

⊂

LS ⊂ PR ∪ ∪ LSE ⊂ E ∩ Ef

∩

⊂

on the Ricci curvature of g, was first considered by Lichnerowicz (see, e.g., [20]). In particular, if (M, g) ∈ E, then (M, g, f = c ∈ R) ∈ Ef , and we can add another inclusion to the previous diagram as follows:

⊂

HC ⊂ Y

Given their importance, we will devote the entire Chapter 7 to Ricci solitons, presenting both “old” and recent results, whereas in Chapter 4 we describe a “potential” generalization of the previous framework, introducing and studying new classes of privileged metrics g on Riemannian manifolds M endowed with smooth potential functions f , which extend the diagram above. The corresponding classes in (1) are recovered when ∇f = 0 on M ; in this latter case, we say that the structure is trivial. How to extend the CM approach It is clear that, in order to widen the CM point of view, we can work in two directions: on the one hand, we can try to prove existence and classification results for critical metrics of already known Riemannian functionals; on the other hand, we can define and study new functionals having, as a subset of their critical points, suitable classical canonical metrics. In Chapter 5 we treat quadratic functionals of the form Z Z 2 Ft = | Ric | dV + t S 2 dV , for some t ∈ R; these functionals have been studied intensively by many authors in the last years (see all the references in Chapter 5). It can be shown that Einstein metrics are always critical points of these functionals (restricted to volume one metrics); we prove that, under some curvature conditions, the converse is also true. An important class of metrics which generalizes the Einstein condition is given by harmonic Weyl metrics, i.e., metrics with divergence-free Weyl tensor, δg Wg = 0 (see again [11]). In fact, it is well known that all Einstein metrics have harmonic Weyl tensor and that, on four-dimensional closed manifolds, there are topological obstructions to the existence of harmonic Weyl metrics with constant scalar curvature (see [19, 70]). In Chapter 8 we consider on a four-dimensional closed smooth manifold M 4 the quadratic scale-invariant functional Z 1 2 |δg Wg |2g dVg . D(g) = Volg (M ) M

Clearly, harmonic Weyl metrics are critical points of D(g), while in general the converse is not true. In the same spirit of the Yamabe problem, we define the

xvi

Introduction

conformal invariant D(M, [g]) := inf D(e g) . g e∈[g]

and we study the existence (and uniqueness) of minimizers in the conformal class for the functional g 7→ D(g). We now describe briefly the content of each chapter of this book. Chapters 1 and 2 are technical, introductory chapters. Specifically, in the first chapter, we recall some important definitions and results of Riemannian Geometry. We follow essentially the first chapter of [1], where the authors exploit a ´ Cartan, but we also use the version of the so-called moving frame method of E. classical coordinate formalism, i.e., the Koszul formalism and Christoffel symbols. In the second chapter, we compute commutation rules for covariant derivatives of functions, of vector fields, and of the geometric tensors introduced in Chapter 1, and we also collect (and compute in some cases) general variations, conformal variations, and a “special” variation of fundamental geometric quantities. Chapter 3 is devoted to one of the main character of this book, the Weyl tensor. We recall the Bianchi identities and the commutation relations for the second and the third covariant derivatives of W . In the subsequent section we focus on the four-dimensional case, introducing the Derdzinski basis and a number of formulas concerning various contractions of W with itself. We conclude the chapter with a refined Kato inequality and with a discussion of a result obtained by Aubin in [4]. The most significant results of this chapter are:  Proposition 3.6 and Lemma 3.8: two algebraic pointwise estimates concerning the Weyl tensor;  Theorem 3.10: every closed manifold admits a metric with nowhere vanishing Weyl tensor;  Lemmas 3.14 and 3.15: two algebraic cubic identities for the Weyl tensor and its derivative, in dimension four;  Lemma 3.17: a quantitative Kato inequality for the the (anti-)self-dual part of the Weyl tensor. In Chapter 4, following the presentation in [50], we introduce a first possible way to define and study canonical metrics on Riemannian manifolds, namely, the one depending on “Curvature Conditions” (CC, for short). Starting from the classical definitions recalled at the beginning of this introduction, we define a number of new classes of canonical metrics “with a potential”, where the conditions generalize the previous ones by involving the curvature of a metric and a potential, that is, a smooth function globally defined on the underlying manifold. In the subsequent sections we state and prove classification results for these new classes, we construct some explicit examples in two cases, and also describe the “nongradient” version of our construction. Finally, we present a list of open problems. The most significant results of this chapter are:

Introduction

xvii

 Propositions 4.6 and 4.7: the classification of f -space forms, f -locally symmetric and f -locally symmetric Einstein metrics;  Theorem 4.8: local characterization of locally conformally flat f -harmonic curvature metrics;  Theorem 4.10: in dimension four, the class of analytic f -harmonic curvature metrics with non-zero signature coincides with the class of gradient Ricci solitons;  Proposition 4.14: a subclass of f -harmonic curvature metrics with positive sectional curvature coincides with the class of gradient Ricci solitons;  Corollary 4.17: a Kazdan-Warner type obstruction to the existence of f Yamabe metrics.  Theorem 4.24: an existence result in dimension four for f -Yamabe metrics conformal to the Euclidean metric. In Chapter 5 we introduce the second possible way to study canonical metrics on Riemannian manifolds, namely the one related to “Critical Metrics of Riemannian functionals” (CM, for short). First we recall the classical Einstein-Hilbert functional and we briefly discuss the celebrated Yamabe problem; then, following [42], we introduce a basis for quadratic curvature functionals, deriving the corresponding Euler-Lagrange equations, and we prove some rigidity results for critical metrics. The most significant results of this chapter are:  Theorem 5.4: critical metrics for a class of quadratic curvature functionals with non-negative sectional curvature are Einstein;  Theorem 5.7: critical metrics for a class of quadratic curvature functionals with non-positive sectional curvature are Einstein;  Proposition 5.17: a pointwise estimate on the curvature for metrics with a lower bound on the sectional curvature. Bochner-Weitzenb¨ock formulas for the Weyl tensor have been exploited in the last decades by a great number of authors: just to quote some of them, focused on the study of Einstein manifolds and related structures, we mention Derdzinski [68], Singer [139], Hebey and Vaugon [93], Gursky [84, 86], Gursky and Lebrun [87], Yang [149] (see also the references therein). In Chapter 6, after proving a very general Bochner-Weitzenb¨ock formula for the Weyl tensor on arbitrary smooth manifolds, we specialize to harmonic Weyl and Einstein metrics proving a classical rigidity result. We then prove a characterization of anti-self-dual metrics arising from an integral Bochner-Weitzenb¨ock identity. Finally, following our work [52], we employ higher-order Bochner-Weitzenb¨ock formulas for four-dimensional closed Einstein manifolds to derive new integral identities and a new rigidity result. The most significant results of this chapter are:  Lemma 6.1: a general Bochner-Weitzenb¨ock formula for the Weyl tensor;

xviii

Introduction

 Corollary 6.4: a first Bochner-Weitzenb¨ock formula on four-manifolds with harmonic Weyl curvature;  Theorem 6.5: a rigidity result for Einstein metrics with positive Yamabe invariant and small L2 -norm of the Weyl tensor;  Theorem 6.7: a characterization of anti-self-dual metrics;  Theorem 6.16: a second Bochner-Weitzenb¨ock formula for four-dimensional Einstein manifolds;  Theorem 6.23: four-dimensional Einstein manifolds with small L2 -norm of the Hessian of the Weyl tensor are locally symmetric. In Chapter 7 we present several selected results, based on our recent research, concerning the classification of Ricci solitons. After recalling some preliminary results in Section 7.1, in Section 7.2 we focus on shrinkers, considering pointwise conditions (e.g., positive sectional curvature or pinched Weyl curvature), and we also recall some classification results for non-necessarily gradient solitons. Section 7.3 deals with integral assumptions (compact shrinkers and non-compact steady and expanding solitons). Section 7.4 explores vanishing conditions related to the Weyl tensor and the integrability conditions for gradient Ricci solitons. Finally, in Section 7.5 we focus further on generalizations of the vanishing conditions introduced previously, defining the so-called Weyl scalars. The most significant results of this chapter are:  Theorem 7.8: complete four-dimensional gradient shrinking solitons with positively pinched sectional curvature are Einstein;  Theorem 7.13: compact four-dimensional gradient shrinking solitons with L2 pinched curvature are quotients of the round sphere;  Theorem 7.17: complete gradient expanding Ricci solitons with non-negative sectional curvature and integrable scalar curvature are quotients of the Gaussian flat space;  Theorem 7.18: complete gradient steady Ricci solitons with non-negative sectional curvature whose scalar curvature satisfy a sharp integral assumption are quotients of the flat space or the product of the cigar soliton with the flat space;  Theorem 7.38: a classification of complete gradient shrinking Ricci solitons with vanishing fourth-order divergence of the Weyl tensor;  Propositions 7.46 and 7.48: rigidity of compact Ricci solitons with vanishing Weyl scalars. In Chapter 8, following the recent work [56], we study the existence of minimizers in the conformal class for the quadratic functional D(g) given by the (rescaled) L2 -norm of the divergence of the Weyl tensor. In general, the problem

Introduction

xix

is degenerate elliptic due to possible vanishing of the Weyl tensor. In order to overcome this issue, we minimize the functional in the conformal class determined by the reference metric constructed by Aubin in [4], with nowhere vanishing Weyl tensor. The most significant results of this chapter are:  Theorem 8.2: on every closed four-dimensional manifold there exists a weak harmonic Weyl metric;  Theorem 8.9: existence of minimizers of the functional;  Theorem 8.13: uniqueness (up to scaling) of minimizers of the functional.

Chapter 1

Basic Concepts of Riemannian Geometry In this first, introductory chapter we recall some important definitions and results of Riemannian Geometry, essentially following [1]. Although we assume the reader to be familiar with the general subject, as presented, e.g., in the standard references [109, 110, 15, 132, 78, 74], several computations and proofs will be provided in full detail. Since they will be extremely useful throughout the book, our presentation will employ both the classical coordinate formalism (i.e., Koszul formalism and Christoffel symbols) and a version of the so called moving frame formalism of ´ Cartan; moreover, unless otherwise stated, all manifolds are assumed to be E. connected. Note that, from now on, we adopt the Einstein summation convention over repeated indices.

1.1

Moving frames: Levi-Civita connection, structure equations and curvatures

In this section we introduce the moving frame method (` a la Cartan) in order to define the Levi-Civita connection forms, the structure equations and various concepts related to curvature.

1.1.1 Levi-Civita connection, first structure equation and covariant derivatives Let (M, g) be a connected Riemannian manifold of dimension n = dim M ≥ 2 with metric g. If p ∈ M and (U, ϕ) is a local chart such that p ∈ U , then at any © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 G. Catino, P. Mastrolia, A Perspective on Canonical Riemannian Metrics, Progress in Mathematics 336, https://doi.org/10.1007/978-3-030-57185-6_1

1

2

Chapter 1. Basic Concepts of Riemannian Geometry

point q ∈ U we have g = gij dxi ⊗ dxj ,

(1.1)

where the dxi ’s denote the differential of the coordinate functions x1 , . . . , x
n and ∂ ∂ gij are the (local) components of the metric, defined by gij = g ∂x i , ∂xj . Applying at q the Gram-Schmidt orthonormalization algorithm, we can find linear combinations of the 1-forms dxi , that we will denote θi , i = 1, . . . , n, such that (1.1) takes the form g = δij θi ⊗ θj , (1.2) where δij is the Kronecker symbol. Since, as q varies in U , the above recipe gives rise to coefficients that are C ∞ functions of q, the set of 1-forms {θi } defines an orthonormal system on U for the metric g, that is, a (local) orthonormal coframe. We can also write n X g= (θi )2 i=1

instead of (1.2). Accordingly, we define the (local) dual orthonormal frame {ei }, i = 1, . . . , n, as the set of vector ei fields on U such that θj (ei ) = δij

(1.3)

(where δij is the “(1, 1)” version of the Kronecker symbol, reflecting the position of the indices in the pairing of θj and ei ). We have the following Proposition 1.1. Let {θi } be a local orthonormal defined on the open set  coframe U ⊂ M ; then on U there exist unique 1-forms θji , i, j = 1 . . . , n, such that dθi = −θji ∧ θj

(1.4)

θji + θij = 0.

(1.5)

and For a proof, see [1]. The forms θji are called the Levi-Civita connection forms associated to the orthonormal coframe {θi }, and equation (1.4) is called the first structure equation. Starting from the Levi-Civita connection forms, we can define a covariant derivative ∇ on every tensor bundle. First we recall that a linear connection on a differentiable manifold M is a map ∇ : X(M ) × X(M ) → X(M ), ∇ : (X, Y ) 7−→ ∇X Y, satisfying the following properties: for any X, Y, X1 , X2 , Y1 , Y2 ∈ X(M ) and any f ∈ C ∞ (M ), i) ∇X1 +X2 Y = ∇X1 Y + ∇X2 Y ;

1.1. Moving frames

3

ii) ∇X (Y1 + Y2 ) = ∇X Y1 + ∇X Y2 ; iii) ∇f X Y = f ∇X Y ; iv) ∇X (f Y ) = f ∇X Y + X(f )Y ; here X(M ) is the set of smooth vector fields on M and C ∞ (M ) denotes the set of all smooth functions on M . Note that, by definition, ∇ is C ∞ (M )-linear on the module X(M ) in the variable X for fixed Y , while it acts as a derivation in the variable Y for fixed X; in particular, ∇ is R-bilinear on the real vector space X(M ). Now, let {ei } and {θi } be an orthonormal frame and its dual coframe, respectively, on the open set U . The connection ∇ induced by the Levi-Civita connection forms is defined by ∇ei = θij ⊗ ej , (1.6) and, for any X, Y ∈ X(U ) (where X(U ) is the set of smooth vector fields on the open set U ), and any f ∈ C ∞ (U ), by the rules ∇(X + Y ) = ∇X + ∇Y,

∇(f X) = df ⊗ X + f ∇X;

(1.7)

the dual connection, acting on forms and still denoted by ∇, is given by the formula ∇θi = −θji ⊗ θj

(which follows imposing the condition ∇θi (ej ) + θi (∇ej ) = ∇ θi (ej ) = d θi (ej ) = 0; see below for the relation between the covariant derivative and the differential of a function). It can be shown that the connection ∇ is globally defined, and independent of the choice of a frame {ei }. For a vector field X ∈ X(M ), which can be locally written as X = X i ei , the covariant derivative ∇X is the tensor field of type (1, 1) ∇X = (dX i ) ⊗ ei + X i ∇ei = (dX i + X j θji ) ⊗ ei . If we set Xki θk = dX i + X j θji , then ∇X can be written as ∇X = Xki θk ⊗ ei ; and Xki is called 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 ), or, in components, ∇Y X = Xki θk (Y )ei = Xki Y k ei .

4

Chapter 1. Basic Concepts of Riemannian Geometry

The divergence of the vector field X ∈ X(M ) is the trace of ∇X, that is, div X = Tr (∇X) = h∇ei X, ei i = Xii .

(1.8)

For a 1-form ω, written locally as ω = ωi θi , the covariant derivative ∇ω is the tensor field of type (0, 2)   ∇ω = (dωi ) ⊗ θi + ωi ∇θi = dωi − ωj θij ⊗ θi . Upon setting ωik θk = dωi − ωj θij , ∇ω can be written as ∇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 ∇Y ω = ωik θk (Y )θi = ωik Y k θi . We can extend ∇ to a generic tensor field T via the Leibniz rule. Recall that a tensor field of of type (r, s) is a law that assigns to each point p ∈ M a multilinear map s times

Tp :

Tp∗ M |

× ··· × {z r times

Tp∗ M

z }| { × Tp M × · · · × Tp M −→ R,

}

where Tp M and Tp∗ M are the tangent and the cotangent space of M at p, respectively, with the usual differentiability requirement with respect to the variable p (see for instance [109]). Thus, for a local orthonormal coframe {θi } with dual frame {ei } on the open set U we have ...ir j1 T = Tji11...j θ ⊗ · · · ⊗ θjs ⊗ ei1 ⊗ · · · ⊗ eir , s

and the covariant derivative of T , ∇T , is then defined on U as the (r, s + 1) tensor field ...ir ∇TU = Tji11...j θk ⊗ θj1 ⊗ · · · ⊗ θjs ⊗ ei1 ⊗ · · · ⊗ eir s ,k where the coefficients are ...ir ...ir i1 ...ir h ...ir Tji11...j θk = dTji11...j − Thj θ − · · · − Tji11...j θh s s ,k 2 ...js j1 s−1 h js i ...i

h

2 ...ir i1 + Tjhi θh + · · · + Tj11...jsr−1 θhir . 1 ...js

(1.9)

1.1. Moving frames

5

We highlight the fact that, by the discussion above, the tensor field ∇T is globally defined and that, by its definition, the operator ∇ satisfies the Leibniz rule and other useful properties (e.g., the commutativity with the trace with respect to any pair of indices). Indeed, one can verify that the previous definition matches the “canonical” one usually given in terms of the Koszul formalism, as we will see in a shortwhile. For a function u ∈ C ∞ (M ), the covariant derivative coincides with the usual differential, i.e. ∇u = u,i θi = du. Indeed, by definition, thinking of u as a (0, 0)-tensor field, u,i θi = du; from now we will simply write du = ui θi .

(1.10)

Remark. In the literature, and also in this book, ∇u usually denotes the gradient ] of u, that is the vector field dual to the 1-form du: more explicitly, ∇u = (du) , ] ∗ where ] is the musical isomorphism : T M → T M (also called sharp map) defined by hdu] , Y i = h∇u, Y i = du(Y ) = Y (u),

∀ Y ∈ X(M ).

i

Note that, in components, (∇u) = δ ij (du)j = δ ij uj = ui , that is, in an orthonormal frame, the differential and thegradient of a function have the same coefficients with respect to the (dual) bases θi and {ei }. It is not difficult to see that this remains true also when we “raise an index” or “lower an index” for higher-order tensors (see e.g. [1]): in an orthonormal frame, writing an index “up” or “down” does not change the numerical value of a component of a tensor (note that this is in sharp contrast with the case of a general non-orthonormal frame, as it will become clear in Section 1.2) . In the rest of the book we choose to maintain the “correct” positions of the indices only to keep in mind the type of the tensors involved in our computations. We remark that (1.5) expresses the “compatibility” of the covariant derivative with the metric (equivalently, the parallelism of the metric with respect to ∇): indeed, since g = δij θi ⊗ θj on the open set U , δij,k θk = dδij − δlj θil − δil θjl = −(θij + θji ), and therefore ∇g ≡ 0 if and only if (1.5) holds. We also note that (1.4) holds if and only if [X, Y ] = ∇X Y − ∇Y X,

∀ X, Y ∈ X(M )

(1.11)

6

Chapter 1. Basic Concepts of Riemannian Geometry

(where [ , ] is the Lie bracket on M ), which means that the Levi-Civita connection is torsion-free: indeed, since the torsion of a generic (linear) connection ∇ on M is the (0, 2)-tensor field Tor(X, Y ) = ∇X Y − ∇Y X − [X, Y ], equation (1.11) is equivalent to the vanishing of Tor. To prove the equivalence, recall that the exterior differential of a 1-form θ is intrinsically defined by dθ(X, Y ) = X(θ(Y )) − Y (θ(X)) − θ([X, Y ]),

∀ X, Y ∈ X(M );

(1.12)

moreover, as a consequence of the definition of the covariant derivative, (∇X θ)(Y ) = X(θ(Y )) − θ(∇X Y ),

(1.13)

so that
X θi (Y ) − θi (∇X Y ) = (∇X θi )(Y ) = −θji (X)θj (Y ), that is,
X θi (Y ) + θji (X)θj (Y ) = θi (∇X Y ).
Then we compute dθi + θji ∧ θj (X, Y ): 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 ]) = θi (Tor(X, Y )), and the claim follows. By the fundamental theorem of Riemannian geometry (see, e.g., [110] or [132]), we deduce that the connection ∇ coincides, as previously asserted, with the Levi-Civita connection of the metric g.

1.1.2 Second structure equation and the Riemann tensor We now introduce a family of 2-forms, the curvature forms {Θij } associated to the orthonormal coframe {θi }, via the second structure equation dθji = −θki ∧ θjk + Θij .

(1.14)

In view of (1.5), it immediately follows that Θij + Θji = 0.

(1.15)

1.1. Moving frames

7

Using the basis {θi ∧ θj }1≤i 0 and M admits a metric with positive scalar curvature (see the seminal work of Kazdan and Warner [101, Theorem 6.4]). 4. It is well known that compact gradient shrinking, steady and expanding Ricci solitons Ef can be characterized as critical points of the F and W, W− Perelman’s functionals, respectively (see e.g. [29]). On the other hand, the class HCf arises naturally in the study of critical metric connections ∇ in a given Riemannian vector bundle π : E → M for the “weighted” Yang-Mills functional Z 1 YMf (∇) := |R∇ |2 e−f dVg , 2 M which leads to the so-called Yang-Mills-dilaton field theory. A simple computation, following the one for YM (see, e.g., [21]), shows that weighted YangMills connections are characterized by the condition d∗f R∇ = 0, where d∗f is the formal adjoint of the exterior differential d with respect to the weighted volume form e−f dVg (see [27]). Note that d∗f becomes the f -divergence operator ef div(e−f ) when E = T M and ∇ is the Levi-Civita connection of g. By the Bianchi identity dR∇ = 0, this means that the curvature of any weighted Yang-Mills connection is weighted harmonic with respect to the weighted Hodge Laplacian ∗ ∗ ∆H f := ddf + df d .

5. In our discussion we have so far considered only the case of dimension greater than three. We observe that in dimension n = 2, the geometry of a Riemann surface (M, g) is encoded by the scalar curvature R. In particular, Ric = R2 g and the equation defining Yf yields
∇ e−f R = 0 ⇐⇒ R = Cef , for some C ∈ R. This is equivalent to the classical problem of prescribing (with sign) the Gaussian (scalar) curvature of a Riemann surface. Thanks to

76

Chapter 4. Curvature Conditions the seminal work of Kazdan and Warner [101], it follows that, on a compact surface M , given any smooth function f , there exists a Riemannian metric g such that (M, g, f ) ∈ Yf (in the genus zero case, a solution is the scalar flat metric).

6. We will see in Propositions 4.6 and 4.7 that, as one can expect, the classes SFf , LSf , LSEf and PRf do not differ too much from their classical counterparts; however, they still contain some interesting Riemannian spaces, such as generalized cylinders (with Gaussian potential) and the Bryant soliton.

4.1.1 Ricci solitons For the reader’s convenience, we recall here some useful equations satisfied by every gradient Ricci soliton (M, g, f ) ∈ Ef . The same results will appear again at the beginning of Chapter 7. By definition, Rij + fij = λgij ,

λ ∈ R,

(4.3)

where fij = ∇i ∇j f are the components of the Hessian of f (see, e.g., [75]). Lemma 4.3. Let (M n , g) be a gradient Ricci soliton of dimension n ≥ 3. Then the following equations holds: Rf := R + ∆f = nλ, ∇R = 2 Ric(∇f, ·), R + |∇f |2 = 2λf + c,

i.e., Ri = 2ft Rit , for some c ∈ R,

Rij,k − Rik,j = −Rtijk,t = −ft Rtijk . The tensor D, denoted here by D∇f to distinguish it from its “generic” counterpart DX (see Section 4.6), was introduced by Cao and Chen in [33] and turned out to be a fundamental tool in the study of the geometry of gradient Ricci solitons (more generally for gradient Einstein-type manifolds, see [59]). In components, ∇f Dijk =

1 1 (fk Rij − fj Rik ) + ft (Rtk δij − Rtj δik ) n−2 (n − 1)(n − 2) R − (fk δij − fj δik ). (n − 1)(n − 2)

(4.4)

∇f The tensor D∇f is skew-symmetric in the second and third indices (i.e., Dijk = ∇f ∇f ∇f ∇f −Dikj ) and totally trace-free (i.e., Diik = Diki = Dkii = 0). Note that our convention for the tensor D differs from that in [33].

4.2. Canonical metrics revisited: equivalent conditions

77

If (M, g, X) is a Ricci soliton structure on (M, g), with X ∈ X(M ), the defining equation becomes 1 Rij + (Xij + Xji ) = λδij . 2 Moreover, we have (see [58]) RX := R + div(X) = nλ;
∇R = 2 Ric(X, ·) + div AX , i.e., Ri = 2Xt Rit + Xit,t − Xti,t , where AX is the antisymmetric part of the covariant derivative of X; in components, (AX )ij = Xij − Xji . Finally, we recall the following formula due to B¨ochner, [150], and rediscovered many times in recent years. Lemma 4.4. Let X be a vector field on the Riemannian manifold (M, g). Then div (LX g)(X) =

1 ∆|X|2 − |∇X|2 + Ric(X, X) + ∇X (div X) , 2

or, in coordinates,
1 Xiji + Xjii Xj = ∆|X|2 − |∇X|2 + Rij Xi Xj + Xjji Xi . 2 For other results concerning Ricci solitons, see Chapter 7.

Canonical metrics revisited: equivalent conditions SF ∩ SFf

⊂ ⊂

PR ∪ ⊂ E ∩ ⊂ Ef ∩ ⊂ PRf ⊂

⊂ ⊂

HC ∩ HCf

Y ∩ ⊂ Yf ⊂

∪

⊂

LS ∪ LSE ∩ LSEf ∩ LSf

∩

⊂

4.2

The aim of this section is to present equivalent conditions characterizing some of the classes in Definition (4.2); for the sake of completeness and to highlight the similarities and the differences with the “potential” counterpart, we report the well-known characterizations of the classical structures. Here (M, g) is a smooth Riemannian manifold of dimension n ≥ 3 with metric g. First, we recall that the decomposition (1.81) can be globally (and orthogonally) written as 1 Riem = W + A ? g, n−2

78

Chapter 4. Curvature Conditions

where A is the Schouten tensor. It this then natural to introduce a new tensor, that we call Af (the f -Schouten tensor), in such a way that   1 ∆f 1 2 Riemf := Riem + ∇ f− g ?g =W + Af ? g. n−2 2(n − 1) n−2 R

f It turns out that Af := Ricf − 2(n−1) g (recall that Ricf = Ric +∇2 f and Rf = R + ∆f ).

The classes SF and SFf A standard computation using Bianchi identities and the constancy of the scalar curvature shows that ( W = 0, λ (M, g) ∈ SF ⇐⇒ Riem = g ? g ⇐⇒ 2(n − 1) Ric = λg, In a similar fashion, using the constancy of Rf , we have (M, g, f ) ∈ SFf

⇐⇒

λ Riemf = g?g 2(n − 1)

⇐⇒

( W = 0, Ricf = λg.

(4.5)

Note that SF ⊂ E and SFf ⊂ Ef ; moreover, in dimension n ≥ 4 every f -space form is a locally conformally flat gradient Ricci soliton (see Proposition 4.6 and Section 4.3 for more details). The classes LS and LSf (and also LSE and LSEf ) One has


1 ∇ A ?g . n−2 Moreover, ∇A = 0 implies the constancy of R, and is thus equivalent to ∇ Ric = 0. By orthogonality, ( ∇W = 0, (M, g) ∈ LS ⇐⇒ ∇ Riem = 0 ⇐⇒ ∇ Ric = 0, ∇ Riem = ∇W +

and analogously (M, g, f ) ∈ LSf

⇐⇒

∇ Riemf = 0

⇐⇒

( ∇W = 0, ∇ Ricf = 0.

(4.6)

Note that LS ⊂ PR and LSf ⊂ PRf . Moreover, since by definition LSE = LS ∩ E and LSEf = LSf ∩ Ef , we get ( ( ∇ Riem = 0, ∇W = 0, (M, g) ∈ LSE ⇐⇒ ⇐⇒ Ric = λg, Ric = λg,

4.2. Canonical metrics revisited: equivalent conditions

79

and analogously (M, g, f ) ∈ LSEf

⇐⇒

( ∇ Riemf = 0, Ricf = λg,

⇐⇒

( ∇W = 0, Ricf = λg.

(4.7)

For a general discussion on the consequences of these equivalences, see again Section 4.3. The classes HC and HCf By the Bianchi identities, div(Riem)ijk = Rtijk,t = Rik,j −Rij,k ; in particular, from the decomposition (1.77) it follows that, on every Riemannian manifold (n ≥ 3), n−3 n−3 Rtijk,t = Wtijk,t + (Rj δik − Rk δij ). n−2 2(n − 1)(n − 2) This implies that (M, g) ∈ HC

⇐⇒

Ric is a Codazzi tensor

⇐⇒

( div (W ) = 0, ∇R = 0.

Moreover, a simple computation shows that (M, g) ∈ HC

⇐⇒

div [E ? g] = 0

where E := Ric − R2 g is the Einstein tensor, which satisfies the condition div(E) = 0. As far as HCf metrics are concerned, we have the Lemma 4.5. The following conditions are equivalent: a) (M, g, f ) ∈ HCf ; b) The Bakry-Emery Ricci tensor, Ricf , is a Codazzi tensor. c) (M, g, f ) satisfies ( ∇f Cijk + ft Wtijk = Dijk , Ri = 2ft Rti ,

(4.8)

where D∇f is the tensor defined in (4.4). Proof. The equivalence a) ⇔ b) follows from the commutation relation fjkt −fjtk = fi Rijkt and the fact that
e−f Rijkt i = e−f (Rijkt,i − fi Rijkt ) = e−f (Rjt,k − Rjk,t − fi Rijkt )
= e−f (Ricf )jt,k − (Ricf )jk,t .

80

Chapter 4. Curvature Conditions

If (M, g, f ) ∈ HCf , we have Rtijk,t − ft Rtijk = 0, that is, Rij,k − Rik,j = −ft Rtijk . Using this relation, the definition of the Cotton tensor C and of D∇f , the decomposition of the Riemann curvature tensor (1.77), and the relation Ri = 2ft Rti , we get the equivalence a) ⇔ c).  If n ≥ 4, Lemma 4.5 and equation (1.82) immediately imply that (M, g, f ) ∈ HCf ⇐⇒ Ricf is a Codazzi tensor   ( ∇f n−3 Wtijk,t = n−2 ft Wtijk − Dijk , ⇐⇒ ∇R = 2 Ric(∇f, ·). R

Let Ef := Ricf − 2f g. ByIn analogy with the classical case, we call Ef the f Einstein tensor. From the commutation rule fijk − fikj = ft Rtijk and since, by the definition of the Kulkarni-Nomizu product, α ? g)tijk,t = αtj,t δik − αtk,t δij + αik,j − αij,k ,

(4.9)

we have       1 1 div(Ef ? g)ijk = ftjt − fttj δik − ftkt − fttk δij + Rik,j − Rij,k 2 2 i 1h + ft Rtikj − (Rf )j δik − (Rf )k δij 2

1 1 = Rk − 2ft Rtk δij − Rj − 2ft Rtj δik 2 2
+ Rik,j − Rij,k + ft Rtikj . Now, if div [Ef ? g] = 0, then taking the trace in this last tracing relation, we obtain ∇R = 2 Ric(∇f, ·). Consequently,
0 = Rik,j − Rij,k + ft Rtikj = ef div(e−f Riem)ijk , i.e., (M, g, f ) ∈ HCf . Note that the converse is also true, and so (M, g, f ) ∈ HCf

⇐⇒

div [Ef ? g] = 0.

(4.10)

Moreover, this equivalence enables us to define the non-gradient counterpart of HCf , as we will see in Section 4.6.

4.3. The rigid classes: SFf , LSf , LSEf and PRf

81

The classes Y and Yf Obviously, the Bianchi identities imply that (M, g) ∈ Y

⇐⇒

∇R = 0

⇐⇒

div(Ric −R g) = 0 .

As far as Yf metrics are concerned, since

Ef ? g isks = (n − 2) Ricf −Rf g ik , we have (M, g, f ) ∈ Yf

⇐⇒

∇R = 2 Ric(∇f, ·)

⇐⇒

div(Ricf −Rf g) = 0

and, again, the latter equivalence enables us to define the non-gradient counterpart of Yf (see again Section 4.6).

The rigid classes: SFf , LSf , LSEf and PRf SF ∩ SF f

⊂ ⊂

PR ∪ ⊂ E ∩ ⊂ Ef ∩ ⊂ PRf ⊂

⊂ ⊂

HC ∩ HCf

Y ∩ ⊂ Yf ⊂

∪

⊂

LS ∪ LSE ∩ LSE f ∩ LS f

∩

⊂

4.3

First of all we observe that, as in the case of PR, if (M, g, f ) ∈ PRf , i.e., ∇ Ricf = 0 on M , then the de Rham decomposition theorem shows that (M, g, f ) is locally a Riemannian product of gradient Ricci solitons (see e.g. [11, Sect. 16.12(i)] for a general splitting result concerning Codazzi tensors with constant eigenvalues). We begin with the classification of f -space forms. Observe that, in dimension n = 3, we have SFf = Ef ; in higher dimensions n ≥ 4, we prove the following Proposition 4.6. Let (M, g, f ) ∈ SFf . Then
• if λ > 0, (M, g, f ) is isometric, up to quotients, to either Sn , gSn , f = c ∈ R ,

or R × Sn−1 , dr2 + gSn−1 , f = λ2 r2 , or Rn , gRn , f = λ2 |x|2 ; • if λ =
0, then (M, g) is isometric, up to quotients, to either Rn , gRn , f = c ∈ R , or the Bryant soliton. • if λ < 0, around any regular point of f the manifold (M, g) is locally a warped product with codimension-one fibers of constant sectional curvature. Moreover, if the Ricci curvature is non-negative, (M, g) is rotationally symmetric.

82

Chapter 4. Curvature Conditions

We recall that the Bryant soliton, constructed in [26], is the unique (up to homotheties) rotationally-symmetric gradient steady Ricci soliton with positive sectional curvature.

Proof. Let (M, g, f ) ∈ SFf . First we observe that, in dimension n = 3, SFf = Ef . Thus, from the classification of three-dimensional gradient shrinking solitons, if λ > 0, then (M, g) is isometric, up to quotients, to either S3 , or R × S2 , or R3 . On the other hand, if n ≥ 4, then in view of the conditions (4.5), (M, g, f ) is a locally conformally flat gradient Ricci soliton. Proposition 4.6 now follows from the classifications results in the shrinking ([126, 151, 134]), steady ([33, 47]) and expanding ([47]) cases. To the best of our knowledge, the complete classification of locally conformally flat, gradient expanding Ricci solitons is still open; however, it is known that around any regular point of f the manifold (M, g) is locally a warped product with codimension one fibers of constant sectional curvature. 

Concerning the classes LSf and LSEf , we note that, in dimension n = 3, LSf = PRf and LSEf = Ef ; in higher dimension n ≥ 4 we prove Proposition 4.7. If (M, g, f ) ∈ LSf , then (M, g, f ) ∈ LS ∪ SFf . Furthermore, if (M, g, f ) ∈ LSEf , then either (M, g, f ) ∈ LSE ∪ SFf , or (M, g, f ) is
isometric, up to quotients, to a Riemannian product Rk × N, gRk + gN , f = λ2 |x|2k , k ≥ 1, with N ∈ LSE being a (n − k)-dimensional locally symmetric Einstein manifold. Proof. Let (M, g, f ) ∈ LSf . As we have already observed, in dimension n = 3, LSf = PRf . If n ≥ 4, equation (4.6) implies that (M, g, f ) ∈ PRf and the Weyl tensor is parallel, ∇W = 0. In particular, by a classical result of Derdzinski and Roter (see [69]), either ∇ Riem = 0, or W = 0. In the first case (M, g, f ) ∈ LS, while in the second case we are dealing with a locally conformally flat manifold with ∇ Ricf = 0. Again, by de Rham decomposition theorem, we have only two possibilities: either (M, g, f ) ∈ Ef with W = 0 and thus, from equation (4.5), (M, g, f ) ∈ SFf ; or (M, g) splits as the product of two locally symmetric factors (a line with a space form or two space forms with sign-opposite constant curvature and same dimension), and then (M, g, f ) ∈ LS. Now let (M, g, f ) ∈ LSEf . In dimension n = 3, LSEf = Ef , while if n ≥ 4, by the previous discussion, either (M, g, f ) ∈ SFf ∩ Ef = SFf , or (M, g, f ) ∈ LS ∩ Ef . In this case, in particular, the manifold is a gradient Ricci soliton which is also locally a product of Einstein metrics. Considering the universal cover and using classical results on concircular (gradient) vector fields (see, e.g., [144]), we see that we can only have two type of factors in the decomposition: the Euclidean space or a (locally symmetric) Einstein manifold. This concludes the proof. 

4.4. The class HCf

The class HCf SF ∩ SFf

⊂ ⊂

PR ∪ ⊂ E ∩ ⊂ Ef ∩ ⊂ PRf ⊂

⊂ ⊂

HC ∩ HC f

Y ∩ ⊂ Yf ⊂

∪

⊂

LS ∪ LSE ∩ LSEf ∩ LSf

∩

⊂

4.4

83

4.4.1 Rigidity and characterization results
First of all, we recall that (M, g, f ) ∈ HCf if and only if div e−f Riem = 0 or, equivalently, by Lemma 4.5, if and only if ( ∇f Cijk + ft Wtijk = Dijk , Ri = 2ft Rti , where ∇f Dijk =

1 1 (fk Rij − fj Rik ) + ft (Rtk gij − Rtj gik ) n−2 (n − 1)(n − 2) R − (fk gij − fj gik ). (n − 1)(n − 2)

The results of the previous section are consequences of the fact that, as we have seen, the equations defining f -space forms and f -locally symmetric metrics impose strong constraints on the Weyl tensor W , since they involve the full f curvature tensor Riemf . On the other hand, when one imposes conditions solely on Ricf , that is on the trace part of Riemf , it is reasonable to expect rigidity only assuming further conditions on the traceless part, i.e., W . The next theorem extends to the HCf class the well known result concerning the local structure of locally conformally flat gradient Ricci solitons. Theorem 4.8. Let (M, g, f ) ∈ HCf . If (M, g) is locally conformally flat, then, around any regular point of f , it is locally a warped product with codimension-one fibers of constant sectional curvature. Proof. Let (M, g, f ) ∈ HCf ; by the assumption of local conformal flatness, both the Cotton and the Weyl tensors vanish on M . By Lemma 4.5, the tensor D∇f also vanishes. Contracting with ∇f and using the equation Ri = 2ft Rti , we obtain 0 = (n − 1)(n − 2)Dijk fk = (n − 1)|∇f |2 Rij − (n − 1)Rik fk fj + Rtk ft fk gij − Rtj ft fi − |∇f |2 R gij + Rfi fj n−1 1 1 = (n − 1)|∇f |2 Rij − |∇f |2 R gij − Ri fj − fi Rj + h∇R, ∇f igij + Rfi fj . 2 2 2

84

Chapter 4. Curvature Conditions

By symmetry, Ri fj = Rj fi , i.e., dR ∧ df = 0. In particular, ∇f is an eigenvector of the Ricci tensor and, from 2 Ric(∇f, ∇f ) = h∇R, ∇f i we obtain 0 = (n − 1)|∇f |2 Rij − |∇f |2 R gij −

n Ri fj + Ric(∇f, ∇f )gij + Rfi fj . 2

Now, around a regular point of f , pick an arbitrary orthonormal frame e1 , . . . , en which diagonalizes the Ricci tensor. Since ∇f is an eigenvector of the Ricci tensor, ∇f without loss of generality we can set e1 = |∇f | . Denote by µk , k = 1, . . . , n the corresponding eigenvalues. Then, for every k ≥ 2, we have
0 = |∇f |2 (n − 1)µk − R + µ1 . 1 Thus, around a regular point of f , one has µk = n−1 (R − µ1 ) for every k ≥ 2. In particular, around a regular point of f , either the Ricci tensor is proportional to the metric, or it has an eigenvalue of multiplicity n − 1 and another eigenvalue of multiplicity 1. Now suppose that f is not constant. We have shown that either the metric is locally Einstein (thus of constant curvature), or the Ricci tensor has two eigenvalues of respective multiplicities 1 and n − 1. In the first case, the manifold must be locally isometric to a space form. In the second case, since the Cotton 1 tensor C vanishes, the Schouten tensor Ric − 2(n−1) R g is a Codazzi tensor with at most two distinct eigenvalues of respective multiplicities 1 and n − 1. Hence, by general results on Codazzi tensors with this property (see [117, 11, 49]), the manifold (M, g) is locally a warped product with codimension one fibers. Since the manifold is locally conformally flat, the fibers must have constant sectional curvature. This concludes the proof of Theorem 4.8. 

It is well known that compact locally conformally flat gradient Ricci solitons have constant curvature (see, e.g., [75]). We will see that this conclusion cannot be extended to manifolds in HCf , since we can construct rotationally-symmetric examples on S1 × Sn−1 (see below). In order to state the next results, we first recall that, as already observed, HC ⊂ Y, i.e., harmonic curvature metrics have constant scalar curvature. This is not true in general for the potential counterpart HCf , but, for instance, on gradient Ricci solitons it holds that Rf = R + ∆f = nλ. Thus, it is natural to introduce the following Definition 4.9. Let (M, g, f ) be a n-dimensional manifold with Riemannian metric g and f ∈ C ∞ (M ). We say that (M, g, f ) ∈ HCλf if (M, g, f ) ∈ HCf and, for some λ ∈ R, Rf := R + ∆f = nλ. Note that Ef ⊂ HCλf ⊂ HCf and also, by a simple computation, PRf ⊂ HCλf . We will see in a short while that under some additional conditions the class HCλf (and HCf , in some cases) coincides with Ef . First, we recall that in dimension four, under the topological condition τ (M ) 6= 0, Bourguignon in [19] proved that

4.4. The class HCf

85

HC = E (where τ is the signature of M ). Moreover, the Hirzebruch signature formula (see (3.24)) says that Z Z 48π 2 τ (M ) = |W + |2 − |W − |2 , M

M

where W + and W − are the self-dual and anti-self-dual parts of the tensor W , respectively. In the next theorem we extend Bourguignon’s result in the HCλf case, and, more generally, in the HCf case, under an additional regularity assumption (which is automatically satisfied by HC metrics, as proved in [73]). Theorem 4.10. Let M be a four-dimensional compact manifold with τ (M ) 6= 0. Then, i) (M, g, f ) ∈ HCf and, in harmonic coordinates, g and f are real analytic if and only if (M, g, f ) ∈ Ef . ii) (M, g, f ) ∈ HCλf if and only if (M, g, f ) ∈ Ef . Note that gradient Ricci solitons satisfy the analyticity assumption, but we do not know in general if this is true for metrics in HCf . Proof. Assume that τ (M ) 6= 0 and let (M, g, f ) ∈ HCf , for some potential function f ; assume also that, in harmonic coordinates, g and f are real analytic. By Lemma 4.5, the Bakry-Emery Ricci tensor Ricf is Codazzi. In particular, the following property holds: Lemma 4.11. Let T be a Codazzi tensor on a four-dimensional Riemannian manifold (M, g). Then, at any point x ∈ M where T is not a multiple of g, the endomorphisms W + of Λ+ and W − of Λ− have equal spectra. This result was proved by Bourguignon [19] (see also [70]) and used in the context of manifolds with harmonic curvature. By analyticity, it implies that either Ricf is proportional to the metric (i.e., (M, g, f ) ∈ Ef ), or W + and W − have equal spectra on M . But this contradicts the topological assumption on τ (M ), and so the first part of Theorem 4.10 is proved. Assume now that (M, g, f ) ∈ HCλf , without imposing extra regularity on g and f . We have that
div e−f Riem = 0 and R + ∆f = nλ . (4.11) By Lemma 4.5, the Bakry-Emery Ricci tensor Ricf is a Codazzi tensor with constant trace. Equivalently, ◦ Rf Ricf := Ricf − g n is a trace-free Codazzi tensor. In particular, we have the following regularity lemma, which follows from a general results of Kazdan [100] (see also [85, 44] for some applications).

86

Chapter 4. Curvature Conditions ◦

Lemma 4.12. Let T be a, non-trivial, trace-free Codazzi tensor on a Riemannian ◦

manifold (M, g) and let Ω0 = { x ∈ M n : |T |(x) 6= 0 }. Then Vol (M \ Ω0 ) = 0. ◦

Using this in combination with Lemma 4.11, we see that either Ricf ≡ 0 (i.e., (M, g, f ) ∈ Ef ), or Z Z |W − |2 ,

|W + |2 =

M

M

which again contradicts the assumption τ (M ) 6= 0, and completes the second part of Theorem 4.10.  The following statement is an immediate consequence of Theorem 4.10 ii) and the classification of half conformally flat gradient Ricci solitons in [63]. Corollary 4.13. Let M be a four-dimensional compact manifold and let (M, g, f ) ∈ HCλf . If (M, g) is half-conformally flat, but not conformally flat, then i) if λ > 0, (M, g) is isometric to CP2 with its canonical metric; ii) if λ = 0, the universal covering of (M, g) is isometric to a K3 surface with the Calabi–Yau metric; iii) if λ < 0, (M, g) ∈ E with negative scalar curvature. In general dimension n ≥ 3 we can prove, assuming positive sectional curvature, the following extension of a results of Berger (see [11]). Proposition 4.14. Let (M, g) be a n-dimensional, n ≥ 3, compact manifold with positive sectional curvature. Then (M, g, f ) ∈ HCλf if and only if (M, g, f ) ∈ Ef . ◦

Proof. Let (M, g, f ) ∈ HCλf . Then Ricf is a trace-free Codazzi tensor. In particular (see [11] or [44]), the following Weitzenb¨ock formula holds: ◦ ◦ ◦ ◦ ◦ ◦ 1 ∆|Ricf |2 = |∇Ricf |2 − Rikjl (Ricf )ij (Ricf )kl + Rjk (Ricf )ij (Ricf )ik . 2

(4.12)

◦

Let ei , i = 1, . . . , n, be the eigenvectors of Ricf and let µi be the corresponding eigenvalues. Let kij be the sectional curvature in the direction of the two-plane spanned by ei and ej . One has ◦

◦

◦

◦

−Rikjl (Ricf )ij (Ricf )kl + Rjk (Ricf )ij (Ricf )ik = −

n X i,j=1

µi µj kij +

n X

µ2i kij

i,j=1

X = (µi − µj )2 kij ≥ 0 , i 0 for all i, j = 1, . . . , n. Using this and integrating the Weitzenb¨ock ◦

formula, we get that Ricf has to be zero on M , i.e., (M, g, f ) ∈ Ef . This proves Proposition 4.14. 

4.4. The class HCf

87

A classical result by Tachibana ([141]) says that if (M, g) ∈ HC, with positive curvature operator, then (M, g) is, up to quotients, the round sphere; in the HCλf case we have the following corollary, which simply follows from Proposition 4.14 and the classification of compact gradient Ricci solitons with positive curvature operator (see [14]). Corollary 4.15. Let (M, g) be a n-dimensional (n ≥ 3) compact manifold with positive curvature operator. If (M, g, f ) ∈ HCλf , then f is constant and (M, g) is isometric, up to quotients, to Sn .

4.4.2 Two examples We construct two examples of Riemannian manifolds in HCf that are not gradient Ricci solitons, following the construction for the harmonic curvature case given by Derdzinski in [67] and using the same notation to highlight the similarities. Let I ⊆ R be an interval, let F ∈ C ∞ (I) be a smooth positive function on I, and let (N, h) be an (n − 1)-dimensional Einstein manifold with constant scalar curvature K. We
consider the warped product manifold M = I × N, g = dt2 + F (t)h . Letting the indices i, j, k run through 1, . . . , n − 1 and given a local chart t = x0 , x1 , . . . , xn−1 for I × N , we have g00 = 1, g0i = 0, gij = F hij and the components of the Ricci tensor Ric and its covariant derivative ∇ Ric are given by  n − 1  00 R00 = − 2q + (q 0 )2 , R0i = 0 , (4.13)  4   K 1  Rij = − eq 2q 00 + (n − 1)(q 0 )2 hij , n−1 4  n − 1  000 ∇0 R00 = − q + q 0 q 00 , ∇0 Ri0 = ∇i R00 = 0 , 2  K 0 1 q 000 n − 1 q 0 00 ∇0 Rij = − q + e q + e q q hij , n−1 2 2   K n − 2 q 0 00 ∇i R0j = − + e q q hij , ∇k Rij = 0 , (4.14) 2(n − 1) 4 where q = log F . Since ∇0 Ri0 = ∇i R00 = Rpi00 fp = 0, the condition div(e−f Riem) = 0 is equivalent to ∇0 Rij − ∇i R0j + R0ji0 ∇0 f = 0 .

(4.15)

Using the expression of the Riemann curvature tensor in terms of the Christoffel symbols and the fact that Γj0i = 12 q 0 hij , Γi00 = Γ0i0 = Γ000 = 0, one has R0ij0 = ∂0 Γji0 − ∂i Γj00 + Γpi0 Γj0p − Γp00 Γjip 1  = ∂0 q 0 hij + Γki0 Γj0k 2  1  00 = 2q + (q 0 )2 hij . 4

88

Chapter 4. Curvature Conditions

Hence, equation (4.15) is equivalent to the following differential equation for the function q  n K −q 0 1  00 q 000 + q 0 q 00 + e q = 2q + (q 0 )2 f 0 . (4.16) 2 n−1 2 First of all, a simple computations shows that the choice n K = 0, q(t) = t2 , f (t) = log(1 + t2 ). 2 gives a solution to the equation. Therefore, given any (n − 1)-dimensional Riemannian Ricci flat manifold (N, h), one has   2 n M = R × N, g = dt2 + et h, f (t) = log(1 + t2 ) ∈ HCf . 2 Now we want to construct a compact example. Integrating equation (4.16), we get Z  n 0 2 K −q 1  00 00 q + (q ) − e = 2q + (q 0 )2 f 0 . (4.17) 4 n−1 2 Now, we suppose that, given a function q defined on some interval I, we can find f solving  1  00 εK 0 −q 2q + (q 0 )2 f 0 = qe , (4.18) 2 n−1 for some ε > 0. Plugging this into (4.17), we reduce the problem to solving the equation n K − ε −q 4 q 00 + (q 0 )2 − e = C, (4.19) 4 n−1 n n for some constant C ∈ R. Letting ϕ := e 4 q , we obtain the ODE ϕ00 −

n(K − ε) 1− 4 ϕ n = Cϕ , 4(n − 1)

(4.20)

for some constant C ∈ R. It was shown in [67, Theorem 1] that, if K > ε and C < 0, this equation has non-constant positive periodic smooth solutions, defined n on R. Now, let ϕ = e 4 q be a solution. Then equation (4.19) implies that   8 n − 2 0 2 2(K − ε) −q 8 2(K − ε) 00 0 2 2q + (q ) = C − (q ) + e ≤ C+ ε > 0. Choose a non-constant, positive, n periodic function F on R such that ϕ = F 4 satisfies (4.20) for some constant C < − 2(K−ε) n−1 , and choose a solution f = f (t) of equation (4.18). Then, following the precise construction in [67, Section 3], we can define a compact Riemannian
f, ge, fe), such that M f is diffeomorphic quotient of R × N, g = dt2 + F (t)h, f (t) , (M 1 f to S × N and ge has weighted harmonic curvature, namely (M , ge, fe) ∈ HCf .

4.5. The class Yf : a possible generalization of the Yamabe problem

The class Yf : a possible generalization of the Yamabe problem

SF ∩ SFf

⊂ ⊂

PR ∪ ⊂ E ∩ ⊂ Ef ∩ ⊂ PRf ⊂

⊂ ⊂

HC ∩ HCf

Y ∩ ⊂ Yf ⊂

∪

⊂

LS ∪ LSE ∩ LSEf ∩ LSf

∩

⊂

4.5

89

In this section we consider the class of Yf Riemannian manifolds, i.e., manifolds (M, g, f ) that satisfy the condition ∇R = 2 Ric(∇f, ·) .

(4.21)

This equation is a meaningful generalization of the equation for Yamabe metrics (Y) and can be seen as a very special constraint on the gradient of the scalar curvature, connecting the Ricci tensor with its trace via the potential function. It is clear that any Ricci flat metric satisfies (4.21), for any function f and, more generally, so does any product of a Ricci flat metric with a metric with constant scalar curvature, for any function f that depends only on the first factor. From this point of view, it is natural to study the following problems on a given manifold M : (A) having fixed f ∈ C ∞ (M ), does there exist a metric g such that (M, g, f ) ∈ Yf ? (B) having fixed f ∈ C ∞ (M ) and a metric g0 , does there exist a conformal metric g ∈ [g0 ] such that (M, g, f ) ∈ Yf ? More generally, one could ask the question (C) does there exist a metric g and a smooth function f ∈ C ∞ (M ) such that (M, g, f ) ∈ Yf ? Clearly the answer to (C) is positive, since it is always possible to construct a (complete) metric with constant (negative) scalar curvature ([5] and [12]). Furthermore, when f is constant, (B) boils down to the well-known Yamabe problem, which is completely solved when M is compact (see e.g. [111]).

4.5.1 An obstruction In the same spirit of the work of Kazdan and Warner (see [102]), we prove here some obstructions to problem (B).

90

Chapter 4. Curvature Conditions

First of all, we recall that a smooth vector field X is a conformal vector field on (M, g) if and only if 2 div (X) LX g = g, (4.22) n where LX g denotes the Lie derivative of the metric in the direction X. Equation (4.21), in conjunction with the the well-known Kazdan-Warner identity (see [102, 18]), Z h∇S, Xi dV = 0

(4.23)

M

for every conformal vector field X, gives the following integral condition for compact f -Yamabe metrics, the simple proof of which is included here for the sake of completeness. Lemma 4.16. If M is compact and (M, g, f ) ∈ Yf , then, for every conformal vector field X on (M, g), Z Ric(∇f, X) dV = 0 . M

Proof. Equation (4.21) and the fact that X satisfies the relation Xij + Xji =

2 div (X) gij , n

imply that Z 2

Z

Z 2n ˚ij,j Xi dV h∇R, Xi dV = R n−2 M M Z
n ˚ij Xij + Xji dV = 0 , =− R n−2 M

Ric(∇f, X) dV = M

where we have used integration by parts and Bianchi identity for the traceless ˚ i.e., in coordinates, R ˚ij = Rij − R δij .  Ricci tensor Ric, n When (M, g0 ) supports a non-trivial (non-vanishing) conformal gradient vector field, the previous lemma gives an obstruction to the existence of an f -Yamabe metric in the conformal class [g0 ]. Corollary 4.17. Let (M, g0 ) be a compact Riemannian manifold and X = ∇f , f ∈ C ∞ (M ), be a non-trivial conformal gradient vector field on (M, g0 ). Then, there are no conformal metrics g ∈ [g0 ] such that (M, g, f ) ∈ Yf . Proof. Let g ∈ [g0 ]. By the conformal invariance of equation (4.22), X = ∇f is also a conformal vector field for (M, g), i.e., the potential function f satisfies ∇2 f =

∆f g, n

4.5. The class Yf : a possible generalization of the Yamabe problem

91

where all the covariant derivatives refer to the metric g. Integrating Bochner’s formula 1 ∆|∇f |2 = |∇2 f |2 + Ric(∇f, ∇f ) + h∇f, ∇∆f i 2 over M , one obtain Z Z Z Z n−1 Ric(∇f, ∇f ) dV = |∆f |2 dV − |∇2 f |2 dV = |∆f |2 dV . n M M M M Suppose now that (M, g, f ) ∈ Yf . Then, using Lemma 4.16 with X = ∇f , we obtain ∆f = 0, i.e., f is constant on M , which is a contradiction.  In particular, from this result we can deduce Proposition 4.18. If f ∈ C ∞ (Sn ) is a first spherical harmonic on the round sphere (Sn , g0 ), then there are no conformal metrics g ∈ [g0 ] such that (M, g, f ) ∈ Yf . Note that, by a classical result of Tashiro [144], every compact manifold supporting a non-trivial (non-vanishing) conformal gradient vector field is conformal to the round sphere Sn .

4.5.2 An example Let I ⊆ R be an interval, let F ∈ C ∞ (I) be a smooth positive function on I, and let (N, h) be an (n−1)-dimensional manifold with Ricci curvature ρ. As in
Section 4.4, we consider the warped product manifold M = I×N, g = dt2 +F (t)h . Letting the indices i, j, k run through 1, . . . , n − 1 and given a local chart t = x0 , x1 , . . . , xn−1 for I × N , we have g00 = 1, g0i = 0, gij = F hij and the components of the Ricci tensor Ric are given by  n − 1  00 R00 = − 2q + (q 0 )2 , R0i = 0 , (4.24) 4   1 Rij = ρij − eq 2q 00 + (n − 1)(q 0 )2 hij . 4 where q = log F . Suppose that (N, h) has constant scalar curvature k. Then, the scalar curvature of (M, g) is given by  n − 1  00 R=− 4q + n(q 0 )2 + ke−q . 4 On the other hand, if the potential function f is radial, then  n − 1  00 Ric(∇f ) = g 00 R00 f 0 = − 2q + (q 0 )2 f 0 . 4 Thus, equation (4.21) is equivalent to the ODE q 000 +

 n 0 00 k 1  00 qq + e−q q 0 = 2q + (q 0 )2 f 0 . 2 n−1 2

92

Chapter 4. Curvature Conditions

Notice that this equation coincide with (4.16). Hence, again the choice k = 0,

q(t) = t2 ,

f (t) =

n log(1 + t2 ) 2

gives a solution to the equation. In this case, given any (n − 1)-dimensional Riemannian scalar flat manifold (N, h), one has   2 n M = R × N, g = dt2 + et h, f (t) = log(1 + t2 ) ∈ Yf . 2 Moreover, if (N, h) is not Ricci flat, it is easy to see that (M, g, f ) ∈ / HCf . On the other hand, following the recipe in Section 4.4, given any compact (n − 1)-dimensional manifold (N, h) with constant positive scalar curvature k > 0, we can construct an f -Yamabe metric on a compact manifold M diffeomorphic to S1 × N . As before, if (N, h) is not Einstein, then this solution (M, g, f ) ∈ / HCf .

4.5.3 An existence result in Euclidean space Here, following [46], we consider the problem (B) stated before at the beginning of Section 4.5, when f is non-constant, on the Euclidean space Rn endowed with the standard flat metric gRn . ODE formulation of the conformal f -Yamabe problem Let (M, g) be a smooth n-dimensional Riemannian manifold, n ≥ 2, and let f ∈ C ∞ (M ). Recall that, if ge = e2w g ∈ [g] for some w ∈ C ∞ (M ), then (2.68) and (2.70) imply that g = Ric −(n − 2)∇2 w + (n − 2)dw ⊗ dw − (∆w) g − (n − 2)|∇w|2 g , Ric   e = e−2w R − 2(n − 1)∆w − (n − 1)(n − 2)|∇w|2 R where ∇2 is the Hessian and ∆ = g ij ∇2ij is the Laplace-Beltrami operator of g. A computation shows that (M, ge, f ) = (M, e2w g, f ) ∈ Yf if and only if the function w solves the system of PDEs  ∇∆w + (n − 2)∇2 w(∇w, ·) − 2∆w + (n − 2)|∇w|2 −

 1 R ∇w n−1

(4.25)

1 ∇R 2(n − 1)  1 n−2 2 1  =− Ric(∇f, ·) + ∇ w(∇f, ·) + ∆w + (n − 2)|∇w|2 ∇f n−1 n−1 n−1 n−2 − h∇w, ∇f i∇w. n−1 −

4.5. The class Yf : a possible generalization of the Yamabe problem

93

In particular, since (Rn , gRn ) is Ricci flat, then (Rn , ge, f ) = (Rn , e2w gRn , f ) ∈ Yf if and only if w solves the system of PDEs   ∇∆w + (n − 2)∇2 w(∇w, ·) − 2∆w + (n − 2)|∇w|2 ∇w (4.26)  n−2 2 1  n−2 = ∇ w(∇f, ·) + ∆w + (n − 2)|∇w|2 ∇f − h∇w, ∇f i∇w. n−1 n−1 n−1 To fully exploit the symmetries of the Euclidean space, it is reasonable to start our analysis by considering radial solutions w = w(r) of (4.26) for a given radial function f = f (r), where r denotes the distance from the origin. In this case, in standard polar coordinates, one has gRn = dr2 + r2 gSn−1 , dw = w0 (r)dr and df = f 0 (r)dr , ∇dw = ∇2 w = w00 (r)dr2 + rw0 (r)gSn−1 , n−1 0 ∆w = w00 (r) + w (r), r and a computation shows that the system (4.26) boils down to the following second order non-linear ODE for the function u(r) := w0 (r): n−1 0 2(n − 1) u (r) − (n − 2)u(r)3 − u(r)2 r r   n−1 u(r) 0 0 − u(r) + (n − 4)u(r)u (r) = u (r) + h(r) , r2 r

u00 (r) +

(4.27)

where h(r) := f 0 (r). Note that, if n = 2, then the cubic term in (4.27) vanishes. We then impose the initial conditions u(0) = 0 ,

u0 (0) = α 6= 0,

(4.28)

which require some explanation, since (4.27) is singular at r = 0. Assume that u ∈ C 2 ([0, ∞)) satisfies (4.28); then u(r) = αr + O(r2 ) as r → 0 and, in turn, n−1 0 n−1 u (r) − u(r) = O(1) , r r2

u(r) = O(1) , r

u(r)2 = o(1) r

as r → 0,

which shows that, by suitably combining the terms in (4.27), we obtain finite limits as r → 0. The existence and uniqueness of a solution of (4.27)-(4.28) can then be proved rigorously by adapting the arguments of Proposition 1 in [127]: one needs to combine the Ascoli-Arzel`a Theorem with the Schauder fixed point theorem in order to establish the existence of a solution. Then the solution is unique as long as it can be continued ([77, Proposition 4.2]). Before stating our existence and non-existence results, let us discuss heuristically the structure of (4.27).

94

Chapter 4. Curvature Conditions

Heuristic preliminaries We first notice that, if n ≥ 3, then there exist exactly two singular (negative) solutions of (4.27) of the type cr−1 , given by u1 (r) = −

1 , r

u2 (r) = −

2 , r

(4.29)

regardless of the explicit form of h. This fact suggests that the “interesting dynamics” for (4.27) occurs when u(r) < 0 and global solutions of (4.27) are more likely to be prevalently negative. If n = 2, then the functions u(r) = cr−1 are singular solutions of (4.27) for any c 6= 0; in particular, there exist infinitely many positive singular solutions and the dynamics appears much more chaotic. It is quite useful to consider the following two functions, defined for all (r, y) ∈ R+ × R:   2(n − 1) n − 1 h(r) P (r, y) = (n − 2)y 2 + y+ + y, r r2 r n−1 Q(r, y) = h(r) − + (4 − n)y . r Then equation (4.27) may be written in normal form as

u00 (r) = Q r, u(r) u0 (r) + P r, u(r) . (4.30) Depending on h ∈ C 0 ([0, ∞)), we define the two regions   Ih := r ≥ 0; (n − 2)rh(r) < n − 1 and I h = r ≥ 0; (n − 2)rh(r) > n − 1 . Clearly, Ih contains a right neighborhood of r = 0 and is therefore nonempty for all h, while I h is empty if rh(r) ≤ n−1 n−2 for all r, in particular if h(r) ≤ 0. It is also straightforward that: • if r ∈ I h , then P (r, y) = 0 if and only if y = 0; moreover P has the same sign as y; • if r ∈ Ih , then we can write " # p n − 1 + n − 1 − (n − 2)rh(r) P (r, y) = (n − 2) y + (n − 2)r " # p n − 1 − n − 1 − (n − 2)rh(r) × y+ y (n − 2)r and hence P (r, y) vanishes if and only if y(r) if of one of the following forms: p n − 1 + n − 1 − (n − 2)rh(r) y = 0 , y = ϕ(r) := − , (n − 2)r p 1 − n + n − 1 − (n − 2)rh(r) y = ψ(r) := . (n − 2)r

4.5. The class Yf : a possible generalization of the Yamabe problem

95

Note that ψ(r) > ϕ(r) for all r ∈ Ih but, while ϕ(r) < 0 for all r ∈ Ih , the sign of ψ(r) may vary and it is the opposite of the sign of rh(r) + n − 1; in particular, ψ(r) < 0 in a right neighborhood of r = 0. One expects that a crucial role for the existence results will be played by the signs of Q and P . However, the overall picture is not completely clear; to see this, consider the trivial case h ≡ 0, in which the function ua (r) = −

2ar 1 + ar2

(4.31)

solves (4.27)-(4.28) with α =p −2a. First notice that if a < 0 (so that u0a (0) > 0), then ua blows up as r → 1/ |a|. Therefore, ua is a global solution of (4.27) if and only if a > 0. Simple computations then show that if n = 3, 4, 5, then ∃ρ > 0 such that ϕ(r) < u1 (r) < ψ(r) ∀r > ρ , if n ≥ 6, then ∃ρ > 0 such that u1 (r) < ϕ(r) ∀r > ρ . These facts are illustrated in Figure 4.1, where the shaded region is Γ := {(r, y) ∈ R+ × R− ; ϕ(r) < y < ψ(r)} . In the left picture we see that the graph of u1 (thick line) eventually lies inside Γ, while in the right picture it eventually lies outside. Therefore, the function P (r, u1 (r)) does not always preserve the same sign as r → ∞. 5

10

15

20

5

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8

-1.0

-1.0

10

15

20

Figure 4.1: Plot of u1 (r) in (4.31) (thick line) and of Γ (shaded region) when n = 3 (left) and n = 8 (right).

Non-existence results We can prove the following (partial) non-existence results. Theorem 4.19. If n ≥ 3, h(r) ≥ − n−1 for all r > 0 and α > 0, then the problem r (4.27)–(4.28) does not admit a global solution. Proof. Throughout this proof we will need the following particular class of test functions.

96

Chapter 4. Curvature Conditions

Definition Let ρ > 0. We say that a non-negative function φ ∈ Cc2 ([0, ∞)) satisfies the ρ-property if for r ∈ [0, ρ] and φ(r) = 0

φ(r) = 1

for r ≥ 2ρ ,

(4.32)

and if Z

2ρ

00 φ (r) +

3/2 2(n−1) 0 φ (r) r



p φ(r)

ρ

dr < ∞ .

(4.33)

It is clear that such functions exist; to see this, it suffices to replace any function φ satisfying (4.32) with a power φk for k sufficiently large so that (4.33) will be satisfied. For the proof of Theorem 4.19, we first observe that, since α > 0 the solution of (4.27)–(4.28) is positive and strictly increasing in a right neighborhood of r = 0, say in some maximal interval (0, R). Clearly, between u and u0 the first one which can vanish is u0 . But if u0 (R) = 0 then, using the lower bound for h, we see that (4.27) yields   2(n − 1) n − 1 h(R) u(R)2 + + u(R) R R2 R 2(n − 1) ≥ (n − 2)u(R)3 + u(R)2 > 0 , R

u00 (R) ≥ (n − 2)u(R)3 +

so we reached a contradiction. Therefore, u0 cannot vanish and two cases may occur: (i) R = ∞ ,

(ii) R < ∞ and lim u(r) = +∞ . r↑R

(4.34)

The proof will be complete if we show that (ii) occurs. At this point, we distinguish two cases. • Case n ∈ {3, 4}. To prove (ii) in (4.34), we argue by contradiction, assuming that R = ∞, so that u, u0 > 0 for all r > 0. From the assumptions and (4.27), we then infer that (recall that n ≤ 4) u00 (r) +

2n − 2 0 u (r) > (n − 2)u(r)3 > 0 ∀r > 0 . r

(4.35)

To reach a contradiction we need the following estimate, inspired by the method developed by Mitidieri and Pokhozhaev [119] (see also the proof of [79, Proposition 5]).

4.5. The class Yf : a possible generalization of the Yamabe problem

97

Lemma 4.20. Assume that w ∈ C 2 ([0, ∞)). Then for any ε > 0, for any ρ > 0 and for all φ satisfying the ρ-property, we have Z 2ρ Z 2n − 2 0 ε 2ρ 2n−2 r2n−2 w00 (r) + w (r) φ(r) dr ≤ r |w(r)|3 φ(r) dr r 3 0 0 00 3/2 Z 2ρ 2n−2 0 2 2n−2 φ (r) + r φ (r) p + √ r dr . 3 ε ρ φ(r) Proof. We will use the Young inequality in the following form: ∀ε > 0 ∀a, b ≥ 0,

ab ≤

εa3 2b3/2 + √ . 3 3 ε

(4.36)

Fix ε > 0 and ρ > 0. We use a PDE approach and introduce the radial C 2 (R2n−1 ) function v given by v(x) = w(|x|) for all x: note that the space dimen0 sion is here 2n − 1 and that ∆v(x) = w00 (|x|) + 2n−2 |x| w (|x|). Next, we multiply ∆v by some function Φ(x) = φ(|x|), where φ satisfies the ρ-property. Since Φ ≡ 1 in Bρ , two integrations by parts and (4.36) yield Z Z Z Z ∆Φ ∆vΦ = v∆Φ = v∆Φ = vΦ1/3 1/3 Φ B2ρ B2ρ B2ρ \Bρ B2ρ \Bρ Z Z ε 2 |∆Φ|3/2 ≤ |v|3 Φ + √ , 3 B2ρ 3 ε B2ρ \Bρ Φ1/2 and back to the radial form of v and Φ this proves the statement.



Take a function φ1 satisfying the 1-property and observe that the function   r φρ (r) := φ1 ∀ρ > 1 ρ satisfies the ρ-property. Therefore, for all ε > 0, we infer from (4.35) and Lemma 4.20 that Z ρ Z 2ρ 2n−2 3 (n − 2) r u(r) dr ≤ (n − 2) r2n−2 u(r)3 φρ (r) dr 0

0 2ρ

 2n − 2 0  r2n−2 u00 (r) + u (r) φρ (r) dr r 0 Z Z 2ρ φ0 (r)|3/2 |φ00ρ (r) + 2n−2 ε 2ρ 2n−2 2 3 p r ρ ≤ r u(r) φρ (r) dr + √ r2n−2 dr . 3 0 3 ε ρ φρ (r) Z

≤

Taking here 0 < ε < 3(n − 2), we obtain 

ε n−2− 3

Z

ρ

r 0

2n−2

2 u(r) dr ≤ √ 3 ε 3

Z

2ρ

r ρ

2n−2

00 φρ (r) + 2n−2 φ0ρ (r) 3/2 r p dr . φρ (r)

98

Chapter 4. Curvature Conditions

With the change of variable r = ρt this becomes 00 Z Z 2  φ (t) + 2n−2 φ01 (t) 3/2 ε  1 2n−2 2 1 p t n−2− t u(ρt)3 dt ≤ √ 3 t2n−2 1 dt , 3 0 3 ερ 1 φ1 (t) Since u is increasing on R+ , we have u(ρt) ≥ u(t) for all ρ > 1 so that the lefthand side of this inequality is positive and increasing for ρ ≥ 1. When ρ → ∞, the right-hand side tends to 0 and this leads to a contradiction which rules out the case (i) . Hence, case (ii) occurs and the solution u of (4.27)-(4.28) with α > 0 cannot be continued to the entire the interval [0, ∞). This completes the proof of Theorem 4.19 in the case n = 3, 4. • Case n ≥ 5. The same arguments leading to (4.35) now yield 0 2n − 2 0 n − 4 u00 (r) + u (r) + u(r)2 > (n − 2)u(r)3 > 0, ∀r > 0, (4.37) r 2 and we need to estimate one more term. The companion of Lemma 4.20 reads Lemma 4.21. Assume that w ∈ C 2 ([0, ∞)). Then for any δ > 0, for any ρ > 0 and for all φ satisfying the ρ-property, we have √ Z Z 2ρ  0 2 δ 2ρ 2n−2 2n−2 2 r w(r) φ(r) dr ≤ r |w(r)|3 φ(r) dr 3 0 0 Z 2ρ 1 |φ0 (r)|3 + r2n−2 dr . 3δ ρ φ(r)2 Proof. We will use the Young inequality in the following form: √ 2 δ a3/2 b3 ∀δ > 0, ∀a, b ≥ 0, ab ≤ + . (4.38) 3 3δ Fix δ > 0 and ρ > 0; then take φ satisfying the ρ-property. An integration by parts yields Z 2ρ Z 2ρ  0   r2n−2 w(r)2 φ(r) dr = − w(r)2 2(n − 1)r2n−3 φ(r) + r2n−2 φ0 (r) dr 0

0

and, since φ ≥ 0, Z 2ρ Z  0 r2n−2 w(r)2 φ(r) dr ≤ −

2ρ

r2n−2 w(r)2 φ0 (r) dr 0 0  Z 2ρ    |φ0 (r)| 4(n−1)/3 ≤ r w(r)2 φ(r)2/3 · r2(n−1)/3 dr φ(r)2/3 0 √ Z Z 2ρ 2 δ 2ρ 2n−2 1 |φ0 (r)|3 3 [by (4.38)] ≤ r |w(r)| φ(r) dr + r2n−2 dr . 3 0 3δ 0 φ(r)2

Since φ0 ≡ 0 on (0, ρ), this completes the proof.



4.5. The class Yf : a possible generalization of the Yamabe problem

99

Take again a function φ1 satisfying the 1-property and let φρ (r) := φ1 (r/ρ) for all ρ > 1, so that φρ satisfies the ρ-property. Multiply (4.37) by φρ and integrate over (0, 2ρ) to obtain ρ

Z

r2n−2 u(r)3 dr ≤ (n−2)

(n−2)

2ρ

Z

0

r2n−2 u(r)3 φρ (r) dr

0

≤

2ρ

 2n − 2 0  u00 (r) + u (r) φρ (r) dr r 0 Z 0 n − 4 2ρ 2n−2  + r u(r)2 φρ (r) dr . 2 0 Z

r

2n−2

Then, by Lemmas 4.20 and 4.21 we infer that √ !Z ρ ε (n − 4) δ n−2− − r2n−2 u(r)3 dr 3 3 0 Z 2ρ Z 2n−2 0 00 |φ (r) + r φρ (r)|3/2 2 n − 4 2ρ 2n−2 |φ0ρ (r)|3 2n−2 ρ p ≤ √ r dr + r dr . 6δ φρ (r)2 3 ε ρ φρ (r) ρ Take ε > 0 and δ > 0 sufficiently small so that Cε,δ := n − 2 − and perform the change of variable r = ρt to obtain Z Cε,δ

1

t 0

2n−2

2 1 u(ρt) dt ≤ √ 3 3 ερ 3

+

Z

2

t2n−2

1

n−4 1 6δ ρ3

Z 1

ε 3

−

√ (n−4) δ 3

>0

|φ001 (t) + 2n−2 φ01 (t)|3/2 p t dt φ1 (t)

2

t2n−2

|φ01 (t)|3 dt . φ1 (1)2

Since u is increasing on R+ , we have u(ρt) ≥ u(t) for all ρ > 1 so that the left-hand side of this inequality is positive and increasing for ρ ≥ 1. When ρ → ∞, the righthand side tends to 0 and this leads to a contradiction which rules out the case (i). Hence, case (ii) occurs and the solution u of (4.27)-(4.28) with α > 0 cannot be continued to the entire interval [0, ∞). This completes the proof of Theorem 4.19 also in the case n ≥ 5.  As a by-product, the same proof of Theorem 4.19 enables us to obtain for all r > 0, then any global solution Theorem 4.22. If n ≥ 3 and h(r) ≥ − n−1 r u of (4.27)–(4.28) is necessarily strictly negative. Indeed, Theorem 4.19 excludes the existence of positive solutions u satisfying u0 (0) > 0. If u0 (0) < 0, then the solution of (4.27) is initially negative and, if it becomes positive, one can argue as before in order to show finite space blow up.

100

Chapter 4. Curvature Conditions

Concerning non-existence of negative solutions, a weaker result holds. First of all, we put together the three static terms 2(n − 1) n−1 u(r)2 + u(r) (4.39) 2 r r√ √    n−1+ n−1 n−1− n−1 = (n − 2) u(r) + u(r) + u(r) . (n − 2)r (n − 2)r

(n − 2)u(r)3 +

This shows that the static term changes sign whenever the graph of u crosses one of the two hyperbolas: √ √ n−1+ n−1 n−1− n−1 h1 (r) = − , h2 (r) = − . (n − 2)r (n − 2)r Then rewrite (4.27) as √   0 n−1+ n−1 n−1 0 r u (r) = (n − 2) u(r) + × rn−1 (n − 2)r √     n−1− n−1 u(r) × u(r) + u(r) − (n − 4)u(r)u0 (r) + u0 (r) + h(r) . (n − 2)r r 1 

If we assume that h(r) ≥ 0 ∀r ≥ 0 ,

n ≥ 4,

(4.40)

and that ∃R > 0 such that u(R) = h1 (R) ,

u0 (R) ≤ 0 ,

(4.41)

then the above equation tells us that the function r 7→ rn−1 u0 (r) is decreasing for r > R. In particular, u0 (r) < 0 and u(r) < h1 (r) for all r > R. Finally, this yields the existence of a γ > 0 such that 1  n−1 0 0 r u (r) ≤ γu(r)3 ∀r > R . rn−1 By arguing as in the proof of Theorem 4.19 (see Lemma 4.20 below) one concludes that ∃R > R such that lim u(r) = −∞ . r→R

Summarizing, we have Proposition 4.23. If (4.40) holds, then the problem (4.27)–(4.28) admits no global solution that satisfies (4.41). Existence results We start with a simple, but interesting example. If  αr  (n − 2)αr2 + n + 2 , h(r) = − 2

4.5. The class Yf : a possible generalization of the Yamabe problem

101

then u(r) = αr is a solution of the problem (4.27)–(4.28). We emphasize that, in any case, the solution u is global and unbounded. Moreover, if α > 0 then h(r) < 0 for all r and the solution of (4.27)–(4.28) is positive, while if α < 0 then h changes sign and the solution is negative. Theorem 4.19 suggests that (4.27) is more likely to have negative solutions whenever h itself is negative. We prove that this is indeed the case, at least in dimension n = 4. Theorem 4.24. In dimension n = 4, if h(r) ≤ 0 for all r > 0, then (4.27) admits infinitely many negative global solutions. More precisely, for any α < 0 the solution of (4.27)–(4.28) is global and it satisfies − 2r < u(r) < 0 for all r > 0. Proof. For the sake of convenience, we introduce the functions Z y (y + 2)2 y 2 h(y) = (y + 2)(y + 1)y and H(y) = h(ξ)dξ = . 4 0 Let v(r) = ru(r). Then v satisfies the equation  
n−3 n−4 n−2 00 v (r) + + v(r) − h(r) v 0 (r) = h v(r) . 2 r r r

(4.42)

If u is a solution of problem (4.27)–(4.28) with α < 0, then u(r) and u0 (r) are strictly negative in a right neighborhood of r = 0. By the definition of v, also v(r) is strictly negative in a right neighborhood of r = 0. We claim that −2 < v(r) < 0 for all r > 0. If not, let R > 0 be the first time where either v(R) = 0,

or v(R) = −2 .

(4.43)

Multiplying (4.42) by r2 v 0 (r) and integrating over [0, R], we obtain Z R    r2 v 00 (r)v 0 (r) + n − 3 + (n − 4)v(r) − rh(r) rv 0 (r)2 dr = 0, 0

because H(0) = H(−2) = 0. An integration by parts then yields Z R   R2 v 0 (R)2 n − 4 + (n − 4)v(r) − rh(r) rv 0 (r)2 dr + = 0. 2 0 If n = 4 and h(r) ≤ 0 we get a contradiction which shows that R does not exist and therefore −2 < v(r) < 0 for all r > 0. This proves the claim. Hence, by (4.42), also v 0 and v 00 remain bounded and the solutions exists. This concludes the proof of Theorem 4.24.  Remarks and open problems Here we discuss some open problems related to conformal f -Yamabe metrics and to solutions of equation (4.25).

102

Chapter 4. Curvature Conditions

1. In Theorem 4.19 we stated a partial non-existence results for radial solutions in the Euclidean space, while Theorem 4.24 provides a general existence result. It would be interesting to prove a sharp condition on the potential function f (or on its derivative) ensuring existence of global solutions to (4.27). 2. It is well known (see [80]) that global positive solutions the Yamabe equation n+2

−∆u = u n−2

on Rn

are necessarily radial (and thus are classified). We could ask the same question for (general) solutions to (4.26). For a given f ∈ C ∞ (Rn ), are there any nonradial solutions w? If f is radial, are all solutions to (4.26) radial? 3. In this section we studied conformal f -Yamabe metrics for (Rn , gRn ). What about other rotationally symmetric spaces? In particular, what we can say for the hyperbolic space (Hn , gHn ) or the round sphere (Sn , gSn )? 4. In the existence result (Theorem 4.24) the dimension n = 4 seems to be peculiar, at least from the analytic point of view. Is there a geometric interpretation of this fact?

4.6

Non-gradient canonical metrics

We provide here the complete generalization of the framework constructed in the previous sections to the non-gradient setting. Again, the starting objects of our analysis are the Ricci solitons, namely Riemannian manifolds (M, g) for which there exists a vector field X ∈ X(M ) such that 1 RicX := Ric + LX g = λg 2 for some constant λ ∈ R, where LX g denotes the Lie derivative of the metric in the direction X. In this we case we say that (M, g, X) ∈ EX . In this section we use the following notation: EX := RicX − R2X g, RX := R + div(X), and AX stands for the antisymmetric part of the ∇X, i.e., in local coordinates, AX ij = Xij − Xji ,
so that ∇X = 12 AX + LX g . If X = ∇f for some smooth potential function f , then the soliton is a gradient Ricci soliton (belongs to the class Ef ); note that, in this case, AX = 0 and 12 LX g = ∇2 f . It follows from the work of Perelman [131] (see [75] for a direct proof) that every compact Ricci soliton is actually a gradient Ricci soliton. In particular, it is well known that, if λ ≤ 0, then (M, g, X) ∈ E. Moreover, Naber [125] has shown that any shrinking (λ > 0) Ricci soliton with bounded curvature has a gradient soliton structure. On the other hand, steady (λ = 0) and expanding (λ < 0) Ricci solitons which do not support a gradient structure were found in [107, 113, 8, 7].

4.6. Non-gradient canonical metrics

103

To define the non-gradient counterparts of the f -canonical metrics that we have introduced in Definition 4.2, we note that the classes HCf and Yf were defined by requiring that the divergence of the “weighted” tensors e−f Riem and e−f Ric vanishes. Fortunately, we have shown in Section 4.2 that these structures can be characterized using the tensor Ricf : this allows us to give the following Definition 4.25. Let (M, g) be a n-dimensional (n ≥ 3) Riemannian manifold with metric g. We say that the triple (M, g, X) belongs to the class • SFX (X-space forms) if there exist X ∈ X(M ) and λ ∈ R such that   1 1 div(X) λ RiemX := Riem + LX g − g ?g = g?g; n−2 2 2(n − 1) 2(n − 1) • LSEX (X-locally symmetric Einstein metrics) if there exist X ∈ X(M ) and λ ∈ R such that
∇ RiemX = 0 and RicX = λg ; • EX (Ricci solitons) if there exist X ∈ X(M ) and λ ∈ R such that RicX = λ g ; • HCX (X-harmonic curvature metrics) if there exist X ∈ X(M ) such that div [EX ? g] = 0 . • YX (X-Yamabe metrics) if there exist X ∈ X(M ) such that ∇R = 2 Ric(X, ·) + div(AX ), where div(AX )i = AX ij,j = Xij,j − Xji,j . Moreover, we say that (M, g, f ) belongs to • LSX (X-locally symmetric metrics) if there exist X ∈ X(M ) such that
∇ RiemX = 0 ; • PRX (metrics with parallel X-Ricci tensor) if there exist X ∈ X(M ) such that
∇ RicX = 0 . Note that, when X = ∇f , we recover the corresponding classes in (4.2); in this latter case, we say that the structure is gradient. In particular, we have

104

Chapter 4. Curvature Conditions

⊂ ⊂

∩

⊂

SFf ∩ SFX

⊂

PRf ∪ ⊂ Ef ∩ ⊂ EX ∩ ⊂ PRX

⊂

⊂ ⊂

HCf ∩ HCX

Yf ∩ ⊂ YX ⊂

∪

LSf ∪ LSEf ∩ LSEX ∩ LSX

4.6.1 The class SFX Using the constancy of RX = R + div(X), which follows by taking the trace twice in the defining equation, we have (M, g, f ) ∈ SFX

⇐⇒

RiemX

λ = g?g 2(n − 1)

⇐⇒

( W = 0, RicX = λg.

Note that SFX ⊂ EX ; moreover, in dimension n ≥ 4 every X-space form is a locally conformally flat Ricci soliton. In particular, using the results in [48], the analogue of Proposition 4.6 holds.

4.6.2 The classes LSX and LSEX One has ∇ RiemX = ∇W +


1 ∇ AX ? g , n−2

RX where AX := RicX − 2(n−1) g. Moreover, ∇AX = 0 implies the constancy of RX , and is thus equivalent to ∇ RicX = 0. By orthogonality,

(M, g, f ) ∈ LSX

⇐⇒

∇ RiemX = 0

⇐⇒

( ∇W = 0, ∇ RicX = 0,

and, obiouvsly, (M, g, f ) ∈ LSEX

⇐⇒

( ∇ RiemX = 0, RicX = λg

⇐⇒

( ∇W = 0, RicX = λg.

Even in this more general situation, the analogue of Proposition 4.7 holds. Note that, for the LSEX , one has to use general results on homothetic vector fields contained, for instance, in [144].

4.6. Non-gradient canonical metrics

105

4.6.3 The class HCX By definition, (M, g, X) ∈ HCX

⇐⇒

div [EX ? g] = 0 ,

where EX = RicX − R2X g and RX = R + div(X). We claim that (M, g, X) ∈ HCX ⇐⇒ RicX is a Codazzi tensor    ( X Wtijk,t = n−3 X W − D t tijk ijk , n−2 ⇐⇒ X ∇R = 2 Ric(X, ·) + div(A ), where X Dijk =

1 1 (Xk Rij − Xj Rik ) + (Xt Rtk δij − Xt Rtj δik ) n−2 (n − 1)(n − 2) R − (Xk δij − Xj δik ) (n − 1)(n − 2) 1 1 + (Xkji − Xjki ) + [(Xtkt − Xktt )δij − (Xtjt − Xjtt )δik ]. 2 2(n − 1)

This definition follows from a previous work of the authors [58], where we derived the so-called integrability conditions for non-gradient Ricci solitons. Assume div [EX ? g] = 0. Equation (4.9) shows that (EX )tj,t δik − (EX )tk,t δij = (EX )ij,k − (EX )ik,j .

Taking the trace yields the equivalence 1 ∇RX ⇐⇒ 2 A simple computation now shows that div(EX ) =

div [EX ? g] = 0

⇐⇒

div(RicX ) = ∇RX .

(RicX )ij,k − (RicX )ik,j = 0 ,

i.e., RicX is a Codazzi tensor. We prove now the second equivalence. Assume that RicX is a Codazzi tensor. Then, by definition, we have

1 1 (RicX )ij,k = (RicX )ik,j ⇐⇒ Rij,k + (Xijk + Xjik = Rik,j + (Xikj + Xkij . 2 2 (4.44) In particular, taking the trace with respect to i, j, we deduce that
Rk = Xktt + Xtkt − 2Xttk
= Xktt + Xtkt − 2Xtkt + 2Xt Rtk
= 2Xt Rtk + Xktt − Xtkt = 2Xt Rtk + AX kt,t ,

106

Chapter 4. Curvature Conditions

i.e., ∇R = 2 Ric(X, ·) + div(AX ) .

(4.45)

Moreover, going back to (4.44), one has Rij,k − Rik,j =


1
1 Xikj − Xijk + Xkij − Xjik . 2 2

Now using again the commutation rule Xijk − Xikj = Xt Rtijk and the Bianchi identities we see that
1 Rij,k − Rik,j = Cijk + Rk δij − Rj δik , 2(n − 1)
1 1 Xikj − Xijk = Xt Rtikj 2 2 and


1
1 Xkij − Xjik = Xkji − Xjki + Xt Rtikj . 2 2

Therefore, Cijk +


1 1 Rk δij − Rj δik = Xt Rtikj + AX . 2(n − 1) 2 kj,i

Inserting here the decomposition of the curvature tensor and using (4.45), we obtain X Cijk + Xt Wtikj = Dijk , since DX can be written using AX as follows X Dijk =

1 1 (Xk Rij − Xj Rik ) + (Xt Rtk δij − Xt Rtj δik ) n−2 (n − 1)(n − 2) R − (Xk δij − Xj δik ) (n − 1)(n − 2)
1 1 X + AX AX kj,i − kt,t δij − Ajt,t δik . 2 2(n − 1)

Equation (1.82) immediately implies that   n − 3 W X Xt Wtijk − Dijk , tijk,t = n−2 RicX is a Codazzi tensor ⇐⇒ ∇R = 2 Ric(X, ·) + div(AX ). From the equivalence (M, g, X) ∈ HCX

⇐⇒

RicX is a Codazzi tensor

and the fact that compact Ricci solitons are gradient, it follows that all the results in Section 4.4 concerning compact HCf metrics can be extended to the non-gradient setting, defining the class HCλX in the natural way.

4.7. Final remarks and open problems

107

4.6.4 The class YX By analogy with the gradient case, a simple computation shows that (M, g, f ) ∈ YX

⇐⇒

div (RicX −RX g) = 0 .

Moreover, we can prove the following obstruction result which extends Corollary 4.17 to the non-gradient setting. Proposition 4.26. Let (M, g0 ) be a compact Riemannian manifold and let X ∈ X(M ) be a non-Killing conformal vector field on (M, g0 ). Then, there are no conformal metrics g ∈ [g0 ] such that (M, g, X) ∈ YX . Proof. Let g ∈ [g0 ]. By the conformal invariance of equation (4.22), X is also a conformal vector field for (M, g). Assume that (M, g, X) ∈ YX , i.e., ∇R = 2 Ric(X, ·) + div(AX ) where div(AX )i = AX ij,j = Xij,j − Xji,j . By the Kazdan–Warner identity (4.23), Z

1 Ric(X, X) dV + 2 M

Z

hdiv(AX ), Xi dV = 0 .

(4.46)

M

Integrating the Bochner formula in Lemma 4.4 and using the conformal vector field equation, we obtain Z Z Z n−2 Ric(X, X) dV = |∇X|2 dV + | div(X)|2 dV . n M M M On the other hand Z Z 1 1 X hdiv(A ), Xi dV = − (Xij − Xji )Xij dV 2 M 2 M Z Z 1 1 =− |∇X|2 dV + Xij Xji dV 2 M 2 M Z Z 1 =− |∇X|2 dV + | div(X)|2 dV . n M M Using these two last expressions in (4.46), we obtain div(X) = 0. Thus X must be a Killing vector field, which contradicts the assumption. 

4.7

Final remarks and open problems

To conclude this chapter, we present a short list of comments and open problems, which could be the subject of further investigations.

108

Chapter 4. Curvature Conditions

1. In Section 4.5 we considered problem (B), which is a possible generalization of the Yamabe problem related to the class Yf . In particular, we proved an obstruction result in Corollary 4.17 and an existence result (in the Euclidean setting) in Theorem 4.24. Are there other obstructions to the latter, different from Corollary 4.17? As far as problem (A) is concerned, are there any obstructions at all? Clearly, all the previous questions could also be asked for the class YX . 2. In Sections 4.4 and 4.5 we constructed some examples of HCf and Yf metrics, respectively, using warped products. Can we construct other examples, apart from gradient Ricci solitons Ef , possibly “non-warped”? Can we construct examples of spaces belonging to HCλf ? Moreover, what can we say in the nongradient cases HCX and YX ? Since compact Ricci solitons EX are gradient, it would be interesting to construct a compact example of HCX metric, with X “genuinely” non-gradient (that is, not of the form X = ∇f + Y , where Y is a Killing vector field). Compare also with 5. below. 3. In the positive (sectional) curvature case we have seen, in Proposition 4.14, that the class HCλf coincides with the one of gradient Ricci solitons Ef . Are there other characterizations? What is the role of the so-called Hamilton 2 identity R + |∇f | − 2λf = C, C ∈ R, which is valid for gradient Ricci solitons? 4. Inspired by the classification results for Ricci solitons, we could study, for instance, the following problems: a. If M is compact and (M, g, f ) ∈ HCλf , with λ ≤ 0, is it true that (M, g) ∈ E? b. If M is three-dimensional, compact and (M, g, f ) ∈ HCλf , with λ > 0, is it true that (M, g) is isometric, up to quotients, to the round sphere S3 ? More generally, can we classify complete three-dimensional manifolds that belong to the class HCλf with λ > 0? c. If (M, g) is compact and locally conformally flat, and (M, g, f ) ∈ HCλf , is it true that (M, g) ∈ SF? Note that, since in Section 4.4 we constructed locally conformally flat examples of HCf metrics, for the latter class this result is clearly false. d. Gradient Ricci solitons can be classified by imposing conditions on the Weyl tensor that are weaker than local conformal flatness (for a detailed description, see Chapter 7, sections 7.4 and 7.5). Can we prove similar results for HCf metrics? 5. As we saw in Section 4.6, compact Ricci solitons EX are gradient. What can we say about the classes HCX , HCλX and YX in the compact case? Are there natural geometric conditions ensuring the “gradientness” for these classes (even in the non-compact setting)?

Chapter 5

Canonical Metrics II: Critical Metrics of Riemannian Functionals In this chapter we introduce a second possible way to study canonical metrics on Riemannian manifolds, namely the one related to “Critical Metrics of Riemannian functionals” (CM, for short). First, we recall the classical Einstein-Hilbert functional and we briefly discuss the celebrated Yamabe problem; then, following [42], we introduce a basis for quadratic curvature functionals, deriving the corresponding Euler-Lagrange equations and we prove some rigidity results for critical metrics.

5.1

The Einstein–Hilbert functional

Given a closed smooth manifold M of dimension n greater than two, it is natural to define canonical metrics on M as critical points of certain Riemmannian functionals defined on the space of metrics. The most famous one is the Einstein-Hilbert action Z n−2 S(g) := Volg (M )− n Rg dVg , M

where Volg (M ) and Rg denote the volume of M and the scalar curvature of g, respectively. All stationary points of S are Einstein metrics, that is (M, g) ∈ E, using the notation of Chapter 4. This follows easily using the variation formulas for the volume form and for the scalar curvature in Chapter 2: more precisely, for © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 G. Catino, P. Mastrolia, A Perspective on Canonical Riemannian Metrics, Progress in Mathematics 336, https://doi.org/10.1007/978-3-030-57185-6_5

109

110

Chapter 5. Critical Metrics of Riemannian Functionals

any symmetric (0, 2)-tensor h, one has 1 HdVg , 2Z 1 (δ Volg (M ))[h] = H dVg , 2 M (δRg )[h] = −∆H + hij,ij − Rij hij , (δdVg )[h] =

and we obtain Z d n−2 − n−2 −1 n S(g + th) =− Volg (M ) (δ Volg (M ))[h] Rg dVg dt n t=0 M Z n−2 + Volg (M )− n [(δRg )[h] dVg + Rg (δdVg )[h]] M Z Z n−2 n−2 =− Volg (M )− n −1 H dVg Rg dVg 2n M M  Z  1 − n−2 n + Volg (M ) −∆H + hij,ij − Rij hij + Rg H dVg . 2 M By the divergence theorem,    Z  n−2 d Rg n−2 S(g + th) = − Volg (M )− n Ricg − − Rg g, h dVg , dt 2 2n t=0 M (5.1) R where Rg = Vol−1 g (M ) M Rg dVg . In particular, this expression vanishes for all h if and only if Rg Rg = Rg and Ric = g. n It is customary to write equation (5.1) in the form Z d S(g + th) = h∇S, hi dVg , dt t=0 M where ∇S, which is a (0, 2)-symmetric tensor, denotes the so-called gradient of the functional S (see also [13]). We will use this notation throughout this chapter. One can obtain the same result by considering instead the unnormalized functional Z g 7−→ Rg dVg M

and restricting it to the submanifold M1 (M n ) of smooth Riemannian metrics of volume one on M n . As we will note later, the existence of Einstein metrics as critical points of S is not guaranteed (in dimension four) due to topological restrictions; on

5.1. The Einstein–Hilbert functional

111

the other hand, a constrained version of the problem always admits a solution. More precisely, Yamabe, Aubin, Trudinger, and Schoen (see [111]) showed that the Yamabe invariant Y(M, [g]) := inf S(e g) g e∈[g]

is always attained in the conformal class  [g] = ge : ge = u2 g, for some u ∈ C ∞ (M ) . The metrics realizing Y(M, [g]) are called Yamabe metrics; one can show that Yamabe metrics necessarily have constant scalar curvature. Conversely, it is not difficult to prove that metrics with constant non-positive scalar curvature are Yamabe metrics (see [111]); an important result of Obata [128] guarantees that Einstein metrics are always Yamabe metrics, even when the scalar curvature is positive. It is easy to see that the Yamabe invariant can be rewritten also as Z n−2 e dVge Y(M, [g]) = inf Volge(M )− 2 R g e∈[g] M Z Z n−2 |∇u|2 dVg + R u2 dVg 4(n − 1) 4(n − 1) M M = inf ; Z (n−2)/n n − 2 u∈W 1,2 (M ) 2n/(n−2) |u| dVg M

this can be obtained using the formulas for the conformal change of the scalar 4 curvature and of the volume (with ge = u n−2 g: see (2.70), where the change is by an exponential factor). As a consequence of the solution of the Yamabe problem, Y(M, [g]) is positive (respectively zero or negative) if and only if there exists a conformal metric in [g] with everywhere positive (respectively, zero or negative) scalar curvature. Moreover, if Y(M, [g]) > 0, then for every u ∈ W 1,2 (M ) the following Yamabe-Sobolev inequality holds, Z  n−2 Z Z n 2n n−2 n−2 2 n−2 Y(M, [g]) |u| dVg ≤ |∇u| dVg + R u2 dVg . 4(n − 1) 4(n − 1) M M M (5.2) These facts will be important in the subsequent chapters. Going back to the obstructions to the existence of Einstein metrics on closed four manifolds, by combining Chern-Gauss-Bonnet formula (3.23) and Hirzebruch signature Theorem (3.24), one has  Z  1 1 2 1 ˚ 2 ± 2 2χ(M ) ± 3τ (M ) = 2|W | + R − | Ric| dV. 16π 2 M 24 2 ˚ = 0, and we have the following In particular, if (M 4 , g) is Einstein, then |Ric| celebrated inequality of Thorpe [145] and Hitchin [94]: χ(M ) ≥

3 |τ (M )| . 2

(5.3)

112

Chapter 5. Critical Metrics of Riemannian Functionals

There are simply connected four-manifolds which do not admit Einstein metrics. For instance, the manifold CP2 ] kCP2 given by the connected sum of the complex projective space CP2 with standard orientation and k-copies of CP2 (the same smooth manifold, but equipped with the opposite orientation), does not admit an Einstein metric if k > 9. More refined obstructions to the existence of Einstein metrics on four-manifolds have been obtained by various authors, especially by C. LeBrun (see the very nice survey [108] and references therein).

5.2

Quadratic curvature functionals

5.2.1 A basis We have seen that Einstein metrics arises as critical points of the Einstein-Hilbert action; thus, it is natural to study canonical metrics which arise as solutions of the Euler-Lagrange equations for more general curvature functionals. In [10], Berger initiated the study of Riemannian functionals that are quadratic in the curvature (see [11, Chapter 4] and [140] for surveys). Let M1 (M n ) denote the space of equivalence classes of smooth Riemannian metrics of volume one on a manifold M n . A basis for the space of quadratic curvature functionals is given by Z W=

|W |2 dV,

Z r=

| Ric |2 dV,

Z S2 =

R2 dV

and, from the decomposition of the Riemann tensor (1.79), one has Z R=

| Riem |2 dV =

Z 

|W |2 +

 2 4 | Ric |2 − R2 dV . n−2 (n − 1)(n − 2)

The Chern-Gauss-Bonnet formula (3.23) implies that the Weyl functional W can be written as a linear combination of the other two (with the addition of a topological term). All such functionals, which also arise naturally as total actions in certain gravitational field theories in physics, have been studied in depth in the last years by many authors. As it will be clear in Section 5.3.5, in dimension greater than four, the Euler-Lagrange equation of the Weyl functional W has a different structure in the zeroth-order curvature terms. In particular, if n > 4, it is not true that Einstein metrics are always critical points for this functional on M1 (M n ). As already observed for the Einstein-Hilbert functional, instead of defining the functionals restricted on M1 (M n ), one could consider the renormalized functionals obtained by multiplying them by an appropriate power of the volume of M n . We will use both points of view throughout this chapter.

5.2. Quadratic curvature functionals

113

5.2.2 Remarks on two special cases The L2 -norm of the scalar curvature Consider the functional

Z

Rg2 dVg .

S2 (g) = M

Using the variational formulas for the scalar curvature in Chapter 2 (see equation (2.54)), it is easily verified that 1 (∇S2 )ij = 2∇2ij R − 2(∆R)gij − 2RRij + R2 gij . 2 Moreover, g is critical for S2 on M1 (M n ) if and only if (∇S2 ) = c g, for some Lagrange multiplier c ∈ R (see [11]). Taking the trace of this equation, we obtain −2(n − 1)∆R +

n−4 2 R = nc . 2

From these, we get that g is critical for S2 on M1 (M n ) if and only if R Ric −∇2 R = and ∆R =

1 2 1 R g − ∆R g , n n


n−4 R2 − R2 , 4(n − 1)

(5.4)

(5.5)

where R2 = S2 (g)/Volg (M n ). Obviously, if a metric is scalar flat or Einstein, then it is critical for S2 on M1 (M n ). We prove that also the converse is true, if we assume that the critical metric has non-negative scalar curvature. We notice that equation (5.5) implies that any four-dimensional critical metric of S2 on M1 (M 4 ) has constant scalar curvature and it is either scalar flat, or Einstein. Proposition 5.1. Any compact critical metric of S2 on M1 (M n ) with non-negative scalar curvature is either scalar flat, or Einstein. Proof. Contracting equation (5.4) with the Ricci tensor, one has 2 1 1 R Ric − R g = Rij ∇ij R − R ∆R . n n Integrating over M n , we get Z Z Z 2 1 1 ij R Ric − R g dVg = R ∇ij R dVg − R∆R dVg n n M M Z Z M 1 1 =− |∇R|2 + |∇R|2 dVg 2 M n M Z n−2 =− |∇R|2 dVg . 2n M Since R ≥ 0, this implies that either R ≡ 0, or the metric g is Einstein.



114

Chapter 5. Critical Metrics of Riemannian Functionals

Note that, in dimension n = 3, we get the same conclusion if we assume that the scalar curvature is strictly negative. In fact, multiplying (5.5) by (R2 − R2 ) and integrating by parts, we obtain Z Z 1 2 2 2 − (R − R ) = −2 R|∇R|2 ≥ 0 . 8 M M Thus R is a negative constant and, from (5.4), g is Einstein (i.e., g has constant negative sectional curvature, since n = 3). These results were proved in dimension n = 3 by Anderson in [2]. Question 5.2. Are there any non-trivial (non-scalar-flat or non-Einstein) critical metrics of S2 on M1 (M n )? Bach-flat metrics On a closed four-dimensional manifold (M, g) we consider the functional Z W(g) = |Wg |2g dVg . This functional has been studied a long time ago by specialists in the theory of relativity, in particular by R. Bach [6] in the early 1920’s in work on conformal relativity. As already observed in Chapter 3, the functional W(g) is conformally invariant, i.e., W(e g ) = W(g), ∀e g ∈ [g]. Moreover, by the Chern-Gauss-Bonnet formula (3.23), 2 W(g) = 32π 2 χ(M ) + r(g) − S2 (g) = 32π 2 χ(M ) + 2F− 13 (g) , 3 where Ft , t ∈ R, is a functional that will be defined in the next section. In particular, from (5.6) we obtain   (∇W)ij = 2 ∇F− 13 ij   2 1 1 4 = −2∆Rij + ∇2ij R + (∆R)gij + | Ric |2 − R2 gij − 4Rikjl Rkl + RRij 3 3 3 3 = −4Bij , where in the last equality we used Lemma 1.5. Equivalently, g is critical for W if and only if g is Bach-flat (see also Chapter 1). We recall that g is Bach-flat if and only if 1 Bij = Wikjl,lk + Rkl Wikjl = 0 . 2 In particular, every locally conformally flat metric (W = 0) is critical, but the following easy observation provides many more examples:

5.3. Some rigidity results for quadratic functionals

115

Lemma 5.3. On a closed four-dimensional manifold, the following two classes of metrics are critical for W: (1) metrics that are conformal to an Einstein metric; (2) half-conformally flat metrics, i.e., such that W + or W − is zero (if M is orientable, see Chapter 3, Section 3.2). Proof. (1) Clearly, if g is Einstein, then Rkl Wikjl = λgkl Wikjl = 0. Moreover, it follows from (1.82) that g has harmonic Weyl tensor, i.e., Wikjl,l = 0, and thus g is Bach-flat. In particular, by the conformal invariance of W (and of the tensor B), the result follows. (2) It can be proved by direct computation. However, we can observe that, from Hirzebruch signature formula (3.24), we have Z Z
W= |W + |2 + |W − |2 dV = 2 |W ± |2 dV ∓ 48π 2 τ (M ) ≥ ∓48π 2 τ (M ), M

M

with equality if and only if g is half-conformally flat. In particular, half conformally flat metrics are global minima of the functional W. 

5.3

Some rigidity results for quadratic functionals

In this section we follow the presentation in [42]. We will consider the curvature functionals Z Z Ft = | Ric |2 dV + t R2 dV , defined for some constant t ∈ R (with t = −∞ formally corresponding to the functional S). Since in dimensions greater than four Ft is not scale-invariant, it is natural to restrict the functional to M1 (M n ). Equivalently, one can consider a modified functional properly normalized with the volume of the manifold (see [89]). It was already observed in [11] that every Einstein metric is critical for Ft on M1 (M n ), for every t ∈ R. The first basic question is whether all critical metrics are necessarily Einstein. Of course, in general this is false. For instance, in dimension four, every Bach-flat metric is critical for F−1/3 , and every Weyl and scalar flat metric is critical for F−1/4 on M1 (M 4 ) (see [11, Chapter 4]). Moreover, Lamontagne in [106] constructed a homogeneous non-Einstein critical metric for R = 4F−1/4 on M1 (S3 ). From this point of view, it is natural to ask under what conditions a critical metric for Ft must be Einstein. Typically, one assumes some curvature conditions (of pointwise or integral type, positivity or negativity of the curvature, etc.) on the critical metric in order to prove rigidity properties. For instance, for S on M1 (M 3 ) in [2, Proposition 1.1] the author assumed that the scalar curvature of the critical metric has definite sign (actually, this holds in every dimension, as we have proved in Proposition 5.1); for F−1/3 on M1 (M 4 ) (variationally equivalent to R) in [105] it is proved that every critical metric with

116

Chapter 5. Critical Metrics of Riemannian Functionals

nonpositive sectional curvature is Einstein; for F−1/3 on M1 (M 3 ) in [143] the author assumed a pointwise pinching condition on the Ricci curvature; finally, for F−3/8 on M1 (M 3 ) (variationally equivalent to the σ2 -functional) in [88] the authors proved that every critical metric must be Einstein (hence a space form), just assuming an integral condition, namely F−3/8 ≤ 0 (this result was extended in dimension greater than four in the locally conformally flat case in [95]). As Corollary 5.13 and Corollary 5.14 shall make clear, for some specific values of the parameter t, critical metrics for Ft inherit additional properties from the EulerLagrange equation, which implies more constraints on the variational solution. For instance, every critical metric has constant scalar curvature if t 6= −1/3 and n = 4, and has constant σ2 -curvature if t = −n/4(n − 1) and n 6= 4. The results just quoted [105, 88, 95] belong to these cases. We prove some rigidity results on critical metrics for Ft on M1 (M n ). First, we characterize critical metrics with non-negative sectional curvature. Theorem 5.4. Let M n be a closed manifold of dimension n ≥ 3. Suppose g is a critical metric for Ft on M1 (M n ) for some t < −1/2. If g has non-negative sectional curvature, then it is an Einstein metric. In the case t = −1/2, we can show the following result. Theorem 5.5. Let M n be a closed manifold of dimension n ≥ 3. If g is a critical metric for F−1/2 on M1 (M n ) with non-negative sectional curvature, then either g is Einstein, or the following possibilities occur:
f3 , ge) is isometric to S2 ×R, a gS2 +b gR , (i) If n = 3, then the universal cover (M for some positive constant a, b > 0. f4 , ge) is isometric to either S2 × (ii) If n = 4, then
the universal cover (M
2 2 2 S , a gS2 + b gS2 , or S × R , a gS2 + b gR2 , for some constants a, b > 0. 2 n−2 fn (iii) If n > 4, , a g S2 +
then the universal cover (M , ge) is isometric to S × R b gRn−2 , for some constants a, b > 0.

Remark 5.6. We note that the condition t < −1/2 in Theorem 5.4 is sharp. In fact, for every t > −1/2, following [106], one can construct non-Einstein critical metrics gt for Ft on M1 (S3 ) (see [89, Section 7]). It turns out that these metrics have non-negative sectional curvature if −1/2 < t ≤ 3/4. On the other hand, a condition on the sectional curvature is necessary too. In fact, recently Gursky and Viaclovsky in [90] constructed critical metrics for Ft on M1 (M 4 ) for t “close” to a given value that depends on the topology of the Einstein building blocks. In particular, they found solutions for t close to −1/2 and in some cases for t < −1/2 (for precise estimates on the critical values, see [146]). As it is clear from the construction, all these metrics have sectional curvature with changing sign. Concerning critical metrics with non-positive sectional curvature, we can extend Lamontagne result [105] in dimension four, proving the following

5.3. Some rigidity results for quadratic functionals

117

Theorem 5.7. Let M 4 be a closed manifold of dimension four. Suppose g is a critical metric for Ft on M1 (M 4 ) for some t ≥ −1/4. If g has nonpositive sectional curvature, then it is an Einstein metric. In Section 5.3.4 we provide a rigidity result for critical metrics for Ft on M1 (M 3 ) for t > −1/2. As observed earlier, one has to assume a stronger condition than non-negative sectional curvature to exclude the non-Einstein examples. Moreover, the estimates used in the proof of Theorem 5.4 are not sufficient in this range of t, due to the presence of bad terms with the wrong sign. We were able to overcome these difficulties by “weighting” the Euler-Lagrange equation, and trying to compensate these quantities. Our result reads as follows: Theorem 5.8. Let M 3 be a closed manifold of dimension three. Suppose g is a critical metric for Ft on M1 (M 3 ) for some −1/3 ≤ t < −1/6. If g has nonnegative scalar curvature, then it has constant positive sectional curvature if |E|2
4, satisfies the Yamabe-type equation 4


n + 4(n − 1)t 1 + 2(n − 1)t σ2 (Ag ) − |Wg |2 = const . 16(n − 2) We hope that this property could help in proving some new variational characterizations of space forms as critical metrics of these functionals. To conclude, we mention that finding conditions that guarantee rigidity of critical metrics for quadratic curvature functionals is also of major interest in the non-compact setting. For instance, Anderson in [3] proved that every complete three-dimensional critical metric for the Ricci functional ρ with non-negative scalar curvature is flat, whereas in [41] we obtained a characterization of complete critical metrics for S with non-negative scalar curvature in every dimension. Recently, in [54], we showed that, in dimension three, flat metrics are the only complete metrics with non-negative scalar curvature that are critical for the σ2 -curvature functional F−3/8 .

5.3.1 The Euler-Lagrange equations In this section we will compute the Euler-Lagrange equation satisfied by critical metrics for the functional Ft on M1 (M n ) (see also [89]). Using formulas (2.53) and (2.54) in Chapter 2 and integration by parts, it is not difficult to show that the gradients of the functionals r and S2 are given by (see also [11, Proposition 4.66]) 1 1 (∇r)ij = −∆Rij − 2Rikjl Rkl + ∇2ij R − (∆R)gij + | Ric |2 gij 2 2 and

1 (∇S2 )ij = 2∇2ij R − 2(∆R)gij − 2RRij + R2 gij , 2 respectively. Hence, the gradient of Ft reads 1 + 4t (∇Ft )ij = −∆Rij + (1 + 2t)∇2ij R − (∆R)gij 2   1 + | Ric |2 + tR2 gij − 2Rikjl Rkl − 2tRRij . 2

(5.6)

5.3. Some rigidity results for quadratic functionals

119

Moreover, g is critical for Ft on M1 (M n ) if and only if (∇Ft ) = c g, for some Lagrange multiplier c ∈ R (see [11]). Taking the trace in this equation, we obtain  n + 4(n − 1)t n − 4 | Ric |2 + tR2 − ∆R = nc . 2 2 This shows that g is critical for Ft on M1 (M n ) if and only if −∆Rij + (1 + 2t)∇2ij R −

and

 2t 2 (∆R)gij + | Ric |2 + tR2 gij n n − 2Rikjl Rkl − 2tRRij = 0 ,

(5.7)

 
n + 4(n − 1)t ∆R = (n − 4) | Ric |2 + tR2 − λ ,

˚ the traceless Ricci tensor, we obtain where λ = Ft (g). Denoting, as usual, with Ric the Euler-Lagrange equation of critical metrics for Ft on M1 (M n ). Proposition 5.11. Let M n be a closed manifold of dimension n ≥ 3. A metric g is critical for Ft on M1 (M n ) if and only if it satisfies the following equations, 1 + 2t ˚kl (∆R)gij − 2Rikjl R n 2 + 2nt ˚ 2 ˚ 2 − RRij + |Ric| gij , n n

(5.8)

   n + 4(n − 1)t ∆R = (n − 4) | Ric |2 + tR2 − λ ,

(5.9)

˚ij = (1 + 2t)∇2ij R − ∆R

and



where λ = Ft (g). In particular, it follows that Einstein metrics are critical (see [11, Corollary 4.67]). Corollary 5.12. Any Einstein metric is critical for Ft on M1 (M n ). From equation (5.9), if n = 4 and t 6= −1/3, we immediately derive the following result. Corollary 5.13. Let M 4 be a closed manifold of dimension four. If g is a critical metric for Ft on M1 (M 4 ) for some t 6= −1/3, then g has constant scalar curvature. Note that, in dimension four, the Chern-Gauss-Bonnet formula (see equation (3.23)) implies that F−1/3 is proportional (plus a constant term) to the Weyl functional W. Hence, critical metrics are Bach-flat and, in general, do not have constant scalar curvature. On the other hand, if t = −n/4(n − 1), then | Ric |2 −

n R2 = −2σ2 (A) , 4(n − 1)

120

Chapter 5. Critical Metrics of Riemannian Functionals

where σ2 (A) is the second elementary symmetric function of the Schouten tensor. Hence, when n 6= 4 and t = −n/4(n − 1), we have Z F−n/4(n−1) = −2 σ2 (A) dV and equation (5.9) implies Corollary 5.14. Let M n be a closed manifold of dimension n 6= 4. If g is a critical metric for F−n/4(n−1) on M1 (M n ), then g has constant σ2 -curvature. ˚ we obtain the following WeitzenNow, contracting equation (5.8) with Ric, b¨ ock-type formula (see Chapter 6 for a detailed discussion). Proposition 5.15. Let M n be a closed manifold of dimension n. If g is a critical metric for Ft on M1 (M n ), then 1 ˚ 2 = |∇Ric| ˚ 2 + (1 + 2t)R ˚ij ∇2 R − 2Rikjl R ˚ij R ˚kl − 2 + 2nt R|Ric| ˚ 2 . (5.10) ∆|Ric| ij 2 n As a consequence, we obtain Corollary 5.16. Let M n be a closed manifold of dimension n. If g is a critical metric for Ft on M1 (M n ), then Z Z   ˚ 2 − (n − 2)(1 + 2t) |∇R|2 dV = 2 ˚ij R ˚kl + 1 + nt R|Ric| ˚ 2 dV. |∇Ric| Rikjl R 2n n Proof. We simply integrate by parts equation (5.21) and use the trace of the second Bianchi identity ˚ij = ∇i Rij − 1 ∇j R = n − 2 ∇j R . ∇i R  n 2n

5.3.2 Proofs of Theorem 5.4 and Theorem 5.5 The first key observations are the following pointwise estimates (see [45]), which are satisfied by every metric with Sec ≥ εR for some ε ∈ R (recall that Sec(p) is the infimum of the sectional curvature at the point p): Proposition 5.17. Let (M n , g) be a Riemannian manifold of dimension n ≥ 3. If the sectional curvature satisfies Sec ≥ εR for some ε ∈ R, then the following two estimates hold: 2 ˚ 2+R ˚ij R ˚ik R ˚jk ˚ik R ˚jl ≤ 1 − n ε R|Ric| Rijkl R n

and 2 2 ˚ik R ˚jl ≤ n − 4n + 2 − n (n − 2)(n − 3)ε R|Ric| ˚ 2 − (n − 1)R ˚ij R ˚ik R ˚jk . Rijkl R 2n

5.3. Some rigidity results for quadratic functionals

121

On four-dimensional manifolds we have ˚ik R ˚jl ≤ 1 − 16ε R|Ric| ˚ 2+R ˚ij R ˚ik R ˚jk , Rijkl R 4 ˚ 2 − 3R ˚ij R ˚ik R ˚jk . ˚ik R ˚jl ≤ 1 − 16ε R|Ric| Rijkl R 4 ˚ and let λi be the correProof. Let {ei }, i = 1, . . . , n, be the eigenvectors of Ric sponding eigenvalues. Moreover, let σij be the sectional curvature defined by the two-plane spanned by ei and ej . Since the sectional curvature satisfies Sec ≥ εR, it is natural to introduce the tensor ε Riem = Riem − R g ∧ g. 2 In particular Ric = Ric −(n − 1)εR g,


R = 1 − n(n − 1)ε R,

and

σ ij = σij − εR ≥ 0 .

Moreover, if µk and µk are the eigenvalues with eigenvector ek of Ric and Ric, respectively, one has X X µk = σik and µk = σ ik . i6=k

i6=k

Denoting the components of Riem by Rijkl , we get ˚ik R ˚jl − Rij R ˚ik R ˚jk = Rijkl R

n X

λi λj σ ij −

i,j=1

=2

X

µk λ2k

k=1

λi λj σ ij −

i 4, then the eigenvalues of the Ricci tensor are equal to µ1 = µ2 = 12 R and µ3 = · · · = µn = 0, where R is a positive constant. Since g has nonnegative sectional curvature, the Ricci flat part has to be flat. Hence, the
fn , ge) is isometric to S2 × Rn−2 , a gS2 + b gRn−2 , for some universal cover (M constants a, b > 0. It is readily seen that all these examples are critical for F−1/2 on M1 (M n ), i.e., they satisfy equation (5.12). For instance, in dimension n > 4, the manifold
S2 × Rn−2 , a gS2 + b gRn−2 has Ricci curvature Ric = gS2 + 0n−2 , where 0n−2 is the zero (n − 2)-dimensional two-tensor, and scalar curvature R = a2 . Clearly, | Ric |2 = 12 R2 and the critical equation (5.12) is satisfied. The cases in dimension n ≤ 4 can be verified in much the same way. This concludes the proof of Theorem 5.5. 

5.3.3 Proof of Theorem 5.7 We begin with several useful estimates that hold for every n-dimensional Riemannian manifold. We have the following formula (see [88, Section 4] for the proof in the three-dimensional case). Proposition 5.20. Let (M n , g) be Then Z  2 ˚ 2 − (n − 2) |∇R|2 − |∇Ric| 4n(n − 1)

a Riemannian manifold of dimension n ≥ 3.

1 2 |C| dV 2 Z   ˚ij R ˚kl − R ˚ij R ˚ik R ˚jl − 1 R|Ric| ˚ 2 dV, = Rikjl R n

where C is the Cotton tensor. Proof. A simple computation shows that 2 1 2 ˚ 2 − (n − 2) |∇R|2 − ∇k R ˚ij ∇j R ˚ik . |C| = |∇Ric| 2 4n2 (n − 1)

(5.13)

Integrating by parts the last term, we get Z Z ˚ ˚ ˚ij ∇k ∇j R ˚ik dV ∇k Rij ∇j Rik dV = − R Z   ˚ij ∇j ∇k R ˚ik + Rkjil R ˚ij R ˚kl + R ˚ij R ˚ik R ˚jl + 1 R|Ric| ˚ 2 dV =− R n Z   n−2˚ ˚ij R ˚kl + R ˚ij R ˚ik R ˚jl + 1 R|Ric| ˚ 2 dV =− Rij ∇i ∇j R − Rikjl R 2n n Z   2 (n − 2) 2 ˚ij R ˚kl − R ˚ij R ˚ik R ˚jl − 1 R|Ric| ˚ 2 dV . = |∇R| + R R ikjl 4n2 n Using that identity and integrating equation (5.13) we get the desired result.



5.3. Some rigidity results for quadratic functionals

127

Proposition 5.21. Let (M n , g) be a Riemannian manifold of dimension n ≥ 3 with non-positive sectional curvature. Then, the following estimate holds: ˚ij R ˚kl + 1 R|Ric| ˚ 2+R ˚ij R ˚ik R ˚jl ≤ 0 . Rikjl R n ˚ = 0. Moreover, if n = 4, equality occurs if and only if |Ric| Proof. As in the proof of Proposition 5.17, we let {ei }, i = 1, . . . , n, be the eigen˚ and Ric and let λi and µi be the corresponding eigenvalues. Morevectors of Ric over, let σij be the sectional curvature defined in the direction of the two-plane spanned by ei and ej . We want to prove that the quantity ˚ij R ˚kl + Rikjl R

n n n X 1 1 X 2 X 3 ˚ 2+R ˚ij R ˚ik R ˚jl = R|Ric| λi λj σij + R λk + λk n n i,j=1 k=1

k=1

is non-positive if σij ≤ 0 for all i, j = 1, . . . , n. First of all, we note that n X

λi λj σij +

i,j=1

n n n n X X 1 X 2 X 3 R λk + λk = 2 λi λj σij + µk λ2k , n i 0 and the pinching assumption (5.14) holds. Hence, Ric This concludes the proof of Theorem 5.8. 

5.3.5 The Euler-Lagrange equation for Ft,s In this section we will compute the Euler-Lagrange equation satisfied by critical metrics for the functional Z Z Z Ft,s (g) = | Ric |2 dV + t R2 dV + s | Riem |2 dV, on M1 (M n ). As already observed in the introduction, this functional substantially differs from Ft only in dimension greater than four. To write the Euler-Lagrange equation for Ft,s , we follow the computations in Section 5.3.1. The gradients of the functionals r, S2 and R are given, using again the variations in Chapter 2 (see also [11, Proposition 4.66] and [11, Proposition 4.70]), by 1 1 (∇r)ij = −∆Rij + ∇2ij R − (∆R)gij − 2Rikjl Rkl + | Ric |2 gij , 2 2 1 (∇S2 )ij = 2∇2ij R − 2(∆R)gij − 2RRij + R2 gij , 2 and 1 (∇R)ij = −4∆Rij + 2∇2ij R − 2Rikpq Rjkpq + | Riem |2 gij − 4Rikjl Rkl + 4Rik Rjk . 2 Hence, the gradient of Ft,s reads (∇Ft,s )ij = −(1 + 4s)∆Rij + (1 + 2t + 2s)∇2ij R 1 + 4t − (∆R)gij − 2Rikjl Rkl − 2tRRij 2   1 + | Ric |2 + tR2 + s| Riem |2 gij 2 − 2sRikpq Rjkpq − 4sRikjl Rkl + 4sRik Rjk .

132

Chapter 5. Critical Metrics of Riemannian Functionals

Moreover, g is critical for Ft,s on M1 (M n ) if and only if (∇Ft,s ) = c g, for some c ∈ R. Taking the trace in this equation, we obtain  n + 4(n − 1)t + 4s n − 4 | Ric |2 + tR2 + s| Riem |2 − ∆R = nc . 2 2 Therefore, a metric g is critical for Ft,s on M1 (M n ) if and only if it satisfies 2t − 2s − (1 + 4s)∆Rij + (1 + 2t + 2s)∇2ij R − (∆R)gij n  2 + | Ric |2 + tR2 + s| Riem |2 gij + 2(1 + 2s)Rikjl Rkl n − 2tRRij − 2sRikpq Rjkpq + 4sRik Rjk = 0 ,

(5.18)

coupled with the scalar equation  
n + 4(n − 1)t + 4s ∆R = (n − 4) | Ric |2 + tR2 + s| Riem |2 − λ , ˚ij = Rij − 1 Rgij , we obtain the Eulerwhere λ = Ft,s (g). Substituting here R n Lagrange equation of critical metrics for Ft,s on M1 (M n ). Proposition 5.23. Let M n be a closed manifold of dimension n ≥ 3. A metric g is critical for Ft,s on M1 (M n ) if and only if it satisfies the equations ˚ij = (1 + 2t + 2s)∇2ij R − 1 + 2t + 2s (∆R)gij − 2(1 + 2s)Rikjl R ˚kl (1 + 4s)∆R n  2 + 2nt − 4s ˚ 2 ˚ 2 − RRij + |Ric| + s| Riem |2 gij (5.19) n n ˚ik R ˚jk , − 2sRikpq Rjkpq + 4sR and  
n + 4(n − 1)t + 4s ∆R = (n − 4) | Ric |2 + tR2 + s| Riem |2 − λ ,

(5.20)

where λ = Ft,s (g). In particular, any Einstein critical metric must satisfy the following pointwise condition (see also [11, Corollary 4.67]) Corollary 5.24. An Einstein metric is critical for Ft,s on M1 (M n ) if and only if it satisfies 1 Rikpq Rjkpq = | Riem |2 gij . n Corollary 5.25. Any space-form metric is critical for Ft,s on M1 (M n ).

5.3. Some rigidity results for quadratic functionals

133

If n 6= 4 and s=−

n + 4(n − 1)t , 4

the quantity | Ric |2 + tR2 −

n + 4(n − 1)t | Riem |2 4

is constant. Since the norm of the Riemann curvature tensor is given by | Riem |2 = |W |2 +

4 2 | Ric |2 − R2 , n−2 (n − 1)(n − 2)

we obtain the following Corollary 5.26. Let M n be a closed manifold of dimension n > 4. If g is a critical metric g for Ft,− n+4(n−1)t on M1 (M n ), then the quantity 4

−


n + 4(n − 1)t 4 |W |2 + 1 + 2(n − 1)t σ2 (A) 4 n−2

is constant on M n . Moreover, we note that, when t = s = −1/4, Corollary 5.26 applies. In this case, the integrand of the curvature functional F− 14 ,− 14 vanishes if n = 3 and it corresponds (in fact, it is proportional) to the Gauss-Bonnet integrand if n = 4. Furthermore, it follows from equation (5.19) that all the second-order terms in the Euler-Lagrange equation vanish. More precisely, one has the following remarkable fact, which, in part, was already observed by Berger in [10, Section 7] (see also [104]). Corollary 5.27. Let M n be a closed manifold of dimension n > 4. A metric g is critical for F− 14 ,− 14 on M1 (M n ) if and only if it satisfies the equation   ˚kl + n − 6 RR ˚ij + 1 4|Ric| ˚ 2 − | Riem |2 gij + 1 Rikpq Rjkpq − R ˚ik R ˚jk = 0. −Rikjl R 2n 2n 2 Moreover, the quantity |W |2 +

2(n − 3) σ2 (A) n−2

is constant on M n . As already suggested by Berger, it will be interesting to have a complete classification of these critical metrics. For the sake of completeness, we compute the pointwise and integral Weitzenb¨ock formulas for critical metrics of the functional Ft,s .

134

Chapter 5. Critical Metrics of Riemannian Functionals

Proposition 5.28. Let M n be a closed manifold of dimension n > 4. If g is a critical metric g for Ft,s on M1 (M n ), then 1 + 4s ˚ 2 = (1 + 4s)|∇Ric| ˚ 2 + (1 + 2t + 2s)R ˚ij ∇2ij R − 2(1 + 2s)Rikjl R ˚ij R ˚kl ∆|Ric| 2 2 + 2nt − 4s ˚ 2 − 2sR ˚ij Rikpq Rjkpq + 4sR ˚ij R ˚ik R ˚jk . (5.21) − R|Ric| n Integrating by parts and using second Bianchi identity, we obtain Corollary 5.29. Let M n be a closed manifold of dimension n > 4. If g is a critical metric for Ft,s on M1 (M n ), then  Z  (n − 2)(1 + 2t + 2s) 2 2 ˚ (1 + 4s)|∇Ric| − |∇R| dV 2n Z  ˚ij R ˚kl + 1 + nt − 2s R|Ric| ˚ 2 =2 (1 + 2s)Rikjl R n  ˚ ˚ ˚ ˚ + sRij Rikpq Rjkpq − 2sRij Rik Rjk dV . In particular, if g is a critical metric for Ft,− n+4(n−1)t on M1 (M n ), then 4

 (n − 2)2 2 2 ˚ − (n − 1)(1 + 4t) |∇Ric| − |∇R| dV 4n(n − 1) Z  n − 2 + 4(n − 1)t ˚ij R ˚kl + n + 2 + 2(3n − 2)t R|Ric| ˚ 2 =2 − Rikjl R 2 2n n + 4(n − 1)t ˚ n + 4(n − 1)t ˚ ˚ ˚  − Rij Rikpq Rjkpq + Rij Rik Rjk dV . 4 2 Z 

Chapter 6

Bochner-Weitzenb¨ock Formulas and Applications Bochner-Weitzenb¨ ock formulas for the Weyl tensor have been widely used in the last decades; to indicate just some of these works, focused on the study of Einstein manifolds and related structures, we mention those of Derdzinski [68], Singer [139], Hebey-Vaugon [93], Gursky [84, 86], Gursky-Lebrun [87], Yang [149] (see also the references therein). In this chapter, after proving a very general BochnerWeitzenb¨ ock formula for the Weyl tensor on every smooth manifold, we specialize to harmonic Weyl and Einstein metrics and prove a classical rigidity result. We then prove a characterization of anti-self-dual metrics arising from an integral Bochner-Weitzenb¨ock identity. Finally, following our work [52], we employ higher-order Bochner-Weitzenb¨ock formulas for four-dimensional closed Einstein manifolds to derive new integral identities and a new rigidity result.

6.1

A general Bochner-Weitzenb¨ock formula

6.1.1 General dimension In this subsection, we prove that on every n-dimensional Riemannian manifold with n ≥ 4, the Weyl tensor satisfies a nice Bochner-Weitzenb¨ock formula. Namely, we have Lemma 6.1. Let (M n , g) be a n-dimensional Riemannian manifold, n ≥ 4. Then n−2 1 2 2 ∆|W |2 = |∇W | − 2 |δW | + 2Rpq Wpikl Wqikl 2 n−3   − 2 2Wijkl Wipkq Wjplq + 12 Wijkl Wijpq Wklpq − 2(Wijkl Cjkl )i .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 G. Catino, P. Mastrolia, A Perspective on Canonical Riemannian Metrics, Progress in Mathematics 336, https://doi.org/10.1007/978-3-030-57185-6_6

135

136

Chapter 6. Bochner-Weitzenb¨ock Formulas and Applications

Proof. The second Bianchi identity for the Weyl tensor (see equation (3.2)) shows that −Wklij,mm − Wklmi,jm + Wmjkl,im = Ψijkl , where 1 (Cljm,m δki + Clmi,m δkj + Clij,k − Ckjm,m δli − Ckmi,m δlj − Ckij,l ). n−2 The previous relation can be rewritten as Ψijkl :=

Wklij,mm = (Wmjkl,mi − Wmikl,mj ) − Ψijkl + (Wklmj,im − Wklmj,mi ) − (Wklmi,jm − Wklmi,mj ) n−3 = (Cikl,j − Cjkl,i ) − Ψijkl + (Wklmj,im − Wklmj,mi ) n−2 − (Wklmi,jm − Wklmi,mj ). Using the commutation relation for the second covariant derivative of the Weyl tensor (see Lemma 3.4) to expand the two terms Wklmj,im −Wklmj,mi and Wklmi,jm − Wklmi,mj , and also the first Bianchi identity for W , we deduce that n−3 (Cikl,j − Cjkl,i ) − Ψijkl (6.1) n−2 + Rip Wpjkl − Rjp Wpikl − 2(Wipjq Wpqkl − Wipql Wjpqk + Wipqk Wjpql )  1  + Rjp Wpikl − Rip Wpjkl − Rlp (Wpikj − Wpjki ) − Rkp (Wpjli − Wpilj ) n−2 1 + Rpq (Wpiql δkj − Wpjql δki + Wpikq δlj − Wpjkq δli ). n−2 Contracting with Wijkl and using again the first Bianchi identity, we obtain formula (6.1).  ∆Wijkl =

6.1.2 Dimension four and an integral identity In dimension four, taking advantage of identities (3.18), (3.19) and the orthogonality of W ± , the formula simplifies to the following Corollary 6.2. Let (M 4 , g) be a four dimensional Riemannian manifold. Then 1 1 2 2 2 ∆|W |2 = |∇W | − 4|δW | + R|W | − 3Wijkl Wijpq Wklpq − 2(Wijkl Cjkl )i . 2 2 As a consequence, if M is closed, then one has the integral identity (see [60, Equation 3.23]): Z   1 |∇W |2 − 4|δW |2 + R|W |2 − 3Wijkl Wijpq Wklpq dV = 0 . 2 M Moreover, Z  M

 1 ± ± ± |∇W ± |2 − 4|δW ± |2 + R|W ± |2 − 3Wijkl Wijpq Wklpq dV = 0 . 2

6.2. Some applications

6.2

137

Some applications

6.2.1 Harmonic Weyl and Einstein manifolds: the first Bochner-Weitzenb¨ock formula In this section we recover the well known Bochner-Weitzenb¨ock formula for manifolds with harmonic Weyl curvature (i.e. div(W ) = 0) (see e.g. [139, 93]). Lemma 6.3. Let (M, g) be a Riemannian manifold of dimension n ≥ 4 with harmonic Weyl tensor. Then ∆Wijkl = Rip Wpjkl − Rjp Wpikl − 2(Wipjq Wpqkl − Wipql Wjpqk + Wipqk Wjpql ) 1 + [Rjp Wpikl − Rip Wpjkl + Rlp (Wpjki − Wpikj ) − Rkp (Wpjli − Wpilj )] n−2 1 + [Rpq (Wpiql δkj − Wpjql δki + Wpikq δlj − Wpjkq δli )] . (6.2) n−2 As a consequence,   1 ∆|W |2 = |∇W |2 + 2Rpq Wpikl Wqikl − 2 2Wijkl Wipkq Wjplq + 12 Wijkl Wijpq Wklpq . 2 (6.3) Proof. Since g has harmonic Weyl tensor, formula (1.82) implies that the Cotton tensor vanishes; the proof of (6.2) now follows immediately from equation (6.1). Equation (6.3) is a consequence of equation (6.2) and Lemma 3.1.  In particular, in dimension four we have Corollary 6.4. On any four-dimensional manifold with harmonic Weyl curvature, ∆Wijkl = Wijkl,tt =

R Wijkl − 2(Wipjq Wpqkl − Wipql Wjpqk + Wipqk Wjqpl ) (6.4) 2

and 1 R ∆|W |2 = |∇W |2 + |W |2 − 3Wijkl Wijpq Wklpq . 2 2

(6.5)

Proof. The proof is just an easy computation using (6.3) in conjunction with equations (3.18) and (3.19).  A computation employing the orthogonality properties of W ± and ∇W ± (see Chapter 3, 3.2) shows that the same equation holds for the self-dual and anti-selfdual parts of the Weyl tensor; namely, on every four-dimensional manifold with half-harmonic Weyl curvature, div(W ± ) = 0, one has 1 R ± ± ± ∆|W ± |2 = |∇W ± |2 + |W ± |2 − 3Wijkl Wijpq Wklpq . 2 2

(6.6)

138

Chapter 6. Bochner-Weitzenb¨ock Formulas and Applications

6.2.2 Rigidity results for Einstein manifolds In this subsection, as a consequence of the Bochner-Weitzenb¨ock formula (6.3), we prove a rigidity result for Einstein metrics with positive Yamabe invariant and small L2 -norm of the Weyl tensor. This result appeared in the works of Singer [139], Hebey and Vaugon [93] and Gursky [83]. As already observed in Chapter 5, on a closed manifold the Yamabe invariant Y(M, [g]) is positive (respectively, zero or negative) if and only if there exists a conformal metric in [g] with everywhere positive (respectively, zero or negative) scalar curvature. In particular, if Y(M, [g]) > 0, then for every u ∈ W 1,2 (M ) the following Yamabe-Sobolev inequality holds: n−2 Y(M, [g]) 4(n − 1)

Z |u|

2n n−2

 n−2 n

Z ≤

dVg

M

|∇u|2 dVg +

M

n−2 4(n − 1)

Z

R u2 dVg .

M

(6.7) Now let (M n , g) be a n-dimensional Einstein manifold with positive scalar curvature. By a result of Obata [128], the metric g is a Yamabe metric. In particular, Y(M, [g]) > 0. The following rigidity result holds: Theorem 6.5. Let (M n , g) be a n-dimensional Einstein manifold with positive scalar curvature. There exists a positive constant A(n) such that if Z

 n2

n

|W | 2 dVg

< A(n) Y(M, [g]) ,

M

then (M n , g) is isometric to a quotient of the round Sn .We can take A(4) = A(5) =

3 32 ,

A(6) =

√ √3 , 5 70

A(n) =

n−2 20(n−1)

if 7 ≤ n ≤ 9, and A(n) =

2 5n

5 √ , 9 6

if n ≥ 10.

Proof. Since the metric g is Einstein, it has harmonic Weyl curvature. From equation (6.3) we deduce that   1 2 1 2 2 2 ∆|W | = |∇W | + R|W | − 2 2Wijkl Wipkq Wpjql + Wijkl Wklpq Wpqij . 2 n 2 Integrating on M n , using the classical Kato inequality |∇W | ≥ |∇|W || (see (3.26) in Chapter 3) and Lemma 3.8, we obtain Z 0≥ M

|∇|W ||2 dVg +

2 n

Z M

R|W |2 dVg − 2C(n)

Z

|W |3 dVg ,

M

where C(n) is defined as in Lemma 3.8. Using the H¨older inequality and (6.7) with

6.2. Some applications

139

u := |W |, we get Z Z 2 0≥ |∇|W ||2 dVg + R|W |2 dVg n M M Z  n2 Z  n−2 n 2n n − 2C(n) |W | 2 dVg |W | n−2 dVg M M Z Z 2 ≥ |∇|W ||2 dVg + R|W |2 dVg n M M Z  n2   Z Z n C(n) 4(n − 1) 2 2 2 −2 |W | dVg |∇|W || dVg + R|W | dVg . Y(M, [g]) n−2 M M M Now, by assumption, Z

 n2

n 2

|W | dVg

< A(n) Y(M, [g]) .

M

It then follows that |W | ≡ 0, so g has constant positive sectional curvature, if A(n) satisfies the following inequalities  n−2  , 2C(n)A(n) ≤ 4(n − 1)  2C(n)A(n) ≤ 2 . n √ 70 √ and 2 3 √ A(6) = 5√370 ,

Since C(5) = 1, C(6) = 3 32 ,

C(n) =

5 2

for n ≥ 7, it is easy to see that we

n−2 20(n−1)

2 can take A(5) = A(n) = if 7 ≤ n ≤ 9, and A(n) = 5n if n ≥ 10. This concludes the proof of the theorem when n 6= 4. In dimension n = 4 we can improve the estimate by using the refined Kato inequality (see Lemma 3.17 and [87])

|∇W |2 ≥

5 |∇|W ||2 . 3

Reasoning as before, we have Z Z 5 1 2 0≥ |∇|W || dVg + R|W |2 dVg 3 M 2 M Z  12  Z  Z C(4) 2 2 2 −2 |W | dVg 6 |∇|W || dVg + R|W | dVg Y(M, [g]) M M M Hence, we need

Since C(4) =

√ 6 4

 5  2C(4)A(4) ≤ , 18  2C(4)A(4) ≤ 1 . 2 we obtain A(4) =

5 √ . 9 6



140

Chapter 6. Bochner-Weitzenb¨ock Formulas and Applications

In dimension four, Theorem 6.5 was substantially improved by Gursky and Lebrun [87]. In fact, using a modified version of the Yamabe problem, they proved the following sharp result: Theorem 6.6. Let (M 4 , g) be a four-dimensional Einstein manifold with positive scalar curvature. If Z Z 1 2 |W | dVg ≤ R2 dVg , 6 M M then (M 4 , g) is isometric to S4 , RP4 , or CP2 with their standard metrics. Recently, a direct proof of this result using equation (6.5) and the improved Kato inequality in Lemma 3.17 appeared in [17].

6.2.3 A general result on four-dimensional manifolds In this section we prove a characterization of anti-self-dual four-dimensional manifolds, i.e., manifolds (M 4 , g) for which Wg+ ≡ 0, assuming the positivity of the Yamabe invariant, a pinching condition on the conformal invariant Z + W (M, [g]) = |Wg+ |2g dVg , M

and the non-positivity of the quadratic functional Z  Z 1 5 − 9α + + 2 + 2 2 Dα (g) := Volg (M ) |δg Wg |g dVg − Rg |Wg |g dVg , 24 M M defined for a given α ∈ [0, 59 ]. In the same spirit we define D− α (g) (note that ± ± D± (g) = D (g): see Chapter 8 for the definition of D (g) and for more details). 5 9 We can show the following: Theorem 6.7. Let (M 4 , g) be a closed Riemannian manifold with positive Yamabe invariant Y(M, [g]) > 0. Then (M 4 , g) is anti-self-dual, i.e., W + ≡ 0, if and only if there exists α ∈ [0, 95 ] such that W+ (M, [g]) ≤

α2 Y(M, [g])2 6

and

D+ g) ≤ 0 α (e

for some ge ∈ [g] .

The same result holds for the anti-self-dual part W − of the Weyl tensor. As a consequence, we can prove the following lower bound for W+ (M, [g]): Corollary 6.8. Let (M 4 , g) be a closed Riemannian manifold with positive Yamabe invariant Y(M, [g]) > 0. Suppose that there exists α ∈ [0, 59 ] such that D± g) ≤ 0 α (e

for some ge ∈ [g] .

6.2. Some applications

141

Then either Wg± ≡ 0, or W± (M, [g]) >

3α2 16π 2 · (2χ(M ) ± 3τ (M )) , 2 1 + 2α 3

where χ(M ) and τ (M ) denote the Euler characteristic and the signature of M , respectively (see Chapter 3, Section 3.2.4). Remarks. 1. Gursky [86] proved that, if δW ± ≡ 0, i.e., D± 5 (g) ≤ 0, on a four-manifold 9

(M 4 , g) with positive Yamabe invariant, then either Wg± ≡ 0, or W± (M, [g]) ≥

16π 2 (2χ(M ) ± 3τ (M )) . 3

The conclusion in this case is stronger than the one in Corollary 6.8. Indeed, the harmonic Weyl condition ensures the validity of the pointwise BochnerWeitzenb¨ ock formula (8.1) which yields the result by using a clever Yamabetype argument. 2. The same estimate as in Corollary 6.8 with α = 1/3 appeared in [36, Theorem 4.1]. The authors proved the lower bound on W± (M, [g]), assuming that (M 4 , g) is a gradient shrinking Ricci soliton (see Chapter 7) which satisfies Z Z 1 |δW ± |2 dV ≤ R|W ± |2 dV , 12 M M i.e., D± 1 (g) ≤ 0. Note that in Corollary 6.8 no curvature condition is imposed 3 on the Ricci tensor. 3. We conjecture that Theorem 6.7 and Corollary 6.8 remain valied if one only assumes one of the conformally invariant conditions ± D± g) ≤ 0 . α (M, [g]) := inf Dα (e g e∈[g]

The conjecture would follow, if one can show that the infimum D± α (M, [g]) is achieved. With some effort, this could be established by arguments similar to those used in the proof of Theorem 7.8, further assuming that |Wge± |ge > 0 for some ge ∈ [g]. Without such an additional condition, the functional and the associated Euler-Lagrange equation are degenerate, thus a completely different analysis is required. Proof of Theorem 6.7 and Corollary 6.8. Let (M 4 , g) be a closed manifold of dimension four with positive Yamabe invariant, Y(M, [g]) > 0. Assume that (M 4 , g) is not anti-self-dual, i.e., W + 6≡ 0, and satisfies the pinching condition W+ (M, [g]) ≤

α2 Y(M, [g])2 , 6

(6.8)

142

Chapter 6. Bochner-Weitzenb¨ock Formulas and Applications

for some α ∈ [0, 59 ]. Obiouvsly, if α = 0 we have a contradiction. Moreover, the case α = 59 was already considered in [86] (see also [93]). Hence we can assume α ∈ (0, 59 ). To prove Theorem 6.7, we will show that D+ g ) > 0 for every ge ∈ [g] . α (e By Lemma 6.2, Z Z + 2 |∇W | dV = M

M



 1 + + + 4|δW + |2 − R|W + |2 + 3Wijkl Wijpq Wklpq dV. 2

(6.9)

On the other hand, since the following sharp inequality holds: 1 + + + Wijkl Wijpq Wklpq ≤ √ |W + |3 , 6

(6.10)

H¨ older’s inequality implies that Z Z 1 + + + Wijkl Wijpq Wklpq dV ≤ √ |W + |3 dV 6 M M Z  12 Z  12 1 + 2 + 4 ≤√ |W | dV |W | dV . 6 M M Moreover, equality is attained if and only if W + has at most two different eigenvalues and |W + | is constant almost everywhere. In particular, in this case, since W + 6≡ 0, |W + | > 0 on W + has exactly two distinct eigenvalues on M 4 . The Yamabe-Sobolev inequality (6.7) applied to u := |W + | yields Z + + + Wijkl Wijpq Wklpq dV M

Z  12  Z  Z 1 + 2 + 2 + 2 ≤√ |W | dV 6 ∇|W | dV + R|W | dV 6 Y(M, [g]) M M M Z Z ∇|W + | 2 dV + α ≤α R|W + |2 dV , 6 M M where in the last inequality we used the assumption (6.8). Let M0 := {x ∈ M : |W + |(x) = 0} . Note that, in general, Vol(M0 ) can be strictly positive (by a unique continuation principle, this is not the case when δW + ≡ 0, see for instance [85]). Then one has Z Z Z + + + ∇|W + | 2 dV + α Wijkl Wijpq Wklpq dV ≤ α R|W + |2 dV . 6 M M \M0 M0

6.2. Some applications

143


Thus, the improved Kato inequality in Lemma 3.17 implies, for every k ∈ 0, 53 , Z Z Z α 8α(k − 1) + + + Wijkl Wijpq Wklpq dV ≤ |∇W + |2 dV + |δW + |2 dV k M \M0 k(5 − 3k) M \M0 M Z α + R|W + |2 dV . 6 M \M0 On the other hand, by Remark 3.18, on M0 we have |∇W + |2 ≥

8 |δW + |2 , 5

hence α k

8α(k − 1) |∇W | dV + k(5 − 3k) M0

Z

+ 2

Z

α |δW | dV + 6 M0 + 2

Z

R|W + |2 dV ≥ 0 .

M0

Combining the above inequalities with (6.9), we conclude that Z Z k − 3α 4k(5 − 3k) + 24α(k − 1) |∇W + |2 dV ≤ |δW + |2 k k(5 − 3k) M M Z 1−α + 2 − R|W | dV . 2 M Now choose k = 3α; then Z Z 5 − 9α |δW + |2 dV ≥ R|W + |2 dV, 24 M M i.e., D+ α (g) ≥ 0 . If D+ α (g) > 0, since all the assumptions are conformally invariant, this estimate holds for all metrics ge in the conformal class [g] and the claim follows. On the other hand, suppose that D+ α (g) = 0. Then Z Z 5 − 9α |δW + |2 dV = R|W + |2 dV. 24 M M From a previous estimate, since k = 3α, we obtain Z Z Z 1 8(3α − 1) + + + Wijkl Wijpq Wklpq dV ≤ |∇W + |2 dV + |δW + |2 dV 3 3(5 − 9α) M M M Z α + 2 R|W | dV + 6 M  Z Z 1 3α − 1 α + 2 = |∇W | dV + + R|W + |2 dV 3 M 9 6 M Z Z 1 9α − 2 ± 2 = |∇W | dV + R|W + |2 dV. 3 M 18 M

144

Chapter 6. Bochner-Weitzenb¨ock Formulas and Applications

Thus, Z

|∇W + |2 dV ≤



M

5 − 9α 1 9α − 2 − + 6 2 6

Z

Z

R|W + |2 dV +

M

Z

|∇W + |2 dV

M

|∇W + |2 dV.

= M

In particular, we have equalities in the previous computations, so |W + | is a positive constant and the equality case in the Yamabe-Sobolev inequality gives that also the scalar curvature R must be a positive constant. Substituting in (6.9), we obtain  Z Z Z 5 − 9α 1 α 1 − 3α − + R|W + |2 dV = R|W + |2 dV |∇W + |2 dV = 6 2 2 3 M M M 1 − 3α + 2 = Vol(M )R|W | . 3 This implies that α ≤ 13 . To conclude, we use the fact that we have equality also in the Kato inequality in Lemma 3.17 with k = 3α, i.e., |∇W + |2 =

8(1 − 3α) |δW + |2 (5 − 9α)

on M 4 , since |W + | > 0. First of all, by the equality in the algebraic estimate (6.10), we know that W + has exactly two distinct eigenvalues. Following the proof of Lemma 3.17, since det(W + ) > 0, we can assume that µ = λ and ν = −2λ. Thus c¯ = 0 and Z = 0. Substituting in (3.27) and (3.29), we obtain ∇W + 2 = 24|dλ|2 + 8|X|2 + 8|Y |2 and δW + 2 = 6|dλ|2 + 2|X|2 + 2|Y |2 − 6 hdλ, Y i + 6 hdλ, Y i − 2 hX, Y i . Thus 0 = |∇W + |2 − =

8(1 − 3α) |δW + |2 (5 − 9α)

8  (9(1 − α)|dλ|2 + 3(1 − α)|X|2 + 3(1 − α)|Y |2 5 − 9α  + 6(1 − 3α) hdλ, Xi − 6(1 − 3α) hdλ, Y i + 2(1 − 3α) hX, Y i .

Using again the notation in Lemma 3.17, the associated matrix is given by   9(1 − α) 3(1 − 3α) −3(1 − 3α) 8  . 3(1 − 3α) 3(1 − α) (1 − 3α) M= 5 − 9α −3(1 − 3α) (1 − 3α) 3(1 − α)

6.3. Higher-order Bochner-Weitzenb¨ock formulas on Einstein manifolds

145

A computation shows that det(M) = 288α(2 − 3α), which has to be zero. This is a contradiction, since 0 < α ≤ 13 This completes the proof of Theorem 6.7. Corollary 6.8 follows from Theorem 6.7 using the lower bound for the Yamabe invariant proved in [83] (see Lemma 3.16). 

6.3

Higher-order Bochner-Weitzenb¨ock formulas on Einstein manifolds

In this section we follow closely the presentation in [52].

6.3.1 Rough formulas The aim of this section is to compute new “rough” Bochner type formulas for the kth covariant derivative of the Weyl tensor on four-dimensional Einstein manifolds. The reason for this terminology is that the proof does not make use of the algebraic properties related to dimension four (see Chapter 3, 3.2), but only exploits the commutation rules for covariant derivatives of W . We first treat the case k = 1. Proposition 6.9. On any four-dimensional Einstein manifold we have 1 R ∆|∇W |2 = |∇2 W |2 + h∇W, ∇∆W i + |∇W |2 + 8Wijkl,s Wrjkl,t Rrist . 2 4 Equivalently, 1 R ∆|∇W |2 = |∇2 W |2 + h∇W, ∇∆W i + |∇W |2 2 4 2 + 8Wijkl,s Wrjkl,t Wrist + R Wijkl,s Wsjkl,i . 3 2

Proof. Since |∇W | = Wijkl,s Wijkl,s , we have |∇W | and so

2
t

= 2Wijkl,s Wijkl,st ,

2 1 2 ∆|∇W | = ∇2 W + Wijkl,s Wijkl,stt . 2

(6.11)

Now we want to write Wijkl,stt in (6.11) as Wijkl,tts plus a remainder; to do so, we observe that Wijkl,stt = (Wijkl,st )t = (Wijkl,ts + R1 )t = Wijkl,tst + (R1 )t = Wijkl,tts + R2 + (R1 )t ,

146

Chapter 6. Bochner-Weitzenb¨ock Formulas and Applications

where R1 and R2 are two terms involving the Weyl tensor and the Riemann curvature tensor. Indeed, using Lemma 3.4 and the fact that (M, g) is Einstein, we have R1 = Wrjkl Rrist + Wirkl Rrjst + Wijrl Rrkst + Wijkr Rrlst , (R1 )t = Wrjkl,t Rrist + Wirkl,t Rrjst + Wijrl,t Rrkst + Wijkr,t Rrlst , and R2 = Wvjkl,t Rvist + Wivkl,t Rvjst + Wijvl,t Rvkst + Wijkv,t Rvlst +

R Wijkl,s . 4

Now, a straightforward computation shows that Wijkl,s R2 = 4Wijkl,s Wvjkl,t Rvist +

R 2 |∇W | , 4

while Wijkl,s (R1 )t = 4Wijkl,s Wvjkl,t Rvist . This concludes the proof of the first formula. The second formula follows simply by the decomposition of the Riemann tensor and the Einstein condition.  Remark 6.10. We explicitly note that in the preceding proof it is not sufficient to assume div(Riem) = 0: we have to require the metric to be Einstein. Following this proof, we obtain a first integral identity which will be used in the proof of Lemma 6.21. Corollary 6.11. On any four-dimensional compact Einstein manifold, Z Z Z 2 1 R Wijkl,s Wrjkl,t Wrist dV = − Wijkl,st − Wijkl,ts dV − |∇W |2 dV. 8 24 Proof. From the proof of Proposition 6.9, it follows that 1 ∆|∇W |2 = |∇2 W |2 + Wijkl,s Wijkl,tst + 4Wijkl,s Wrjkl,t Rrist 2 R Wijkl,s Wsjkl,i 3 R = |∇2 W |2 + Wijkl,s Wijkl,tst + 4Wijkl,s Wrjkl,t Wrist + |∇W |2 , 6 = |∇2 W |2 + Wijkl,s Wijkl,tst + 4Wijkl,s Wrjkl,t Wrist +

where in the last equality we used Lemma 3.3. Now, noting that 1 |Wijkl,st − Wijkl,ts |2 = |∇2 W |2 − Wijkl,st Wijkl,ts 2 and integrating the preceding equation over M , we obtain the result.



6.3. Higher-order Bochner-Weitzenb¨ock formulas on Einstein manifolds

147

The general Bochner formula for the k-th covariant derivative of the Weyl tensor, k ≥ 2, is contained in the next proposition. Proposition 6.12. On any four-dimensional Einstein manifold, for every k ∈ N, k ≥ 2, we have 1 R ∆|∇k W |2 = |∇k+1 W |2 + h∇k W, ∇∆∇k−1 W i + |∇k W |2 2 4 + 8Wαβγi0 ,i1 i2 ···ik−1 ik Wαβγj0 ,i1 i2 ···ik−1 jk Rj0 i0 ik jk +2

k−1 X

(6.12)

Wαβγδ,i1 i2 ···ih ···ik−1 ik Wαβγδ,i1 i2 ···jh ···ik−1 jk Rjh ih ik jk .

h=1

Proof. We follow the proof of Proposition 6.9. Since k 2 ∇ W = Wαβγδ,i1 ···i Wαβγδ,i1 ···i , k k we have k 2
∇ W = 2Wαβγδ,i1 ···ik Wαβγδ,i1 ···ik t t and thus 1 k 2 k+1 2 ∆∇ W = ∇ W + Wαβγδ,i1 ···ik Wαβγδ,i1 ···ik tt . 2

(6.13)

Now we want to write Wαβγδ,i1 ···ik tt as Wαβγδ,i1 ···ik−1 ttik plus a remainder, using Lemma 3.5; to do so, we observe that Wαβγδ,i1 ···ik tt = (Wαβγδ,i1 ···ik t )t = Wαβγδ,i1 ···ik−1 tik + R1


t

= Wαβγδ,i1 ···ik−1 tik t + (R1 )t = Wαβγδ,i1 ···ik−1 ttik + R2 + (R1 )t , where R1 and R2 are two terms involving the Weyl tensor and the Riemann curvature tensor. Indeed, using Lemma 3.5 and the fact that (M, g) is Einstein (and thus div(W ) = 0), we have R1 = Wαβγδ,i1 ···ik t − Wαβγδ,i1 ···tik = Wpβγδ,i1 ···ik−1 Rpαik t + Wαpγδ,i1 ···ik−1 Rpβik t + . . . + Wαβγδ,i1 ···ik−2 p Rpik−1 ik t , | {z } k+3 terms

(R1 )t = Wpβγδ,i1 ···ik−1 t Rpαik t + Wαpγδ,i1 ···ik−1 t Rpβik t + . . . + Wαβγδ,i1 ···ik−2 pt Rpik−1 ik t ; | {z } k+3 terms

148

Chapter 6. Bochner-Weitzenb¨ock Formulas and Applications

and R2 = Wαβγδ,i1 ···ik−1 tik t − Wαβγδ,i1 ···ik−1 ttik = Wpβγδ,i1 ···ik−1 t Rpαik t + Wαpγδ,i1 ···ik−1 t Rpβik t + . . . + Wαβγδ,i1 ···ik−2 pt Rpik−1 ik t + Wαβγδ,i1 ···ik−1 p Rptik t | {z } k+3 terms

= (R1 )t +

R Wαβγδ,i1 ···ik−1 ik , 4

and thus (6.13) becomes 1 k 2 k+1 2 ∆∇ W = ∇ W 2   R + Wαβγδ,i1 ···ik Wαβγδ,i1 ···ik−1 ttik + 2(R1 )t + Wαβγδ,i1 ···ik . 4 Now, a lengthy computation shows that Wαβγδ,i1 ···ik (R1 )t = 4Wαβγi0 ,i1 i2 ···ik−1 ik Wαβγj0 ,i1 i2 ···ik−1 jk Rj0 i0 ik jk +

k−1 X

Wαβγδ,i1 i2 ···ih ···ik−1 ik Wαβγδ,i1 i2 ···jh ···ik−1 jk Rjh ih ik jk ,

h=1

which implies equation (6.12).



Remark 6.13. To help the reader, we provide the expressions for k = 2, 3, 4: 1 R ∆|∇2 W |2 = |∇3 W |2 + h∇2 W, ∇∆∇W i + |∇2 W |2 + 8Wijkl,tr Wpjkl,ts Rpirs 2 4 + 2Wijkl,tr Wijkl,ps Rptrs , 1 R ∆|∇3 W |2 = |∇4 W |2 + h∇3 W, ∇∆∇2 W i + |∇3 W |2 + 8Wijkl,trs Wpjkl,tru Rpisu 2 4 + 2Wijkl,pru Wijkl,trs Rptsu + 2Wijkl,tpu Wijkl,trs Rprsu , 1 R ∆|∇4 W |2 = |∇5 W |2 + h∇4 W, ∇∆∇3 W i + |∇4 W |2 2 4 + 8Wijkl,trsu Wpjkl,trsv Rpiuv + 2Wijkl,trsu Wijkl,prsv Rptuv + 2Wijkl,trsu Wijkl,tpsv Rpruv + 2Wijkl,trsu Wijkl,trpv Rpsuv . Remark 6.14. We note that, with suitable changes, Proposition 6.12 holds in every dimension. To conclude this section, we observe that, with no changes in the proofs, all the previous formulas hold also for the self-dual and anti-self-dual parts of the Weyl tensor:

6.3. Higher-order Bochner-Weitzenb¨ock formulas on Einstein manifolds

149

Proposition 6.15. On any four-dimensional Einstein manifold, for every k ∈ N, k ≥ 2, we have 1 R ∆|∇k W ± |2 = |∇k+1 W ± |2 + h∇k W ± , ∇∆∇k−1 W ± i + |∇k W ± |2 2 4 ± ± + 8Wαβγi W R j 0 i0 ik j k αβγj0 ,i1 i2 ···ik−1 jk 0 ,i1 i2 ···ik−1 ik +2

k−1 X

(6.14)

± ± Wαβγδ,i Wαβγδ,i Rjh ih ik jk . 1 i2 ···ih ···ik−1 ik 1 i2 ···jh ···ik−1 jk

h=1

6.3.2 The second Bochner-Weitzenb¨ock formula In this subsection we first prove the following second Bochner-type formula: Theorem 6.16. Let (M, g) be a four-dimensional Einstein manifold. Then the Weyl tensor satifies the equation 1 13 ∆|∇W |2 = |∇2 W |2 + R|∇W |2 − 10 Wijkl Wijpq,t Wklpq,t . 2 12 Proof. By Proposition 6.9, 1 R ∆|∇W |2 = |∇2 W |2 + h∇W, ∇∆W i + |∇W |2 + 8Wijkl,s Wrjkl,t Wrist 2 4 2 + R Wijkl,s Wsjkl,i . 3 Now observe that, using Lemma 3.3, one has Wijkl,s Wsjkl,i = Wijkl,s Wijks,l =

1 |∇W |2 . 2

Moreover, since renaming indices we have Wijkl,s Wrjkl,t Wrist = Wijkl Wjpqt,k Wipqt,l , Lemma 3.14 shows that 1 Wijkl,s Wrjkl,t Wrist = − Wijkl Wijpq,t Wklpq,t . 2 Now, Theorem 6.16 follows from the next lemma. Lemma 6.17. On every four-dimensional Riemannian manifold with harmonic Weyl curvature, one has h∇W, ∇∆W i =

1 R|∇W |2 − 6Wijkl Wijpq,t Wklpq,t . 2

150

Chapter 6. Bochner-Weitzenb¨ock Formulas and Applications

Proof. First we observe that, using the first Bianchi identity, equation (6.4) can be rewritten as R ∆Wijkl = Wijkl,tt = Wijkl − Wijpq Wklpq − 2(Wipkq Wjplq − Wiplq Wjpkq ), 2 which thanks to the symmetries of the Weyl tensor implies that   R h∇W, ∇∆W i = Wijkl,t Wijkl − Wijpq Wklpq − 2(Wipkq Wjplq − Wiplq Wjpkq ) 2 t R 2 = |∇W | − Wijkl,t (Wijpq Wklpq )t − 4Wijkl,t (Wipkq Wjplq )t 2 R 2 = |∇W | − Wijkl,t Wijpq,t Wklpq − Wijkl,t Wijpq Wklpq,t 2 − 4Wijkl,t Wipkq,t Wjplq − 4Wijkl,t Wipkq Wjplq,t R 2 = |∇W | − Wpqkl,t Wpqij,t Wklij − Wijpq,t Wijkl Wklpq,t 2 − 4Wjilk,t Wpiqk,t Wjplq − 4Wjilk,t Wjplq Wpiqk,t , where in the last line the change of indices exploits again the symmetries of W . Therefore, R 2 |∇W | − 2Wijpq,t Wijkl Wklpq,t − 8Wjilk,t Wpiqk,t Wjplq , 2 which immediately implies the needed relation using Lemma 3.15. h∇W, ∇∆W i =

This concludes the proof of Theorem 6.16.

 

6.3.3 A rigidity result for Einstein manifolds Starting from Theorem 6.16, we derive here some new integral identities for the Weyl tensor on four-dimensional Einstein manifolds. First of all we have the following identity. Proposition 6.18. On any four-dimensional compact Einstein manifold, Z Z Z 5 R 2 2 2 |∇ W | dV − |∆W | dV + |∇W |2 dV = 0 . 3 4 Proof. We simply integrate over M the second Bochner type formula and use Lemma 6.17 to get Z Z 2 |∆W | dV = − h∇W, ∇∆W idV Z Z (6.15) R =− |∇W |2 dV + 6 Wijkl Wijpq,t Wklpq,t dV , 2 i.e., Z 10

Wijkl Wijpq,t Wklpq,t dV =

5 3

Z

5 |∆W |2 dV + R 6

Z

|∇W |2 dV.



6.3. Higher-order Bochner-Weitzenb¨ock formulas on Einstein manifolds

151

Remark 6.19. We will see in the next section that this formula also holds for W ± . Now we want to estimate the Hessian in terms of the Laplacian of Weyl. Of course, one has 1 |∇2 W |2 ≥ |∆W |2 . 4 In the next proposition we will show that on compact Einstein manifolds one has an improved estimate in the L2 -integral sense. Theorem 6.20. On any four-dimensional compact Einstein manifold, Z Z 5 |∇2 W |2 ≥ |∆W |2 , 12 with equality if and only if ∇W ≡ 0. Proof. In some local basis, using the fact that for a 4 × 4 matrix A it holds that |A|2 ≥ (trace A)2 /4, one has |∇2 W |2 =

X

2 Wijkl,st

ijklst

2 1 X  Wijkl,st − Wijkl,ts + 4 ijklst 2 1 X  ≥ Wijkl,st − Wijkl,ts + 4 =

ijklst

2 1 X  Wijkl,st + Wijkl,ts 4 ijklst

1 |∆W |2 , 4

with equality if and only if Wijkl,st + Wijkl,ts =



1 1 trace Wijkl,st + Wijkl,ts δst = ∆Wijkl δst 4 2

(6.16)

at every point. The final estimate now follows from the following lemma. Lemma 6.21. On any four-dimensional Einstein manifold, Z Z Wijkl,st − Wijkl,ts 2 dV = 2 |∆W |2 dV . 3 Proof. By Corollary 6.11, Z Z Z 2 1 R Wijkl,s Wrjkl,t Wrist dV = − Wijkl,st − Wijkl,ts dV − |∇W |2 dV. 8 24 Moreover, Lemma 3.14 shows that 1 Wijkl,s Wrjkl,t Wrist = − Wijkl Wijpq,t Wklpq,t . 2

152

Chapter 6. Bochner-Weitzenb¨ock Formulas and Applications

Therefore, Z Z Wijkl,st − Wijkl,ts 2 dV = 4 Wijkl Wijpq,t Wklpq,t dV Z 2 = |∆W |2 dV, 3

−

R 3

Z

|∇W |2 dV

where in the last equality we used equation (6.15).



This concludes the proof of the inequality case. As for the equality, equation (6.16) shows that, at every point, Wijkl,st + Wijkl,ts =


1 ∆Wijkl δst . 2

Taking the divergence with respect to the index t and using the second commutation formula in Lemma 3.4 and the fact that Weyl tensor is divergence free, we obtain 1 Wijkl,tts + Wijkl,stt + Wvjkl,t Rvist + Wivkl,t Rvjst 2 + Wijvl,t Rvkst + Wijkv,t Rvlst +

R Wijkl,s = 0 . 4

Contracting with Wijkl,s and using the decomposition of the Riemann tensor, we obtain 1 h∇W, ∇∆W i + h∇W, ∆∇W i + 4Wijkl,s Wrjkl,t Wrist 2 R 1 + Wijkl,s Wsjkl,i + R|∇W |2 3 4 1 1 = h∇W, ∇∆W i + ∆|∇W |2 − |∇2 W |2 + 4Wijkl,s Wrjkl,t Wrist 2 2 R 1 + Wijkl,s Wsjkl,i + R|∇W |2 3 4 21 = R|∇W |2 − 15Wijkl Wijpq,t Wklpq,t , 12

0=

where we have used Lemma 3.3 and 3.14, Theorem 6.16 and Lemma 6.17. Hence, we have proved that, at every point, one has Wijkl Wijpq,t Wklpq,t =

7 R|∇W |2 . 60

The second Bochner formula (Theorem 6.16) yields 1 1 ∆|∇W |2 = |∇2 W |2 − R|∇W |2 . 2 12

(6.17)

6.3. Higher-order Bochner-Weitzenb¨ock formulas on Einstein manifolds

153

In particular, since R is constant, if R ≤ 0, then integrating over M we obtain ∇2 W ≡ 0, which implies ∆W ≡ 0 and, by compactness, ∇W ≡ 0. Now assume that R > 0. Let α be the 2-form (pqt)

αij

:=

Wijpq,t , |∇W |

defined where ∇W 6= 0. Note that |α| = 1. By (6.17), (pqt) (pqt) αkl

Wijkl αij

=

7 R 60

at every point where ∇W 6= 0. In particular, by normalization, the positive constant 7R/120 is an eigenvalue of W viewed as an oper 7 7 R, or ν := ν + + ν − = R. 120 120 First of all we, claim that µ cannot be positive. Indeed, if µ > 0, then det(W ) = λµν has to be negative where ∇W 6= 0. Since W is trace-free, this is equivalent to saying that Wijkl Wijpq Wklpq < 0 . either µ := µ+ + µ− =

From equation (6.6) one has 1 R ∆|W |2 = |∇W |2 + |W |2 − 3Wijkl Wijpq Wklpq . 2 2 Assume that ∇W 6≡ 0. Let Mε := {p ∈ M : |∇W |2 (p) ≤ ε}. Since g is Einstein, in harmonic coordinates g is real analytic (see [11]), and so is the function |∇W |2 . In particular, Vol(Mε ) → 0 as ε → 0. Integrating over M the Bochner formula (6.6) for W + , we obtain Z Z Z R 2 2 0= |∇W | dV + |W | dV − 3 Wijkl Wijpq Wklpq dV 2 M M M \Mε Z −3 Wijkl Wijpq Wklpq dV Mε Z Z   R 2 ≥ |∇W | dV + |W |2 dV − 3 sup |W |3 Vol(Mε ) , 2 M M M where we have used the fact that Wijkl Wijpq Wklpq < 0 on M \ Mε . Letting ε → 0, we obtain W ≡ 0, hence ∇W ≡ 0, so we reached a contradiction. This argument shows that, necessarily, 7 ν= R, 120 where ∇W 6= 0. In particular ν < R/6, which implies that, where ∇W 6= 0, g has strictly positive isotropic curvature (see [118]). Assume that ∇W 6≡ 0. By

154

Chapter 6. Bochner-Weitzenb¨ock Formulas and Applications

analyticity this is true on a dense subset. Thus, by continuity, (M, g) is an Einstein manifold with positive isotropic curvature, hence isometric to a quotient of the round sphere S4 (see again [118]). In particular, ∇W ≡ 0, a contradiction. This concludes the proof of the equality case.  Propositions 6.18 and 6.20 immediately yield the following gap result in the form of a Poincar´e type inequality. Corollary 6.22. On any four-dimensional Einstein manifold with positive scalar curvature R, Z Z R |∇2 W |2 dV ≥ |∇W |2 dV , 12 with equality if and only if ∇W ≡ 0. As a consequence, using the classification of four-dimensional locally symmetric Einstein manifolds with positive scalar curvature (which follows from the general classification of locally symmetric spaces by Cartan [38]), we get the following rigidity result: Theorem 6.23. Let (M 4 , g) be a four-dimensional Einstein manifold with positive scalar curvature R. If Z Z R 2 2 |∇ W | dV ≤ |∇W |2 dV, 12 then (M 4 , g) is isometric, up to quotients, to S4 , CP2 , or S2 ×S2 with their standard metrics. Equivalently, we can reformulate Corollary 6.22 as follows: Corollary 6.24. On any compact four-dimensional Einstein manifold, Z Z 7R |∇W |2 dV − 6 Wijkl Wijpq,t Wklpq,t dV ≤ 0 , 10 with equality if and only if ∇W ≡ 0. We conclude this section by combining Propositions 6.18 and 6.20 in order to obtain the following L2 -bounds: Corollary 6.25. On any four-dimensional Einstein manifold with positive scalar curvature R, Z Z Z 5 5 |∆W |2 dV ≤ |∇2 W |2 dV ≤ |∆W |2 dV , 12 3 with equalities if and only if ∇W ≡ 0. To conclude this subsection, we show that all the previous integral identities hold separately for the self-dual and anti-self-dual parts. First, we state the following

6.3. Higher-order Bochner-Weitzenb¨ock formulas on Einstein manifolds

155

Theorem 6.26. Let (M 4 , g) be a compact four-dimensional Einstein manifold. Then Z Z Z 5 R 2 ± 2 ± 2 |∇ W | dV − |∆W | dV + |∇W ± |2 dV = 0 . 3 4 Proof. We will prove the assertion for the self-dual case W + . By the rough second Bochner formula in Proposition 6.15, 1 R ∆|∇W + |2 = |∇2 W + |2 + h∇W + , ∇∆W + i + |∇W + |2 2 4 2 + + + + 8Wijkl,s Wrjkl,t Wrist + R Wijkl,s Wsjkl,i . 3 Now observe that, using Lemma 3.3, we have + + + Wijkl,s Wsjkl,i = Wijkl,s Wijks,l =

1 |∇W + |2 . 2

+ + + + Moreover, renaming indices we have Wijkl,s Wrjkl,t Wrist = Wijkl Wjpqt,k Wipqt,l . Next, by equation (3.22), + + + + + Wijkl Wjpqt,k Wipqt,l = Wijkl Wjpqt,k Wipqt,l

and using Lemma 3.14, we obtain 1 + + + + + Wijkl,s Wrjkl,t Wrist = − Wijkl Wijpq,t Wklpq,t . 2 Since the Hessian of W decomposes as ∇2 W = ∇2 W + + ∇2 W − , we have Z Z h∇W + , ∇∆W + idV = − |∆W + |2 dV Z Z + = − ∆Wijkl ∆Wijkl dV = h∇W + , ∇∆W idV. By orthogonality, a simple computation shows that Z Z + 2 |∆W | dV = − h∇W + , ∇∆W idV Z Z R + + + =− |∇W + |2 dV + 6 Wijkl Wijpq,t Wklpq,t dV 2 and the result follows.

(6.18)



In particular, using the previous formulas and readapting the computations in Theorem 6.20, it is not difficult to prove the following

156

Chapter 6. Bochner-Weitzenb¨ock Formulas and Applications

Theorem 6.27. Let (M 4 , g) be a four-dimensional Einstein manifold with positive scalar curvature R. Then Z Z R 2 ± 2 |∇ W | dV ≥ |∇W ± |2 dV, 12 with equality if and only if ∇W ± ≡ 0. Lemma 6.28. Let (M 4 , g) be a four-dimensional Einstein manifold. Then Z Z  1  ± ± ± ± ± Wijkl,s Wrjkl,t Wrist dV = R Wijkl Wijpq Wklpq − |W ± |4 dV. 8 Equivalently, Z Z Z   R 1 ± ± + 2 Wijkl,s Wrjkl,t Wrist dV = |∇W | dV − |W + |2 6|W + |2 − R2 dV. 24 48 Proof. First of all, by equation (3.22), + + + + + Wijkl,s Wrjkl,t Wrist = Wijkl,s Wrjkl,t Wrist .

Moreover, integrating by parts and using the commutation formula in Lemma 3.4, we obtain Z + + + 2 Wijkl,s Wrjkl,t Wrist Z + + + = − 2 Wijkl,st Wrjkl Wrist Z + + + + = − (Wijkl,st − Wijkl,ts )Wrjkl Wrist Z  + + + + =− Wpjkl Wpist + Wipkl Wpjst + Wijpl Wpkst + Wijkp Wplst R + + + + Wsjkl git − Wtjkl gis + Wiskl gjt − Witkl gjs 12 
+ + + + + + + Wijsl gkt − Wijtl gks + Wijks glt − Wijkt gls Wrjkl Wrist +

Therefore Z + + + 2 Wijkl,s Wrjkl,t Wrist Z
+ + + + + + =− Wpjkl Wpist + Wipkl Wpjst + Wijpl Wpkst + Wijkp Wplst Wrjkl Wrist Z
R + + + + + + + + + − Wiskl Wrjkl Wrisj + Wijsl Wrjkl Wrisk + Wijks Wrjkl Wrisl 6

6.3. Higher-order Bochner-Weitzenb¨ock formulas on Einstein manifolds

157

Z


+ + + + + + + + + + Wpjkl Wpist + Wipkl Wpjst + Wijpl Wpkst + Wijkp Wplst Wrjkl Wrist Z
R + + + + + + + 2Wijkl Wipkq Wjplq + 12 Wijkl Wijpq Wklpq 6 Z R + + + = Wijkl Wijpq Wklpq 4 Z
+ + + + + + + + + + − Wpjkl Wpist + Wipkl Wpjst + Wijpl Wpkst + Wijkp Wplst Wrjkl Wrist .

=−

To conclude the proof, we have to establish the fourth-order identity
+ 1 + + + + + + + + + Wpjkl Wpist + Wipkl Wpjst + Wijpl Wpkst + Wijkp Wplst Wrjkl Wrist = |W + |4 . 4 Define
+ + + + + + + + + + Q := Wpjkl Wpist + Wipkl Wpjst + Wijpl Wpkst + Wijkp Wplst Wrjkl Wrist . Note that + + + + + + + + + + + + Q = Wpjkl Wrjkl Wpist Wrist + Wipkl Wrjkl Wpjst Wrist + 2Wijpl Wrjkl Wpkst Wrist

=: Q1 + Q2 + 2Q3 . From equation (3.18), we have 1 +2 1 + + |W | δpr Wpist Wrist = |W + |4 . 4 4 So, it remains to show that Q2 = Q3 = 0 on M . Following the notation in Chapter 3, we easily get Q1 =

+ + Wipkl Wrjkl = λ2 ωip ωrj + µ2 ηip ηrj + ν 2 θip θrj .

Thus, + + + + Q2 = Wipkl Wrjkl Wpjst Wrist

= λ2 ωip ωrj + µ2 ηip ηrj + ν 2 θip θrj




λ2 ωpj ωri + µ2 ηpj ηri + ν 2 θpj θri




= −2(λ4 + µ4 + ν 4 ) + 4(λ2 µ2 + λ2 ν 2 + µ2 ν 2 ) = 0 , since λ + µ + ν = 0. A similar computation shows that + + + + Q3 = Wijpl Wrjkl Wpkst Wrist  1 2 = (λ + µ2 + ν 2 )δir δpk + 2λµθir θpk + 2λνηir ηpk + 2µνωir ωpk 4
× λ2 ωpk ωri + µ2 ηpk ηri + ν 2 θpk θri
= −2 λµν 2 + λνµ2 + µνλ2 = −2λµν(λ + µ + ν) = 0 .

This concludes the proof of the first identity in the lemma. The second identity simply follows from the Bochner identity (6.6). 

158

Chapter 6. Bochner-Weitzenb¨ock Formulas and Applications

Putting together Lemma 6.28, equation (6.18), Lemma 3.14 and Lemma 6.26, we obtain the following: Proposition 6.29. Let (M 4 , g) be a four-dimensional Einstein manifold. Then Z Z Z   1 ± 2 ± 2 |∆W | dV + R |∇W | dV = |W ± |2 6|W ± |2 − R2 dV 4 and Z

|∇2 W ± |2 dV +

23 R 12

Z

|∇W ± |2 dV =

5 12

Z

  |W ± |2 6|W ± |2 − R2 dV .

Chapter 7

Ricci Solitons: Selected Results Let (M n , g) be a Riemannian manifold of dimension n; a Ricci soliton structure on M is a choice of a smooth vector field X (if any) satisfying for some λ ∈ R the soliton equation 1 Ric + LX g = λ g (7.1) 2 where LX is the Lie derivative of the metric g in the direction of X; in an orthonormal frame this reads as 1 Rij + (Xij + Xji ) = λδij . 2 A Ricci soliton (M, g, X) is called shrinking, steady or expanding if λ is > 0, = 0 or < 0, respectively. If X is the gradient of a smooth function f on M , the soliton is called a gradient Ricci soliton, f is the potential and equation (7.1) becomes Ric + Hess(f ) = λg,

(7.2)

Rij + fij = λδij .

(7.3)

or, in an orthonormal frame,

As in Chapter 4, we denote the spaces of Ricci solitons and of gradient Ricci solitons by EX and Ef , respectively. A soliton X is trivial if X is a Killing vector field, or if ∇f is parallel in the gradient case: clearly, a trivial Ricci soliton is an Einstein manifold. Ricci solitons generate self-similar solutions of the Ricci flow (see Theorem 7.5) and play a fundamental role in the formation of singularities. They were studied by a number of authors for their connection with the celebrated Poincar´e conjecture, settled by Perelman, who employed Hamilton’s Ricci flow theory (see for instance the monograph [120] and references therein). For (partial but) really nice surveys on Ricci solitons (examples, properties,. . . ) see also [29] and [30]. The © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 G. Catino, P. Mastrolia, A Perspective on Canonical Riemannian Metrics, Progress in Mathematics 336, https://doi.org/10.1007/978-3-030-57185-6_7

159

160

Chapter 7. Ricci Solitons: Selected Results

Ricci soliton equation (7.1) can also be interpreted as a particular, global, “prescribed Ricci curvature problem”; this kind of problem already appeared in the pioneering work of Lichnerowicz (see the discussion in [20], Section 3). From the classification point of view, it is known that: • All compact Ricci solitons are gradient: to be precise, in the compact case the vector field X can always be writen as X = ∇f + Y , where f ∈ C ∞ (M ) and Y is a Killing vector field (see [131]). In the complete non-compact case, the same result was proved by Naber [125] for shrinking Ricci solitons with bounded curvature. • G. Perelman has shown, [131], that there are no non-trivial compact steady and expanding Ricci solitons. • Three-dimensional complete gradient shrinking Ricci solitons are classified: indeed, it is well known that they are finite quotients of either the round sphere   3 2 S , gS3 , f = const. , λ or the Gaussian shrinking soliton   λ R3 , gR3 , f (x) = |x|2 , 2 or the round cylinder   1 λ S2 × R, g = gS2 + dr2 , f (r) = r2 λ 2 (see Ivey [98] for the compact case and Perelman [131], Ni and Wallach [126] and H.-D. Cao, B.-L. Chen, and X.-P. Zhu [32] for the complete case). • In the steady three-dimensional case known examples are quotients of the trivial flat R3 , the product space R × Σ2 , where Σ2 is the so-called cigar soliton, and the rotationally symmetric example constructed by Bryant [26]. In [24], Brendle showed that the Bryant soliton is the only non-flat, k-noncollapsed, steady soliton, proving a famous conjecture by Perelman [131]. It is still an open problem to classify three-dimensional steady solitons that do not satisfy the k-non-collapsing condition. • The case of expanding solitons is far less rigid; however, some properties and classification theorems have been proved in the recent years by various authors, see for instance [133], [114], [136], [62], [138], [64], [72] and references therein. In dimension greater than four, the situation is still wide open: recently we have seen numerous interesting results concerning the classification of complete Ricci

7.1. Preliminary results

161

solitons satisfying both pointwise and integral curvature conditions. Clearly, it is impossible to summarize all known results concerning the classification (let alone the existence) of Ricci solitons, due to the enormous amount of works that are, also currently, shaping this vast panorama. In this chapter we present several selected results, based on our recent research, which pursue this direction: after recalling some preliminary results in Section 7.1 (see also Chapter 4), in Section 7.2 we focus on shrinkers, considering pointwise conditions (e.g., positive sectional curvature or pinched Weyl curvature), and we also recall some classification results for non-necessarily gradient solitons; in Section 7.3 we deal with integral assumptions (compact shrinkers and non-compact steady and expanding solitons); in Section 7.4 we explore vanishing conditions involving the Weyl tensor and the integrability conditions for gradient Ricci solitons; finally, in Section 7.5 we further concentrate on generalizations of the vanishing conditions previously introduced, defining the so-called Weyl scalars.

7.1

Preliminary results

7.1.1 Fundamental formulas for Ricci solitons We recall here, for the sake of completeness, some useful formulas that are consequences of the defining equation for Ricci solitons. We only consider the gradient case: at the end of Section 7.2 we will recall, without proofs, some results dealing with the general case. The interested reader can find the extension of the next lemmas, for instance, in [57] and [115]. Note that some of the results presented here already appeared in some sections of Chapter 4. Lemma 7.1. Let (M n , g) be a gradient Ricci soliton. Then the following formulas hold: R + ∆f = λn, (7.4) Ri = 2ft Rit , R + |∇f |2 − 2λf = C,

for some C ∈ R,

(7.5) (7.6)

∆f R = 2λR − 2|Ric|2 ,

(7.7)

Rij,k − Rik,j = −Rtijk,t = −ft Rtijk ,

(7.8)

∆f Rik , = 2λRik − 2Rijkl Rjl = 2λRik − 2 Wijkl Rjl (7.9)
2 + R2 gik − nR Rik + 2(n − 1)Rij Rjk − (n − 1)|Ric|2 gik . (n − 1)(n − 2) Here ∆f denotes the f -Laplacian, ∆f = ∆ − g(∇f, ·).

162

Chapter 7. Ricci Solitons: Selected Results

For the proof, see for instance [75] or [57]; note that, in global notation, equation (7.5) becomes dR = 2 Ric (∇f, ·). ˚ij = Rij − 1 Rgij and In particular, since R n ˚ij R ˚jk R ˚ik + 3 R|Ric|2 − 2 R3 , Rij Rjk Rik = R n n2 a simple computation yields the following equation for the f -Laplacian of the squared norm of the traceless Ricci tensor: Lemma 7.2. Let (M n , g) be a gradient Ricci soliton. Then 1 ˚ 2 = |∇Ric| ˚ 2 + 2λ|Ric| ˚ 2 − 2Rijkl R ˚ik R ˚jl − 2 R|Ric| ˚ 2 ∆f |Ric| 2 n ˚ 2 + 2λ|Ric| ˚ 2 − 2Wijkl R ˚ik R ˚jl = |∇Ric| +

4 ˚ ˚ ˚ 2(n − 2) ˚ 2. Rij Rjk Rik − R|Ric| n−2 n(n − 1)

It follows from equation (7.7) and the maximum principle that every compact shrinking solitons has positive scalar curvature (see [98]). Moreover, we have the following strong maximum principle for the scalar curvature, which holds on every complete shrinker (see [136]). Lemma 7.3. Let (M n , g) be a complete gradient shrinking Ricci soliton. Then either g is flat, or its scalar curvature is positive. Finally, as a consequence of the above formulas, we obtain the following integral identities in the compact case. Lemma 7.4. Let (M n , g) be a compact gradient Ricci soliton. Then Z Z Z Z n−4 n−4 2 2 3 ˚ 2 dV . |∇R| dV = λ R dV − R dV + 2 R|Ric| 2 2n In particular, in dimension four Z Z ˚ 2 dV . |∇R|2 dV = 2 R|Ric| Proof. Integrating by parts and using Lemma 7.1 we obtain Z Z |∇R|2 dV = − R∆R dV Z Z Z Z 1 ˚ 2 dV + 2 =− h∇R2 , ∇f i dV − 2λ R2 dV + 2 R|Ric| R3 dV 2 n Z Z Z Z 1 2 2 2 2 ˚ = R ∆f dV − 2λ R dV + 2 R|Ric| dV + R3 dV 2 n Z Z Z n−4 n−4 ˚ 2 dV . = λ R2 dV − R3 dV + 2 R|Ric|  2 2n

7.1. Preliminary results

163

We show that gradient Ricci solitons give rise to solutions of the Ricci flow, that is ∂ g = −2 Ric . (7.10) ∂t Although the proof is classical, we include it for the reader’s convenience. Theorem 7.5. If (M, g0 , f0 ) is a complete gradient Ricci soliton with constant λ, then there exist (i) a family of metrics g(t), solution of the Ricci flow (7.10), with g(0) = g0 , (ii) a family of diffeomorphisms φ(t, · ) : M → M , with φ(0, · ) = idM , (iii) a family of functions f (t, · ) : M → R with f (0, · ) = f0 (·), defined for every t such that τ (t) := −2λt+1 > 0, and with the following properties: 1. the family φ(t, · ) is generated by the vector field ∇g0 f0 , eventually scaled by the inverse of τ (t): ∂φ 1 (t, · ) = (∇g0 f0 )(φ(t, · )) ; ∂t τ (t)

(7.11)

2. the metric g(t) is given by pull-back under φ(t, · ) and rescaling by τ (t): g(t) = τ (t) φ(t, ·)∗ g0 ;

(7.12)

3. the function f (t) is also given as well by pull-back, namely f (t, ·) = (f0 ◦ φ)(t, · ) .

(7.13)

Proof. First of all, recall that if (M, g) is a complete gradient Ricci soliton, then ∇f is a complete vector field (see [152]). Set τ (t) = −2λt + 1. As ∇g0 f0 is a complete vector-field, there exists a 1-parameter family of diffeomorphisms φ(t, · ) : M → M generated by the time-dependent family of vector fields X(t, · ) :=

1 ∇g0 f0 (φ(t, · )), τ (t)

for every t such that τ (t) > 0. We also set f (t, ·) = (f0 ◦ φ)(t, · ) and g(t) = τ (t) φ(t)∗ g0 . Then ∂ 2λ ∂ g(t) = − g(t) + τ (t) φ(t, ·)∗ g0 . ∂t τ (t) ∂t ∂ By the definition of the Lie derivative, ∂t φ(t, ·)∗ g0 = L(φ(t)−1 )∗ ∂ φ(t,·) φ(t, ·)∗ g0 . ∂t On the other hand, equation (7.11) implies that

∂φ 1 1 (t, · ) = (∇g0 f0 )(·) = φ(t, · )∗ ∇g(t) f (t, · ) , ∂t τ (t) τ (t)

164

Chapter 7. Ricci Solitons: Selected Results ∗

where we used the fact that φ(t, ·)∗ ∇g0 f0 = ∇φ(t,·) Combining these two facts, we have that

g0

φ(t, ·)∗ f0 = ∇g(t) f (t, ·).

∂ 2λ 1 g(t) = − g(t) + L g(t) g(t) . ∂t τ (t) τ (t) ∇ f (t,·) With this at hand, we compute   1 − Ric(g(t)) = φ(t, ·)∗ (− Ric(g0 )) = φ(t, ·)∗ L∇g0 f0 g0 − λ g0 2   1 1 2 = L∇g(t) f (t,·) g(t) − λ g(t) 2 τ (t) τ (t) 1 ∂ = g(t) 2 ∂t and we observe that R(τ (t)−1 g(t)) = τ (t) R(g(t)). Therefore, ∂ g(t) = −2 [ Ric(g(t))] , ∂t and the proof is complete.



To conclude this section, we turn our attention to the regularity of gradient Ricci solitons. We recall that, in harmonic coordinates (see [11, Eq. 5.22]), 1 Ric = − ∆(gij ) + Qij (g −1 , ∂g) , 2

(7.14)

where Q is a quadratic form in the coefficients of g −1 and the first derivatives of the coefficients of g. Theorem 7.6. Let (M n , g, f ), n ≥ 3, be a gradient Ricci soliton. Then, in harmonic coordinates, the metric g and the potential function f are real analytic. Proof. First, taking the divergence of equation (7.3) we get 1 ∇j ∇j ∇i f = −∇j Rij = − ∇i R. 2 Hence, using equation (7.5), we obtain −∆∇f = Ric(∇f, ·) . To prove our statement, it is useful to consider the system ( Ric + ∇2 f − λ g = 0, −∆∇f − Ric(∇f, · ) = 0 ,

7.1. Preliminary results

165

with respect to the unknowns (g, ∇f ). According to (7.14), in harmonic coordinates the scalar curvature is given by 1 R = − g ij ∆(gij ) + g ij Qij (g, ∂g) . 2 Thus, the linearization of the above system in the direction of (h, W ) ∈ S 2 T ∗ M ⊕ T M is given by  1 ∂ 2 hij   − g rs r s + l.o.t. = 0   2 ∂x ∂x  2 2    −g rs ∂ Wi + 1 g rs ∂ hij ∇j f + l.o.t. = 0 , ∂xr ∂xs 2 ∂xr ∂xs where l.o.t. denotes terms involving only W , h, or their first derivatives. Therefore, the principal symbol σζ : S 2 T ∗ M ⊕ T M → S 2 T ∗ M ⊕ T M is given by   1 2 (h, W ) 7−→ σζ (h, W ) = |ζ|g h , |ζ|2g W − Lζ h , 2 where Lζ h is a linear function of h. If σζ (h, W ) = 0 and ζ 6= 0, then h = 0, which implies W = 0. This shows that, if ζ 6= 0, the symbol σζ is an automorphism of S 2 T ∗ M ⊕ T M , and this in turn implies that the linearization of the system is elliptic. If the pair (g, ∇f ) has C 2 -regularity in harmonic coordinates, we can apply Morrey’s interior regularity theorem [28, Theorem 6.7.6] and since our system of equations is analytic in both its dependent and independent variables, the solutions are real analytic as well. We observe that, in general, (g, ∇f ) may be only C 1,α after passing to harmonic coordinates. To overcome this difficulty, we apply Theorem 9.19 in [81] to the components of the system, and conclude that (g, ∇f ) are in fact C 2,α . 

7.1.2 The tensor D and the integrability conditions In [33], H.-D. Cao and Q. Chen introduced for the first time a new tensor, D, which turned out to be a fundamental tool in the study of the geometry of gradient Ricci solitons (and, more generally, of gradient Einstein-type manifolds, see [59]). In components, D is defined as Dijk =

1 1 (fk Rij − fj Rik ) + ft (Rtk gij − Rtj gik ) n−2 (n − 1)(n − 2) R − (fk gij − fj gik ). (n − 1)(n − 2)

(7.15)

The tensor D is skew-symmetric in the second and third indices (i.e., Dijk = −Dikj ) and totally trace-free (i.e., Diik = Diki = Dkii = 0). Note that our

166

Chapter 7. Ricci Solitons: Selected Results

convention for the tensor D differs from that in [33]. In the last section of the chapter, the contraction with the Cotton tensor will be abbreviated as CD = DC := Cijk Dijk . We now recall the four so-called integrability conditions for gradient Ricci solitons of dimension n ≥ 3. Proposition 7.7. If (M n , g) is a gradient Ricci soliton with potential function f , then the Cotton tensor, the Bach tensor and the tensor D satisfy the following conditions Cijk + ft Wtijk n−3 (n − 2)Bij − ft Cjit n−2 Rkt Ckti 1 2 |C| + Rkt Ckti,i 2

= Dijk ,

(7.16)

= Dijk,k ,

(7.17)

= (n − 2)Ditk,tk ,

(7.18)

= (n − 2)Ditk,tki .

(7.19)

Proof. We begin with the conditions (7.16) and (7.17), which were first obtained in [34]. Taking the covariant derivative of the soliton equation (7.3), we obtain Rij,k + fijk = 0. Skew-symmetrizing, recalling the commutation relation fijk − fikj = ft Rtijk and using (1.77) we deduce that Rij,k − Rik,j = ft Rtikj n = ft Wijkj +

(7.20) 1 (Rtk gij − Rtj gik + Rij gtk − Rik gtj ) n−2 o R − (gtk gij − gtj gik ) (n − 1)(n − 2) 1 = −ft Wtijk + (ft Rtk gij − ft Rtj gik + fk Rij − fj Rik ) n−2 R − (fk gij − fj gik ). (n − 1)(n − 2)

Now we insert in the previous equation the expressions defining the Cotton tensor (1.83) and of the tensor D, deducing (7.16). In order to prove (7.17), we take the divergence of (7.16): Cijk,k + ftk Wtijk + ft Wtijk,k = Dijk,k .

(7.21)

Inserting in (7.21) the definition of the Bach tensor (1.86), the soliton equation (7.3) and (1.82), and employing the fact that the Weyl tensor is totally trace-free, equation (7.17) follows immediately.

7.2. Rigidity I: pointwise conditions

167

To show (7.18), we take the covariant derivative of equation (7.17), obtaining (n − 2)Bij,k −

n−3 (ftk Cjit + ft Cjit,k ) = Dijt,tk , n−2

which in view of the soliton equation and the fact that Cjit = −Cjti implies that (n − 2)Bij,k −

n−3 (λCjik + Rtk Cjti + ft Cjit,k ) = Dijt,tk . n−2

Taking the trace with respect to j and k and using equation (1.84) and the fact that the Cotton tensor is totally trace-free we get (n − 2)Bik,k −

n−3 Rtk Ckti = Dikt,tk . n−2

Now using (1.89) and the fact that Dijk = −Dikj in the previous relation, we obtain (7.18). Finally, to prove (7.19) we take the divergence of (7.18), Rkt,i Ckti + Rkt Ckti,i = (n − 2)Ditk,tki , and then use the symmetry of the Cotton tensor to obtain 1 (Rkt,i − Rki,t )Ckti + Rkt Ckti,i = (n − 2)Ditk,tki . 2 This immediately yields (7.19).

7.2



Rigidity I: pointwise conditions

7.2.1 Compact shrinkers with strictly positive sectional curvature In this first section we investigate the case of gradient shrinking Ricci solitons with positive sectional curvature; the main reference is the paper [45]. It is well known that (compact) Einstein manifolds can be classified, if they are sufficiently positively curved. Sufficient conditions are non-negative curvature operator (S. Tachibana [141]), non-negative isotropic curvature (M. J. Micallef and Y. Wang [118] in dimension four and S. Brendle [23] in every dimension), and weakly 14 -pinched sectional curvature [9] (if Sec and R denote the sectional and the scalar curvature, respectively, this condition in dimension four is implied by 1 Sec ≥ 24 R). Moreover, in dimension four, it was proved by D. Yang [149]) that four-dimensional Einstein manifolds satisfying Sec ≥ √εR are isometric to either S4 , RP4 , or CP2 with their standard metrics, if ε = 1249−23 . The lower bound 480 √ 2− 2 1 was improved to ε = 24 by E. Costa [66] and, more recently, to ε = 48 by E. Ribeiro [137]. It is conjectured in [149] that the result should be true if one assumes positive sectional curvature.

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Chapter 7. Ricci Solitons: Selected Results

In dimension n ≤ 3, complete shrinking Ricci solitons are classified. The last years have seen a lot of interesting results concerning the classification of positively-curved shrinking Ricci solitons. For instance, it follows by the work of C. B¨ ohm and B. Wilking [14] that the only compact shrinking Ricci solitons with positive (2-positive) curvature operator are quotients of Sn . In dimension four, A. Naber [125] classified complete shrinkers with non-negative curvature operator. Four-dimensional shrinkers with non-negative isotropic curvature were classified by X. Li, L. Ni and K. Wang [112]. Recently, O. Munteanu and J. P. Wang [124] showed that every complete shrinking Ricci soliton with positive sectional curvature is compact. It is natural to ask the following question: given ε > 0, are there four-dimensional non-Einstein shrinking Ricci solitons satisfying Sec ≥ εR?. Here we give an answer to this question, proving the following Theorem 7.8. Let (M 4 , g) be a four-dimensional complete gradient shrinking Ricci 1 soliton with Sec ≥ 24 R. Then (M 4 , g) is necessarily Einstein, thus isometric to 4 either R (and quotients), S4 , RP4 or CP2 with their standard metrics. Note that, by the work of S. Brendle and R. Schoen [25], using the Ricci flow, one can show that compact Ricci shrinkers with weakly 14 -pinched sectional curvature are isometric to S4 , RP4 , or CP2 with their standard metrics. The 1 condition Sec ≥ 24 R is a little stronger, but the proof of Theorem 7.8 that we present is completely “elliptic”. Proof. By Lemma 7.3, either g is flat or R > 0. In this second case, by a result in [124], we know that M 4 must be compact. From now on we can assume that (M 4 , g) is compact with Sec ≥ εR > 0. Lemma 7.2 gives 1 ˚ 2 = |∇Ric| ˚ 2 + 2λ|Ric| ˚ 2 − 2Rijkl R ˚ik R ˚jl − 1 R|Ric| ˚ 2. ∆f |Ric| 2 2 Integrate over M 4 and using equation (7.4) we obtain Z Z ˚ 2 , ∇f i dV + |∇Ric| ˚ 2 dV + 2 λ|Ric| ˚ 2 dV h∇|Ric| Z Z 1 ˚ ˚ ˚ 2 dV − 2 Rijkl Rik Rjl dV − R|Ric| 2 Z Z ˚ 2 dV − 2 Rijkl R ˚ik R ˚jl dV . = |∇Ric|

0=

1 2

Z

(7.22)

On the other hand, given a1 , a2 , b1 , b2 , b3 ∈ R, we define the three-tensor ˚ij + a1 ∇j R ˚ik + a2 ∇i R ˚jk + b1 ∇k Rgij + b2 ∇j Rgik + b3 ∇i Rgjk . Fijk := ∇k R

7.2. Rigidity I: pointwise conditions

169

˚ij = 1 ∇j R gives A computation using the Bianchi identity ∇i R 4 ˚ 2 + 2(a1 + a2 + a1 a2 )∇k R ˚ij ∇j R ˚ik |F |2 = (1 + a21 + a22 )|∇Ric|  1 + a1 (b1 + b3 ) + a2 (b1 + b2 ) + b2 + b3 + 8(b21 + b22 + b23 ) 2  + 4(b1 b2 + b1 b3 + b2 b3 ) |∇R|2 .

In particular, Z ˚ 2 dV = |∇Ric|

Z Z 1 2(a1 + a2 + a1 a2 ) 2 ˚ij ∇j R ˚ik dV |F | dV − ∇k R 1 + a21 + a22 1 + a21 + a22 Z Q0 − |∇R|2 dV 2(1 + a21 + a22 ) Z Z 1 2(a1 + a2 + a1 a2 ) 2 ˚ij ∇j R ˚ik dV = |F | dV − ∇k R (7.23) 1 + a21 + a22 1 + a21 + a22 Z ` ˚ 2 dV , − R|Ric| 1 + a21 + a22

where for brevity we denoted Q0 := a1 (b1 + b3 ) + a2 (b1 + b2 ) + b2 + b3 + 8(b21 + b22 + b23 ) + 4(b1 b2 + b1 b3 + b2 b3 ) and in the last equality we used Lemma 7.4. On the other hand, integrating by parts and commuting the covariant derivatives, one has Z Z ˚ij ∇j R ˚ik dV = − R ˚ij ∇k ∇j R ˚ik dV ∇k R Z   ˚ij ∇j ∇k R ˚ik + Rkjil R ˚ij R ˚kl + Rij R ˚ik R ˚jl dV =− R  Z  1˚ 1 2 ˚ ˚ ˚ ˚ ˚ ˚ =− Rij ∇i ∇j R − Rijkl Rik Rjl + Rij Rik Rjl + R|Ric| dV 4 4  Z  1 1 2 2 ˚ ˚ ˚ ˚ ˚ ˚ = |∇R| + Rijkl Rik Rjl − Rij Rik Rjl − R|Ric| dV 16 4  Z  ˚ik R ˚jl − R ˚ij R ˚ik R ˚jl − 1 R|Ric| ˚ 2 dV . = Rijkl R (7.24) 8 From equation (7.23), we obtain Z Z 1 ˚ 2 dV = |∇Ric| |F |2 dV 1 + a21 + a22 Z  2(a1 + a2 + a1 a2 )  ˚ik R ˚jl − R ˚ij R ˚ik R ˚jl dV − R R ijkl 1 + a21 + a22 Z ˚ 2 dV, + Q1 R|Ric|

170

Chapter 7. Ricci Solitons: Selected Results

with Q1 :=

a1 + a2 + a1 a2 Q0 − . 2 2 4(1 + a1 + a2 ) 1 + a21 + a22

Using this inequality in (7.22), we obtain that Z Z 1 2(1 + a21 + a22 + a1 + a2 + a1 a2 ) 2 ˚ik R ˚jl dV 0= |F | dV − Rijkl R 1 + a21 + a22 1 + a21 + a22 (7.25) Z Z 2(a1 + a2 + a1 a2 ) ˚ij R ˚ik R ˚jl dV + Q1 R|Ric| ˚ 2 dV . + R 1 + a21 + a22 By Corollary 5.18, ˚ik R ˚jl ≤ 1 − 16ε R|Ric| ˚ 2 − (3 − 4s)R ˚ij R ˚ik R ˚jk Rijkl R 4

(7.26)

for every s ∈ [0, 1]. Thus, if a1 + a2 + a1 a2 ≥ 0, then for every s ∈ [0, 1], estimate (7.25) gives Z 1 0≥ |F |2 dV 1 + a21 + a22   2 (3 − 4s)(1 + a21 + a22 ) + 4(1 − s)(a1 + a2 + a1 a2 ) Z ˚ij R ˚ik R ˚jl dV + R 1 + a21 + a22 Z ˚ 2 dV + Q2 R|Ric| (7.27) with (1 − 16ε)(1 + a21 + a22 + a1 + a2 + a1 a2 ) 2(1 + a21 + a22 ) a1 + a2 + a1 a2 (1 − 16ε)(1 + a21 + a22 + a1 + a2 + a1 a2 ) Q0 = − − . 2 2 4(1 + a1 + a2 ) 2(1 + a21 + a22 ) 1 + a21 + a22

Q2 := Q1 −

Now, choose a1 = a2 = 1 and b1 = b2 = b3 =: b. Then Q2 = −12b2 − 2b + 16ε −

3 . 4

In particular, the maximum is attained for b = −1/12 and is given by Q2 =

48ε − 2 . 3

(7.28)

Actually, a (long) computation gives that the maximum of the function Q2 defined for general variables (a1 , a2 , b1 , b2 , b3 ) is attained at the point   1 1 1 (a1 , a2 , b1 , b2 , b3 ) = 1, 1, − , − , − (7.29) 12 12 12

7.2. Rigidity I: pointwise conditions

171

and is given by the value (7.28). Moreover, under the choice (7.29), one has   2 (3 − 4s)(1 + a21 + a22 ) + 4(1 − s)(a1 + a2 + a1 a2 ) = 2(7 − 8s) . 1 + a21 + a22 In particular, choosing s=

7 , 8

we obtain from (7.27) that Z Z 1 48ε − 2 ˚ 2 dV . 0≥ |F |2 dV + R|Ric| 3 3 ˚ ≡ 0, i.e., (M 4 , g) is Einstein. By Berger’s classification Thus, if ε > 1/24, then Ric result [9], we conclude the proof of Theorem 7.8 in this case. If ε = 1/24, then Q1 = 1/3, Q2 = 0 and all previous inequalities become equalities. In particular, F ≡ 0. Moreover, from (7.26), we get ˚ik R ˚jl ≡ 1 R|Ric| ˚ 2 Rijkl R 12

˚ij R ˚ik R ˚jk ≡ 0 . and R

(7.30)

Equation (7.24) and Lemma 7.4 imply that Z Z Z 1 1 2 ˚ ˚ ˚ ∇k Rij ∇j Rik dV = − R|Ric| dV = − |∇R|2 dV. 24 48 Thus, equation (7.23) gives Z

˚ 2 dV = 1 |∇Ric| 12

Z

|∇R|2 dV.

(7.31)

Now, to complete the proof we have to use the fact that F ≡ 0, i.e., ˚ij + ∇j R ˚ik + ∇i R ˚jk − 1 (∇k Rgij + ∇j Rgik + ∇i Rgjk ) . 0 = ∇k R 12 ˚ij , we obtain Taking the divergence in k and contracting with R   1 ˚ ˚ ˚ ˚ 0 = Rij ∆Rij + ∇k ∇j Rik + ∇k ∇i Rjk − (∆Rgij + 2∇i ∇j R) 12   1 1 2 2 ˚ ˚ ˚ ˚ ˚ ˚ik R ˚jl = ∆|Ric| − |∇Ric| + Rij ∇j ∇k Rik + ∇i ∇k Rjk − ∇i ∇j R − 2Rijkl R 2 6 1 ˚ 2 − |∇Ric| ˚ 2 + 1R ˚ij ∇i ∇j R − 2Rijkl R ˚ik R ˚jl = ∆|Ric| 2 3 1 ˚ 2 − |∇Ric| ˚ 2 + 1R ˚ij ∇i ∇j R − 1 R|Ric| ˚ 2, = ∆|Ric| 2 3 6

172

Chapter 7. Ricci Solitons: Selected Results

where we used (7.30). Integrating by parts over M , using (7.31), we obtain Z Z 1 1 ˚ 2 dV 0=− |∇R|2 dV − R|Ric| 6 6 ˚ ≡ 0, i.e., (M 4 , g) is Einstein and the assertion follows again by which implies Ric Berger’s result. This concludes the proof of Theorem 7.8. 

7.2.2 Further results in the non-necessarily gradient case In this short section we recall, without proof, some other results concerning the classification of non-necessarily gradient Ricci solitons satisfying some pointwise curvature conditions. We refer to [57] and [115] for details. Contrary to the case of gradient solitons, in the literature relatively little is known about generic Ricci solitons, that is, when X is not necessarily the gradient of a potential f . For instance, as we recalled before, it is well known that generic expanding and steady compact Ricci solitons are trivial. It is also relevant pointing out that on a compact manifold shrinking Ricci solitons always support a gradient soliton structure, [131], and that every complete non-compact shrinking Ricci soliton with bounded curvature supports a gradient soliton structure, [125]. Ricci solitons that do not support a gradient soliton structure were found by J. Lauret, [107], P. Baird [7] and P. Baird and L. Danielo, [8]. A first important difference is that, in the general case, we cannot make use of the weighted manifold structure (M, g, e−f dV ) which arises naturally when dealing with gradient solitons. There are also limitations to the applicability of analytical tools, such as the weak maximum principle for the diffusion operator ∆f , weighted Lp Liouville-type theorems and a priori estimates that have been considered in previous investigations (see, e.g., [136], [135]). Nevertheless, in the general case the soliton structure is encoded in the geometry of an appropriate operator ∆X , which we call “X–Laplacian”, and which is defined for u ∈ C 2 (M ) by ∆X u = ∆u − hX, ∇ui . Clearly, if X is the gradient of a function f , ∆X reduces to the f -Laplacian. On the other hand, for generic Ricci solitons neither (7.5) nor Hamilton’s identity (7.6) are available. In the case of (7.5) this is technically due to the fact that, in the generic setting, the symmetry of Hess(f ), i.e., Hess(f )(Y, Z) = Hess(f )(Z, Y ) for every smooth vector fields Y and Z, is replaced by the much more involved “commutation rule” g(∇Y X, Z) = 2λg(Y, Z) − 2 Ric(Y, Z) − g(∇Z X, Y ). Nevertheless, even in this more general situation, some important equations valid for gradient solitons still hold, basically in the same form: indeed we have (see Lemma 2.1 in [57])

7.2. Rigidity I: pointwise conditions

173

Lemma 7.9. Let (M, g, X) be a generic Ricci soliton. Then 1 R2 ˚ 2, ∆X R = λR − | Ric |2 = λR − − |Ric| 2 n   1 ˚ 2 = |∇Ric| ˚ 2 + 2 λ − n − 2 R |Ric| ˚ 2 ∆X |Ric| 2 n(n − 1) 4 ˚ ˚ ˚ ˚ki R ˚sj Wksij . + Rij Rjk Rki − 2R n−2 The main results in [57] are the following: Theorem 7.10. Let (M, g, X) be a complete, generic Ricci soliton of dimension n and scalar curvature R. Set R∗ = inf M S. (i) If λ < 0, then nλ ≤ R∗ ≤ 0. Furthermore, if S(x0 ) = R∗ = nλ for some x0 ∈ M , then (M, g) is Einstein and X is a Killing field; on the other hand, if R(x0 ) = R∗ = 0 for some x0 ∈ M , then (M, g) is Ricci flat and X is a homothetic vector field. (ii) If λ = 0, then R∗ = 0. Furthermore, if R(x0 ) = R∗ = 0 for some x0 ∈ M , then (M, g) is Ricci flat and X is a Killing field. (iii) If λ > 0, then 0 ≤ R∗ ≤ nλ. Furthermore, if R(x0 ) = R∗ = 0 for some x0 ∈ M , then (M, g) is flat and X is a homethetic vector field, while R∗ < nλ unless M is compact, Einstein and X is a Killing field. Theorem 7.11. Let (M, g, X) be a complete generic shrinking Ricci soliton of dimension three. Furthermore, if M is non-compact, assume that the scalar curvature is bounded and |∇X| = o(|X|) as r → ∞. Then (M, g) is isometric to a finite quotient of either S3 , R × S2 , or R3 (here r is the distance function from a fixed origin o ∈ M ). In higher dimensions Theorem 7.11 generalizes to Theorem 7.12. Let (M, g, X) be a complete generic shrinking Ricci soliton of dimension n > 3. Furthermore, if M is non-compact, assume that the scalar curvature is bounded and |∇X| = o(|X|) as r → ∞. If, for some Λ > 0, | Ric | ≤ Λ S and !2 r 2(n − 1) 1 ˚ −p |W | R ≤ |Ric| R , (7.32) n−2 n(n − 1) then (M, g) is isometric to a finite quotient of either Sn , R × Sn−1 , or Rn . Theorems 7.11 and 7.12 extend to the non-gradient case the previous results in [131], [32] and [40], and provide results in the non-conformally flat case, which was treated in [48].

174

7.3

Chapter 7. Ricci Solitons: Selected Results

Rigidity II: integral conditions

7.3.1 The compact case: integral pinching conditions In this section we investigate compact gradient shrinking Ricci solitons satisfying an Ln/2 -pinching condition; we follow the presentation in [43]. The idea is to extend the technique used in Chapter 6 to the case of Einstein manifolds. In dimension four we have the following L2 -pinching result: Theorem 7.13. Every four-dimensional compact shrinking Ricci soliton satisfying Z Z 2 ˚ 2 dVg < 1 Y (M, [g])2 |W | dVg + |Ric| 48 M M is isometric to a quotient of the round S4 . As a corollary, using a lower bound for the Yamabe invariant proved in [83] (see Lemma 3.16 in Chapter 3), we get the following result: Corollary 7.14. Every four-dimensional compact shrinking Ricci soliton satisfying Z Z Z 5 1 2 2 ˚ |W | dVg + |Ric| dVg ≤ R2 dVg 4 48 M M M is isometric to a quotient of the round S4 . Remark 7.15. We will prove, at the end of the section, that the pinching condition in Corollary 7.14 is equivalent to Z Z 2 160 2 |W |2 dVg + R2 dVg ≤ π χ(M ) , 39 13 M M where χ(M ) is the Euler-Poincar´e characteristic of M . We notice that in Corollary 7.14 and Remark 7.15 we only have to assume the not-strict inequality. In fact, when equality occurs we can show that (M n , g) has to be conformally Einstein, in particular Bach-flat, and using [34] the conclusion follows. Theorem 7.13 is the four-dimensional case of the following result which holds in every dimension 4 ≤ n ≤ 6. Theorem 7.16. Let (M n , g) be a n-dimensional (4 ≤ n ≤ 6) compact shrinking Ricci solitons satisfying ! n2 s √ n/2 Z 2 2 (n − 4)2 (n − 1) ˚ W + √ dVg Ric ∧ g + λV (M ) n 8(n − 2) n(n − 2) M s n−2 < Y (M, [g]) . 32(n − 1) Then (M n , g) is isometric to a quotient of the round Sn . Moreover, in dimension 5 ≤ n ≤ 6, the same result holds only assuming the not-strict inequality.

7.3. Rigidity II: integral conditions

175

It is easy to see that, on a shrinking Ricci soliton of dimension n ≥ 7, the pinching condition does not hold, since λV (M )

2 n

2−n 1 = V (M ) n n

Z R dVg ≥ M

1 Y (M, [g]) . n

The proof of Theorem 7.16 is inspired by the classification of Einstein (or locally conformally flat) metrics satisfying an Ln/2 -pinching condition (see Chapter 6, Theorem 6.5). More precisely, we use the soliton equation to obtain an elliptic PDE ˚ 2 ; since every compact shrinking soliton has positive scalar curvature, the for |Ric| positivity of the Yamabe invariant Y (M, [g]) implies a Sobolev-type inequality on ˚ which, combined with the PDE, allows us to get an Ln/2 -estimate on the |Ric| curvature on every non-Einstein shrinking Ricci solitons. In doing this, we prove an algebraic curvature estimate (see Proposition 3.6) which holds in every dimension and was first observed in dimension four in [16]. The pinching assumption of Theorem 7.16 implies that the manifold (M n , g) has to be Einstein, with “small” Ln/2 -norm of the Weyl tensor. The final result then follows using again Theorem 6.5. Proof of Theorem 7.16. Let us assume that 4 ≤ n ≤ 6 and (M n , g) satisfies the integral pinching condition as in Theorem 7.16 ! n2 s √ n/2 2 2 (n − 4)2 (n − 1) ˚ W + √ Ric ∧ g dVg + λV (M ) n 8(n − 2) n(n − 2) M s n−2 < Y (M, [g]) , 32(n − 1)

Z

with V (M ) :=

R M

(7.33)

dVg . By Lemma 7.2,

1 ˚ 2 = |∇Ric| ˚ 2 + 2λ|Ric| ˚ 2 − 2Wijkl R ˚ik R ˚jl ∆f |Ric| 2 4 ˚ ˚ ˚ 2(n − 2) ˚ 2. + Rij Rjk Rik − R|Ric| n−2 n(n − 1) ˚ 2 ≥ ∇|Ric| ˚ 2 at every point where |Ric| ˚ 6= Using the Kato inequality (3.26) (|∇Ric| 0) and Proposition 3.6, we obtain 1 ˚ 2 + |∇|Ric|| ˚ 2 + 2λ|Ric| ˚ 2 0 ≥ − ∆f |Ric| 2 r  1/2 2(n − 2) 8 2 2 ˚ ˚ 2 − 2(n − 2) R|Ric| ˚ 2. − |W | + |Ric| |Ric| n−1 n(n − 2) n(n − 1)

176

Chapter 7. Ricci Solitons: Selected Results

Integrating by parts over M n and using equation (7.4) we obtain Z Z Z 1 ˚ 2 ∆f dVg + ˚ 2 dVg + 2λ ˚ 2 dVg 0≥− |Ric| |∇|Ric|| |Ric| 2 M M M r 1/2 Z  2(n − 2) 8 ˚ 2 ˚ 2 dVg − |W |2 + |Ric| |Ric| n−1 M n(n − 2) Z 2(n − 2) ˚ 2 dVg − R|Ric| n(n − 1) M Z Z n−4 2 ˚ ˚ 2 dVg = |∇|Ric|| dVg − λ |Ric| 2 M M r 1/2 Z  2(n − 2) 8 ˚ 2 ˚ 2 dVg |W |2 + |Ric| |Ric| − n−1 M n(n − 2) Z n2 − 5n + 8 ˚ 2 dVg + R|Ric| 2n(n − 1) M ˚ we Using the Yamabe-Sobolev inequality (5.2) from Chapter 5 with u := |Ric| further get Z  n−2 Z n 2n n−2 n−4 ˚ ˚ 2 dVg 0≥ Y (M, [g]) |Ric| n−2 dVg − λ |Ric| 4(n − 1) 2 M M r 1/2 Z  2(n − 2) 8 2 2 ˚ ˚ 2 dVg − |W | + |Ric| |Ric| n−1 M n(n − 2) Z (n − 4)2 ˚ 2 dVg + R|Ric| 4n(n − 1) M Now, since λ > 0 and n ≥ 4, H¨older’s inequality yields h n−2 2 n−4 0≥ Y (M, [g]) − λV (M ) n 4(n − 1) 2 ! n2 r n/4 Z  i 2(n − 2) 8 2 2 ˚ − |W | + |Ric| dVg n−1 n(n − 2) M Z  n−2 Z n 2n (n − 4)2 ˚ ˚ 2 dVg . n−2 · |Ric| dVg + R|Ric| 4n(n − 1) M M ˚ ≡ 0, i.e., (M n , g) is Einstein, or the following estimate holds: Thus, either |Ric| ! n2 s n/4 Z  2 8 (n − 4)2 (n − 1) 2 2 ˚ |W | + |Ric| dVg + λV (M ) n n(n − 2) 8(n − 2) M s n−2 ≥ Y (M, [g]) . 32(n − 1)

7.3. Rigidity II: integral conditions

177

Since W is totally trace-free, one has 2 √ 2 8 ˚ ˚ 2 Ric ∧ g = |W |2 + |Ric| W + √ n(n − 2) n(n − 2) and the pinching condition (7.33) implies that (M n , g) is Einstein. Moreover, observe that in dimension n = 5, 6, we reach the same conclusion if we assume just the not-strict inequality in (7.33). Since g is Einstein, by Obata’s Theorem [128], g is a (the) Yamabe metric of [g], i.e. Z 2−n n Y (M, [g]) = V (M ) R dVg . M

Moreover, integrating equation (7.4) one has Z 2−n 2 1 1 n n λV (M ) = V (M ) R dVg = Y (M, [g]) . n n M Hence, the pinching condition (7.33) implies that Z  n2 n 8n − n2 − 8 2 p |W | dVg ≤ Y (M, [g]) . 4n 2(n − 1)(n − 2) M The conclusion now follows from Theorem 6.5.

(7.34) 

Proof of Corollary 7.14 and Remark 7.15. Using the lower bound for the Yamabe invariant given in Lemma 3.16 of Chapter 3, we get Z Z ˚ 2 dVg − 1 Y (M, [g])2 |W |2 dVg + |Ric| 48 M M Z Z Z 5 ˚ 2 dVg − 1 ≤ |W |2 dVg + |Ric| R2 dVg . 4 M 48 M M Moreover, the inequality is strict unless (M 4 , g) is conformally Einstein. In the first case, Theorem 7.13 immediately implies Corollary 7.14. In the second case, the fact that g is conformally Einstein implies that (M 4 , g) is Bach flat (see Lemma 5.3 in Chapter 5). Since M 4 is compact, by a result of H.-D. Cao and Q. Chen [34] it follows that (M 4 , g) has to be Einstein. Following the proof of Theorem 7.16, we can see that the pinching condition together with Theorem 6.5 concludes the proof of Corollary 7.14. Finally, observe that by the Chern-Gauss-Bonnet formula (3.23), the righthand side can be written as Z Z Z 5 ˚ 2 dVg − 1 |W |2 dVg + |Ric| R2 dVg 4 48 M M ZM Z 13 1 = |W |2 dVg + R2 dVg − 20π 2 χ(M ) . 8 M 12 M This proves Remark 7.15.



178

Chapter 7. Ricci Solitons: Selected Results

7.3.2 The non-compact case: L1 conditions and integral curvature decay In this section we prove, following [53], new classification results of gradient expanding and steady solitons in dimension three and above, under integral assumptions on the scalar curvature. Note that similar conditions have been considered by Deruelle [71] in the steady case (although he employs a completely different approach; see also the recent [122]). More precisely, we prove the following Theorem 7.17. Let (M n , g) be a complete gradient expanding Ricci soliton of dimension n ≥ 3 with non-negative sectional curvature. If R ∈ L1 (M n ), then M n is isometric to a quotient of the Gaussian soliton Rn . Theorem 7.18. Let (M n , g) be a complete gradient steady Ricci soliton of dimension n ≥ 3 with non-negative sectional curvature. Suppose that Z 1 lim inf R = 0. r→+∞ r B (o) r Then, M n is isometric to a quotient of Rn or Rn−2 × Σ2 , where Σ2 is the cigar soliton. In the three-dimensional case we can prove an analogous results under weaker assumptions.
Theorem 7.19. Let M 3 , g be a three-dimensional complete gradient expanding Ricci soliton with non-negative Ricci curvature. If R ∈ L1 (M 3 ), then M 3 is isometric to a quotient of the Gaussian soliton R3 . In particular, in dimension three the non-negativity assumption on the curvature is automatically satisfied (see [61]), implying
Theorem 7.20. Let M 3 , g be a three-dimensional complete gradient steady Ricci soliton. Suppose that Z 1 lim inf R = 0. r→+∞ r B (o) r Then M 3 is isometric to a quotient of R3 or R × Σ2 , where Σ2 is the cigar soliton. Remark 7.21. As it will be clear from the proofs of Theorems 7.17 and 7.19, instead of R ∈ L1 (M n ) we can assume that Z lim inf R = 0. r→+∞

B2r (o)\Br (o)

Remark 7.22. The quantity 1 lim inf r→+∞ r

Z R Br (o)

(7.35)

7.3. Rigidity II: integral conditions

179

that appears in Theorems 7.18 and 7.20 is independent of the choice of the center o ∈ M n . Moreover, note that our assumptions in the steady case do not imply a priori that the scalar curvature goes to zero at infinity, in contrast with the results in [71]. In fact, in [71] it is assumed that R ∈ L1 (M n ). This, under the hypothesis that the steady Ricci soliton has non-negative sectional curvature, implies that the scalar curvature is non-negative, bounded, and globally Lipschitz, and thus that R → 0 at infinity. As a consequence of the integral decay estimate in [71] (see Lemma 7.34), the assumption in Theorems 7.18 and 7.20 holds if g has less than quadratic volume growth, i.e., Vol (Br (o)) = o(r2 ) as r → +∞. This immediately implies the following Corollary 7.23. The only complete gradient steady Ricci solitons of dimension n ≥ 3 with non-negative sectional curvature and less than quadratic volume growth are quotients of Rn or of Rn−2 × Σ2 . Corollary 7.24. The only three-dimensional complete gradient steady Ricci solitons with less than quadratic volume growth are quotients of R3 or of R × Σ2 . We note that the conditions in Theorem 7.20 and Corollary 7.24 are sharp: in fact, the steady Bryant soliton has positive sectional curvature, linear curvature decay and quadratic volume growth, hence Z 1 lim inf R r→+∞ r B (o) r is finite and strictly positive. The three-dimensional rotationally symmetric expanding example constructed by Bryant in [26] (see also the appendix in [64]) has positive sectional curvature, quadratic curvature decay at infinity and Euclidean volume growth. Thus, Z R ≥ Cr Br (o)

for some positive constant C. This suggests that a sharp condition under which one could hope to improve Theorem 7.19 would be Z 1 lim inf R = 0. r→+∞ r B (o) r We emphasize that Theorems 7.18 and 7.20 improve a result in [71], while to the best of our knowledge the results in the expanding case are completely new and should be compared with [136, Theorem 4] and [133, Theorem 4.5], where the required integral conditions involve the measure e−f dµ, and the weight e−f , under mild assumptions on the curvature, has exponential growth (see, e.g., [31, Lemma 5.5]). We also note that, in dimension three, the condition Z 1 lim inf R≥k>0 (7.36) r→+∞ r B (o) r

180

Chapter 7. Ricci Solitons: Selected Results

implies the k-non-collapsing of balls with sufficiently large radii, a priori nonuniformly with respect to the center. It would be extremely interesting either to show that the only three-dimensional gradient steady Ricci soliton satisfying (7.36) is, up to scaling, the Bryant soliton, or to construct a (k-collapsed) counterexample. The first case, together with Theorem 7.20, would complete the classification of steady solitons in dimension three. One of the main tools in our analysis is a geometric (0, 2)-tensor that we call b defined as the weighted Einstein tensor E,
b = Ric − 1 R g e−f , E (7.37) 2 where f is the soliton potential. The weighted Einstein tensor appeared for the b is a Codazzi tensor on every first time in [49], where the authors observed that E gradient three-dimensional Ricci soliton. Here we prove, in Section 7.3.2, that b satisfies the Weitzenb¨ock formula every gradient Ricci soliton E 1 b2 b 2 − 1 h∇|E| b 2 , ∇f i − (n − 2)λ|E| b 2 + Q, ∆|E| = |∇E| 2 2

(7.38)

where Q is a cubic curvature term. Quite surprisingly, we are able to show that this quantity enjoys nice algebraic properties under suitable curvature assumptions; namely, we prove that Q ≥ 0 if the sectional curvature (or the Ricci curvature, in dimension three) is non-negative, and we completely characterize the equality case. We highlight the fact that equation (7.38) is only effective when λ ≤ 0; this feature allows us to exploit, in the expanding and steady cases, a technique reminiscent of those used to prove earlier results concerning gradient shrinking Ricci solitons (see e.g. [126], [134], [40], [123], [148]). A Weitzenb¨ ock formula for the weighted Einstein tensor b defined in (7.37) we prove the following For the weighted Einstein tensor E Proposition 7.25. Let (M n , g) be a complete gradient Ricci soliton of dimension n ≥ 3. Then 1 b2 b 2 − 1 h∇|E| b 2 , ∇f i − (n − 2)λ|E| b 2 − 2 Rm(E, b E) b ∆|E| = |∇E| 2 2 
n−2 b 2 − 1 tr(E) b 2 , − R |E| 2 n−2

(7.39)

b E) b = Rijkl E bik E bjl and tr denotes the trace with respect to the metric where Rm(E, g. Proof. From (7.37) we have, in a local orthonormal frame, bij = Rij − 1 Rδij , ef E 2

7.3. Rigidity II: integral conditions

181

which implies, taking the covariant derivative, that   bij + E bij,k = Rij,k − 1 Rk δij . e f fk E 2 Taking the divergence in (7.40) we get   1 2b b b b ef |∇f | E + 2f E + ∆f E + E ij k ij,k ij ij,kk = Rij,kk − ∆Rδij . 2

(7.40)

(7.41)

b (which implies that Rij = Using Lemma 7.1 in equation (7.41), the definition of E 2 (n−4) 2 1 f b 2f b 2 e Eij + Rδij and |Ric| = e |E| − R ) and simplifying we deduce that 2

4



bij,kk + fk E bij,k + (n − 2)λE bij ef E




bkt Rikjt − (n − 2) R2 δij + e2f |E| b 2 δij . = −2ef E 4

(7.42)

bij , observing that tr(E) b =E btt = − (n−2) Re−f , and Now we contract (7.42) with E 2 obtain bij E bij,kk = − 1 h∇|E| b 2 , ∇f i − (n − 2)λ|E| b 2 − 2 Rm(E, b E) b E 2  (7.43)  2  (n − 2) b2− 1 b − R |E| tr(E) , 2 n−2 b 2 = |∇E| b 2+E bij E bij,kk . which easily implies (7.39) because 12 ∆|E| 
b 2 = |Ric|2 + (n−4) R2 e−2f Proposition 7.25 can in princiRemark 7.26. Since |E| 4 ple be related to the evolution equation of the squared norm of the Einstein tensor under the Ricci flow which is essentially contained in [91]. Corollary 7.27. Let (M n , g) be a complete gradient Ricci soliton of dimension n ≥ 3. Then 1 b2 b 2 − 1 h∇|E| b 2 , ∇f i − (n − 2)λ|E| b 2 + Q, ∆|E| = |∇E| 2 2

(7.44)

where Q := e

−2f



 (n − 2)3 3 (n − 2)(n − 4) 2 ˚ ˚ ˚ R − 2Rikjt Rij Rkt − R|Ric| , 4n2 2n

˚ is the traceless Ricci tensor. and Ric b we deduce that Proof. Using the definition of E, 2 bij E bkt e2f = −2Rikjt R ˚ij R ˚kt + 2(n − 2) R|Ric| ˚ 2 − (n − 2) R3 . −2Rikjt E n 2n2

Now the claim follows by inserting this expression into (7.39).



182

Chapter 7. Ricci Solitons: Selected Results In the particular case of dimension three we have

Corollary 7.28. Let (M 3 , g) be a three-dimensional complete gradient Ricci soliton. Then 1 b2 b 2 − 1 h∇|E| b 2 , ∇f i − λ|E| b 2 + Q, ∆|E| = |∇E| (7.45) 2 2 where   7 3 Q = e−2f 4Rij Rjk Rki − R| Ric |2 + R3 . 2 4 Proof. The proof of the corollary is a simple computation using the fact that, in dimension three, ˚ 2 = | Ric |2 − 1 R2 , |Ric| 3 ˚ij R ˚kt = Rijkt Rij Rkt − 2 R|Ric|2 + 1 R3 Rijkt R 3 9 and Rijkt = Rik δjt − Rit δjk + Rjt δik − Rjk δit −

R (δik δjt − δit δjk ). 2



In the next proposition we prove the main integral estimate that will be used in the proof of our results. Proposition 7.29. Let (M n , g) be a complete gradient Ricci soliton of dimension n ≥ 3. Then, either (M n , g) is Ricci flat or, for every non-negative cutoff function ϕ with compact support in M n , we have Z M

  Z b 2 ϕ3 ef Q − (n − 2)λ|E| b ∇ϕiϕ2 ef dV. dV ≤ −3 h∇|E|, b |E| M

(7.46)

Proof. For every ε ≥ 0, define b Ωε := {x ∈ M n | |E(x)| ≥ ε} and let ( b |E(x)|, if x ∈ Ωε , hε (x) := ε, if x ∈ M \ Ωε . Let ϕ be a smooth non-negative cutoff function with compact support in M . 3 f n Multiplying equation (7.44) by h−1 ε ϕ e and integrating on M , we obtain 1 2

Z M

b 2 ϕ3 ef ∆|E| dV = hε

Z b ∇hε i|E|ϕ b 3 ef b ∇ϕi|E|ϕ b 2 ef h∇|E|, h∇|E|, dV − 3 dV 2 hε hε M M Z b ∇f i|E|ϕ b 3 ef h∇|E|, − dV. hε M Z

7.3. Rigidity II: integral conditions

183

b on Ωε and ∇hε = 0 on M \ Ωε , Since hε = |E| Z Z Z b 2 ϕ3 ef b ∇ϕi|E|ϕ b 2 ef 1 ∆|E| |∇hε |2 ϕ3 ef h∇|E|, dV = dV − 3 dV 2 M hε hε hε M M Z b ∇f i|E|ϕ b 3 ef h∇|E|, − dV. hε M Equation (7.44) and Kato’s inequality (3.26) yield Z Z Z b 2 ϕ3 ef b ∇ϕi|E|ϕ b 2 ef |∇E| |∇hε |2 ϕ3 ef h∇|E|, 0= dV − dV + 3 dV hε hε hε M M M  Z  b 2 ϕ3 ef Q − (n − 2)λ|E| + dV hε M Z Z Z b 2 ϕ3 ef b ∇ϕi|E|ϕ b 2 ef |∇|E|| |∇hε |2 ϕ3 ef h∇|E|, ≥ dV − dV + 3 dV hε hε hε M M M  Z  b 2 ϕ3 ef Q − (n − 2)λ|E| + dV hε M Z Z b 2 ϕ3 ef b ∇ϕi|E|ϕ b 2 ef |∇E| h∇|E|, = dV + 3 dV hε hε M \Ωε M  Z  b 2 ϕ3 ef Q − (n − 2)λ|E| + dV hε M  Z Z  b 2 ϕ3 ef b ∇ϕi|E|ϕ b 2 ef Q − (n − 2)λ|E| h∇|E|, ≥3 dV + dV. hε hε M M By Theorem 7.6, every complete Ricci soliton is real analytic in suitable coordib ≡ 0 or the nates and, by the unique continuation property, one has that either |E| b zero set of |E| has measure zero. In the first case, the Bianchi identity implies that b −1 g is Ricci flat, while in the second case, taking the limit as ε → 0, since |E|h ε →1 n almost everywhere on M , inequality (7.46) follows.  Expanding case: proof of Theorems 7.17 and 7.19 The n-dimensional case. Let (M n , g) be a complete gradient expanding Ricci soliton of dimension n ≥ 3 with non-negative sectional curvature and assume that R ∈ L1 (M n ). By Proposition 7.29, either the soliton is Ricci flat (hence flat, since g has non-negative sectional curvature), or  Z  Z b 2 ϕ3 ef Q − (m − 2)λ|E| b ∇ϕiϕ2 ef dV dV ≤ −3 h∇|E|, (7.47) b | E| M M for every non-negative smooth cutoff function ϕ with compact support in M n . Since (M n , g) has non-negative sectional curvature, | Rm | ≤ αR, for some positive

184

Chapter 7. Ricci Solitons: Selected Results

constant α (see, e.g., [22]) Moreover, for expanding solitons with non-negative Ricci curvature the scalar curvature R is bounded, see for instance [72, Proposition 2.4]. Thus, g has bounded curvature. From Hamilton’s identity (7.6), and since R ≥ 0, we deduce that |∇f |2 ≤ 2λf +c. By [31, Lemma 5.5], there exist positive constants c1 , c2 , c3 such that −

λ λ (r(x) − c1 )2 − c2 ≤ −f (x) ≤ − (r(x) + c3 )2 , 2 2

(7.48)

where r(x) = dist(x, o) for some fixed origin o ∈ M n ; in particular, f is proper and, up to translation, we can assume that −f ≥ 0. We define, for t  1, Ωt = {x ∈ M n : −f (x) ≤ t}. We choose ϕ(x) = ψ(−f (x)), where ψ(s) = η(s/t) with η ≡ 1 on [0,p 1], positive, √ decreasing and with support in [0, 2]. Since in Ωt we have |∇f | ≤ c |f | ≤ c t, we deduce that c |∇ϕ| ≤ |ψ 0 ||∇f | ≤ √ in Ωt ; (7.49) t moreover, since ∆f = nλ − R ≤ nλ on M , c 2 |∆ϕ| ≤ ψ 0 ∆f + ψ 00 |∇f | ≤ t

in Ωt .

(7.50)

Then, integrating by parts we have Z Z   2 b ∇ϕiϕ2 ef dV = b f dV. h∇|E|, |∆ϕ|ϕ2 + 2ϕ|∇ϕ| + |∇f ||∇ϕ|ϕ2 |E|e Ω2t \Ωt

M

b and the non-negative curvature assumption one has E b ef ≤ By the definition of E cR, and from the previous estimates (7.49) and (7.50) we get Z Z b ∇ϕiϕ2 ef dV ≤ c h∇|E|, R. M

Ω2t \Ωt

By (7.48) and since R ∈ L1 (M n ), the left-hand side tends to zero as t → +∞, and from (7.47) we obtain, applying Fatou’s lemma, Z M

  b 2 ef Q − (m − 2)λ|E| dV ≤ 0. b |E|

Now we use the fact that under our assumptions Q is non-negative (see Proposition b ≡ 0. By the Bianchi identity, 7.35 below), and since λ is strictly negative, we get E we get R ≡ 0, and so g is flat thanks to the non-negative curvature assumption. This concludes the proof of Theorem 7.17. 

7.3. Rigidity II: integral conditions

185

The 3-dimensional case. The proof of Theorem 7.19 in dimension three is formally the same as the higher-dimensional case, with some minor modifications. In fact, under the weak assumption of non-negativity of the Ricci curvature we still have that the full curvature tensor is controlled by the scalar curvature, i.e. | Rm | ≤ αR for some constant α. Hence, following the proof in the previous subsection, we conclude that either (M 3 , g) is flat, or  Z  b 2 ef Q − λ|E| dV ≤ 0. b |E| M Now, we use the fact that under our assumptions Q is non-negative (see Proposib ≡ 0. By Bianchi identity tion 7.37 below), and since λ is strictly negative we get E we get R ≡ 0, and so g is flat by the non-negative curvature assumption. This concludes the proof of Theorem 7.19.  Remark 7.30. Theorems 7.17 and 7.19 can be proved also using a L1 -Liouville property for the operator ∆−f . In fact, since Q ≥ 0 it follows from equation (7.44) b ≥ 0 in a distributional sense. Moreover, the non-negativity of the that ∆−f |E| Ricci curvature implies that Ric−f = Ric −∇2 f = 2 Ric −λg ≥ 0. Therefore, since
b ∈ L1 ef dµ, M n , we can apply [147, Theorem 1.5], which R ∈ L1 (M n ) implies |E| asserts that on a complete Riemannian manifold (M n ,
g) with Ric−f ≥ 0 every positive solution u of ∆−f u ≥ 0 with u ∈ L1 ef dµ, M n must be constant. Thus b is constant, and therefore equal to zero in view of equation (7.44). |E| Steady case: proof of Theorems 7.18 and 7.20 The n-dimensional case. Let (M n , g) be a complete gradient steady Ricci soliton of dimension n ≥ 3 with non-negative sectional curvature and assume that Z 1 lim inf R = 0. r→+∞ r B r In particular, there exists a sequence {ri }, i ∈ N, of positive radii converging to +∞ such that Z 1 lim R = 0. (7.51) i→+∞ ri B ri By Proposition 7.29, either the soliton is Ricci flat (hence flat, since g has nonnegative sectional curvature), or Z Z Z Q ϕ3 ef 2 f b ∇ϕiϕ2 ef dV ≤ 3 b dV ≤ −3 h∇|E|, |∇|E|||∇ϕ|ϕ e dV (7.52) b |E| M M M for every non-negative smooth cutoff function ϕ with compact support in M n . Since (M n , g) has non-negative sectional curvature, | Rm | ≤ αR, for some positive constant α. Hamilton’s identity R + |∇f |2 = c

186

Chapter 7. Ricci Solitons: Selected Results

implies that both the scalar curvature R and |∇f |2 are bounded. Moreover, it follows, e.g., from [57], that either R > 0, or the soliton is Ricci flat, thus flat. So from now on we will assume that the scalar curvature is strictly positive. Using √ Kato’s inequality (3.26) and the fact that |∇ Ric | ≥ |∇R|/ n, we get   b ≤ |∇E| ≤ |∇ Ric | + n |∇R| + |∇f || Ric − 1 Rg| e−f |∇|E||
2
2 ≤ c |∇ Ric | + |∇f |R e−f ≤ c |∇ Ric | + R e−f for some positive constant c. Hence, the left-hand side of (7.52) can be estimated as Z Z Z 2 f b |∇|E|||∇ϕ|ϕ e dV ≤ |∇ Ric ||∇ϕ|ϕ2 dV + R|∇ϕ|ϕ2 dV . M

M

M

Now fix an index i and choose ϕ with support in B2ri = B2ri (o) for some origin o ∈ M n and such that ϕ ≡ 1 in Bri , |∇ϕ| ≤ 2/ri on M n . Then, by (7.51), the second term in the left-hand side tends to zero as i → +∞. By the H¨older inequality and the fact that R > 0, the remaining term can be estimate as Z

|∇ Ric ||∇ϕ|ϕ2 dV ≤

M

Z M

|∇ Ric |2 ϕ2 dV R

1/2 Z

R|∇ϕ|2 ϕ2 dV

1/2 .

M

(7.53) To conclude the estimate we need the following lemma: Lemma 7.31. Let (M n , g) be a complete, non-flat, gradient steady Ricci soliton of dimension n ≥ 3 with non-negative sectional curvature. Then, for every nonnegative cutoff function ϕ with compact support in M n , there exists a positive constant c such that Z Z   |∇ Ric |2 ϕ2 dV ≤ c Rϕ2 + R|∇ϕ|2 dV. R M M Proof. First of all, in some local frame we have |∇ Ric |2 =

1 |∇k Rij − ∇j Rik |2 + ∇k Rij ∇j Rik . 2

From the soliton equation and the commutation rule of covariant derivatives, one has ∇k Rij − ∇j Rik = Rkijl ∇l f . Since | Rm | ≤ αR and |∇f |2 ≤ c for some α, c > 0, we obtain |∇ Ric |2 ≤ cR2 + ∇k Rij ∇j Rik .

(7.54)

Hence, to finish the proof we have to estimate the right-hand side. Integrating by

7.3. Rigidity II: integral conditions

187

parts, commuting indices and using Young’s inequality we get Z Z Z ∇k Rij ∇j Rik ϕ2 Rij ∇k ∇j Rik ϕ2 Rij ∇j Rik ∇k ϕϕ dV = − dV − 2 dV R R R M M M Z Rij ∇j Rik ∇k Rϕ2 + dV R2 M Z Z Rij ∇j ∇k Rik ϕ2 (Rkjil Rij Rkl + Rij Ril Rjl )ϕ2 ≤− dV − dV R R M M Z Z  |∇ Ric |2 ϕ2 |∇R|2 2  +ε dV + c(ε) R|∇ϕ|2 dV + ϕ dV, R R M M for every ε > 0 and some constant c(ε). Using the Bianchi identity, the fact that | Rm | ≤ αR and the soliton identity ∇R = 2 Ric(∇f ), we obtain Z Z Z 1 |∇ Ric |2 ϕ2 ∇k Rij ∇j Rik ϕ2 Rij ∇i ∇j Rϕ2 dV ≤ − dV + ε dV R 2 M R R M M Z   + c(ε) Rϕ2 + R|∇ϕ|2 dV M Z Z Z   1 |∇R|2 ϕ2 |∇ Ric |2 ϕ2 = dV + ε dV + c1 (ε) Rϕ2 + R|∇ϕ|2 dV 4 M R R M M Z Z   2 2 |∇ Ric | ϕ ≤ε dV + c2 (ε) Rϕ2 + R|∇ϕ|2 dV, R M M for every ε > 0 and some constant c2 (ε). Choosing ε  1, this estimate and (7.54) conclude the proof of the lemma.  Now we can return to the proof of Theorem 7.18. Using the previous lemma and (7.53), we obtain Z  1/2 Z  1/2 Z |∇ Ric ||∇ϕ|ϕ2 dV ≤ c Rϕ2 + R|∇ϕ|2 dV R|∇ϕ|2 ϕ2 dV M M ZM c ≤ RdV ri B2ri which in view of (7.51) tends to zero as i → +∞. Applying Fatou’s lemma, from (7.52), we get Z Q ϕ3 ef dV ≤ 0. b |E| M Hence, Proposition 7.35 implies that Q ≡ 0 on M . In the equality case the Ricci tensor at every point has at most two distinct eigenvalues Λ = 0 with multiplicity n − 2 and Υ = 12 R with multiplicity two. To complete the proof we need the following general result (see Lemma 3.2 in [134]) for constant rank, symmetric, non-negative tensors. We recall that for a smooth vector field X the X-Laplacian is defined as ∆X = ∆ − g(X, ·) (see, e.g., [57]).

188

Chapter 7. Ricci Solitons: Selected Results

Lemma 7.32. Let T be a constant rank, symmetric, non-negative tensor on some tensor bundle. If (∆X T)(V, V ) ≤ 0 for V ∈ Ker T and X is a vector field, then the kernel of T is a parallel subbundle. Now let e1 , . . . , en be a local orthonormal frame such that Ric (e1 , e1 ) = Ric (e2 , e2 ) = 21 R and Ric (ek , ·) = 0 for every k = 3, . . . , n. In particular, if σij denotes the sectional curvature in the direction of the two-plane spanned by ei and ej , then the only non-zero such curvature is σ12 . Then, since ∆∇f Rij = 2λRij − 2Riljt Rlt = −2Riljt Rlt , we see that, for every fixed k = 3, . . . , n, we have ek ∈ Ker Ric and X (∆∇f Ric)(ek , ek ) = −2 Rm (ek , ei , ek , ei ) Ric (ei , ei ) = −R(σ1k + σ2k ) = 0. i

Moreover, since g has non-negative sectional curvature, 12 Rg − Ric is non-negative, and a simple computation shows that       1 1 ∆∇f Rg − Ric (e1 , e1 ) = ∆∇f Rg − Ric (e2 , e2 ) = 0. 2 2 Now Lemma 7.32 applies, and, by de Rham Decomposition Theorem (see for instance [103], Chapter 1, Section 6), the metric splits and Theorem 7.18 follows since, as asserted in the Introduction, the cigar soliton Σ2 is the only complete two-dimensional steady soliton with positive curvature. Remark 7.33. Note that, under our assumptions, f in general is not proper, thus one cannot resort to the argument used in the expanding case involving a cutoff function depending on the potential f . The three-dimensional case. The proof of Theorem 7.20 in dimension three follows the lines of the higher-dimensional case. First of all, by Chen’s work [61], g 2 must have non-negative sectional curvature, and since R + |∇f | = c, g has also bounded curvature. Hamilton’s strong maximum principle (see, e.g., [65]) shows
that M 3 , g either is, or it splits as a product R × Σ2 (where Σ2 is again the cigar steady soliton), or it has strictly positive sectional curvature. In the latter case, following the proof of Theorem 7.18 we obtain Z Qϕ3 ef dV ≤ 0. b |E| M Now, we use the fact that under our assumptions Q is non-negative (see Proposition 7.37 below), and we get Q ≡ 0. The equality case in Proposition 7.37 implies that the Ricci curvature has a zero eigenvalue, a contradiction. This concludes the proof of Theorem 7.19.  Proof of Corollary 7.23. The corollary is a direct consequence of the following

7.3. Rigidity II: integral conditions

189

Lemma 7.34 (Lemma 4.3 in [71]). Let (M n , g) be a complete gradient steady Ricci soliton with non-negative Ricci curvature. Then, for every o ∈ M n and every r > 0, Z √ Vol (Br (o)) RdV ≤ n c , r Br (o) 2

where c is the constant in Hamilton’s identity R + |∇f | = c. Proof. Integrating the equation R + ∆f = 0 one has Z Z Z RdV = − ∆f dV ≤ |∇f | ≤ CA(∂Br (o)), Br (o)

Br (o)

∂Br (o)

where A(∂Br (o)) is the (n − 1)-dimensional volume of the geodesic sphere ∂Br (o). Now, since (M n , g) has non-negative Ricci curvature, the Bishop-Gromov theorem (see for instance [65]) implies that for every o ∈ M and every r > 0 one has rA(∂Br (o)) ≤ n, Vol (Br (o)) which yields the result.



Auxiliary results. We provide here the proof of some algebraic curvature estimates needed for the proof of the main theorems. Proposition 7.35. Let (M n , g) be a Riemannian manifold of dimension n ≥ 3 with non-negative sectional curvature. Then P :=

(n − 2)3 3 ˚ij R ˚kt − (n − 2)(n − 4) R|Ric| ˚ 2 ≥ 0, R − 2Rikjt R 4n2 2n

with equality if and only if the Ricci tensor has at most two distinct eigenvalues, Λ = 0 of multiplicity n − 2 and Υ = 12 R of multiplicity two. Proof. From Lemma 5.17 we deduce that ˚ij R ˚kl ≥ − n − 2 R|Ric| ˚ 2; −2Rikjl R n using the previous estimate in the definition of P , we get  3 2 2  (n − 2) 3 (n − 2) ˚ 2 = (n − 2) R (n − 2) R2 − |Ric| ˚ 2 . (7.55) P ≥ R − R| Ric| 4n2 2n 2n 2n By the non-negativity assumption on the sectional curvature, 2

|Ric| ≤ which implies that

1 2 R , 2

˚ 2 ≤ n − 2 R2 . |Ric| 2n

190

Chapter 7. Ricci Solitons: Selected Results

Inserting the previous relation in (7.55) we get P ≥ 0. If P = 0 at a point, we have | Ric |2 = 12 R2 and the equality case in Lemma 5.17. Hence, the Ricci tensor has at most two distinct eigenvalues, Λ of multiplicity n − 2 and Υ of multiplicity two. In particular, R = (n − 2)Λ + 2Υ. Combining this with the identity 1 | Ric |2 = (n − 2)Λ2 + 2Υ2 = R2 , 2 we obtain Λ2 =

2 RΛ. n

Now either Λ = 0 and Υ = 12 R, or Λ = n2 R and Υ = − n−2 2n R. But, since g has non-negative sectional curvature, this second case implies R = 0 and so g is flat. In both cases we have the splitting result and this concludes the proof of the proposition.  In the three-dimensional case we need the following algebraic lemma. Lemma 7.36. For x, y, z ≥ 0, let P (x, y, z) = 5(x3 + y 3 + z 3 ) − 5(x2 y + xy 2 + x2 z + xz 2 + y 2 z + yz 2 ) + 18xyz. Then P (x, y, z) ≥ 3xyz ≥ 0 and P (x, y, z) = 0 if and only if x = 0 and y = z, or y = 0 and x = z, or z = 0 and x = y. Proof. It is easy to see that P (x, y, z) = 5x(x − z)(x − y) + 5y(y − z)(y − x) + 5z(z − x)(z − y) + 3xyz. (7.56) Since P (x, y, z) is symmetric in (x, y, z), i.e., it is invariant under any permutation of the variables x, y, z, without loss of generality we can assume that 0 ≤ x ≤ y ≤ z. Hence, P¯ (x, y, z) := 5x(x − z)(x − y) + 5y(y − z)(y − x) + 5z(z − x)(z − y) = 5x(x − z)(x − y) + 5(z − y)2 (z + y − x) ≥ 0.

(7.57)

From (7.56) and (7.57) we conclude that, for every x, y, z ≥ 0 P (x, y, z) = P¯ (x, y, z) + 3xyz ≥ 3xyz ≥ 0.

(7.58)

If P (x, y, z) = 0, then by (7.58) we have that xyz = 0. If x = 0 then 0 = P (0, y, z) = 5(y 3 + z 3 − y 2 z − yz 2 ) = 5(y − z)2 (y + z), and so y = z. The cases when y = 0 or z = 0 can be obtained by permutation of the variables x, y, z. 

7.4. Rigidity III: vanishing conditions on the Weyl tensor

191


Proposition 7.37. Let M 3 , g be a three-dimensional Riemannian manifold with non-negative Ricci curvature. Then 7 3 2 P := 4Rij Rjk Rki − R|Ric| + R3 ≥ 0, 2 4 with equality if and only if the Ricci tensor has at most two distinct eigenvalues, Λ = 0 and Υ = 12 R, of multiplicity two. Proof. Let e1 , e2 , e3 be a local orthonormal frame such that Ric(ei , ·) = µi ei for i = 1, 2, 3. Then

3 4P = µ31 + µ32 + µ33 − 14(µ1 + µ2 + µ3 ) µ21 + µ22 + µ23 + 3(µ1 + µ2 + µ3 ) = 5(µ31 + µ32 + µ33 ) − 5(µ21 µ2 + µ1 µ22 + µ21 µ3 + µ1 µ23 + µ22 µ3 + µ2 µ23 ) + 18µ1 µ2 µ3 . Now the proposition follows from Lemma 7.36.

7.4



Rigidity III: vanishing conditions on the Weyl tensor

As it is clear from the definition, the equation of Ricci solitons can be interpreted as a constraint on the Ricci tensor of g, that is on the trace part of the Riemann tensor (see for instance the interesting paper [20]). Thus, we can expect classification results for these structures only if one assumes further conditions on the traceless part of the Riemann tensor, i.e., on the Weyl tensor W , if n ≥ 4. Three-dimensional complete gradient shrinking Ricci solitons are classified and indeed it is well known that they are finite quotients of either the round sphere S3 , or the Gaussian shrinking soliton R3 , or the round cylinder S2 × R (see Ivey [98] for the compact case and Perelman [131], Ni–Wallach [126] and H.-D.Cao, B.-L.Chen and X.-P. Zhu [32] for the complete case). In higher dimensions, classification results for Ricci shrinkers were obtained by several authors under curvature conditions on the Weyl tensor. Z.-H.Zhang [151], based on the work of Ni and Wallach [126], showed that complete locally conformally flat gradient shrinking Ricci solitons, i.e., with Wikjl = 0 , are isometric to finite quotients of either Sn , Rn , or Sn−1 × R (see also the works of Eminenti, La Nave and Mantegazza [75], Petersen and Wylie [134], X. Cao, B.Wang and Z. Zhang [37]). Other rigidity results were obtained under suitable pointwise or integral pinching conditions on the Weyl tensor by Catino [40, 43] and X. Cao-Tran [36]. In dimension four, X. Chen and Y. Wang [63] (see also H.-D. Cao and Q.Chen [34]) proved that half-conformally flat (i.e. W ± = 0) gradient shrinking Ricci solitons are finite quotients of S4 , CP2 , R4 , or S3 × R.

192

Chapter 7. Ricci Solitons: Selected Results Under the weaker condition that the harmonic Weyl tensor is harmonic, i.e., div(W ) = ∇l Wikjl = 0 ,

Fernandez-Lopez and Garcia-Rio [76] and Munteanu and Sesum [121] proved that n-dimensional complete gradient shrinking solitons are either Einstein, or finite quotients of N n−k × Rk , (k > 0), the product of a Einstein manifold N n−k with the Gaussian shrinking soliton Rk . In the case n = 4, a stronger result was obtained by J.-Y. Wu, P. Wu and Wylie [148], assuming the Weyl tensor is half- harmonic (i.e. div(W ± ) = 0). It is interesting to observe that the aforementioned results can be interpreted as rigidity results under zero- and first-order vanishing conditions on the Weyl tensor. In 2013, H.-D. Cao and Q. Chen [34] showed that Bach-flat gradient shrinking Ricci solitons, i.e., with Bij =

1 1 ∇k ∇l Wikjl + Rkl Wikjl = 0 , n−3 n−2

are either Einstein, or finite quotients of Rn or N n−1 ×R, where N n−1 is an (n−1)dimensional Einstein manifold. In the same spirit as before, this can be seen as a vanishing condition involving second- and zero-order terms in the Weyl tensor, which a posteriori captures a more rigid class of solitons than in the harmonic Weyl case. Gradient steady Ricci solitons are less rigid, but many results in the same spirit have been obtained. It is well known that compact gradient steady solitons must be Ricci flat. In dimension n = 2, the only gradient steady Ricci soliton with positive curvature is Hamilton’s cigar Σ2 , see Hamilton [92]. In dimension three, the classification of complete gradient steady Ricci solitons is still open. Known examples are given by quotients of R3 , Σ2 × R, and the rotationally-symmetric example constructed by Bryant [26]. In the paper by Brendle [24] it was shown that the Bryant soliton is the only non-flat, k-non-collapsed, steady soliton, proving a famous conjecture by Perelman [131]. Other results in the steady three dimensional case have been obtained in H.-D. Cao, Catino, Q. Chen, Mantegazza and Mazzieri [31] and Catino, Mastrolia and Monticelli [53]. In particular, in [31] the authors showed rigidity just assuming that the Bach tensor is divergence-free, which is equivalent to a second-order vanishing condition on the Cotton tensor. In higher dimensions, H.-D. Cao and Q.Chen proved in [33] that complete n-dimensional (n ≥ 3) locally conformally flat gradient steady Ricci solitons are isometric to either a finite quotient of Rn , or the Bryant soliton. The same result for n ≥ 4 was proved independently by Catino and Mantegazza in [47] by using different methods. When n = 4, X. Chen and Y. Wang [63] showed that any fourdimensional complete half-conformally flat gradient steady Ricci soliton is either Ricci flat, or isometric to the Bryant soliton. Again, these are rigidity results under zero-order conditions on the Weyl tensor.

7.4. Rigidity III: vanishing conditions on the Weyl tensor

193

Classification results were obtained in [31] for Bach flat steady solitons case in dimension n ≥ 4. In particular, it follows that Bach flatness implies local conformal flatness. It is still an open question if similar results can be obtained under firstorder vanishing conditions on the Weyl tensor. The case of expanding solitons is clearly the less rigid. However, several interesting results under vanishing conditions on the Weyl tensor were obtained, see for instance [31, 47]. We remark that all the aforementioned results rely on vanishing conditions involving zero-, first- or specific second-order derivatives of the Weyl tensor. The aim of this paper is to obtain a classification of gradient Ricci solitons under much weaker assumptions, only requiring the vanishing of a fourth-order divergence of the Weyl tensor. Since, in dimension three, Weyl tensor is identically null, our results in this case will require the vanishing of a third-order divergence of the Cotton tensor. Hence, in every dimension, we only require a scalar condition on the Weyl (Cotton) curvature of the soliton in order to conclude. In order to precisely state our results, which are contained in [55], we introduce the following definitions div4 (W ) = ∇k ∇j ∇l ∇i Wikjl , div3 (C) = ∇i ∇j ∇k Cijk , where W and C are the Weyl and the Cotton tensors, respectively. Recall that, by (1.82) in Chapter 1, in dimension n ≥ 4, div4 (W ) = 0 if and only if div3 (C) = 0. Our first main result is the following classification theorem for gradient shrinking Ricci solitons of dimension n ≥ 4 with div4 (W ) = 0. Theorem 7.38. Every complete gradient shrinking Ricci soliton of dimension n ≥ 4 with div4 (W ) ≡ 0 on M is either Einstein, or isometric to a finite quotient of of N n−k × Rk (k > 0), the product of a Einstein manifold N n−k with the Gaussian shrinking soliton Rk . This theorem improves the results on gradient shrinking solitons with harmonic Weyl tensor in [76, 121]. In the case of steady and expanding solitons, under natural assumptions on the Ricci curvature, we show that the soliton has harmonic Weyl curvature. Namely, we have the following theorems. Theorem 7.39. Let (M n , g), n ≥ 4, be a complete gradient steady Ricci soliton with positive Ricci curvature and such that the scalar curvature attains its maximum at some point. If div4 (W ) = 0 on M , then (M n , g) has harmonic Weyl curvature. Theorem 7.40. Let (M n , g), n ≥ 4, be a complete gradient expanding Ricci soliton with non-negative Ricci curvature. If div4 (W ) = 0 on M , then (M n , g) has harmonic Weyl curvature. In dimension three, in the steady and expanding cases, we can prove stronger results. Namely, we have the following theorems.

194

Chapter 7. Ricci Solitons: Selected Results

Theorem 7.41. Every three-dimensional complete gradient steady Ricci soliton with div3 (C) = 0 on M , and such that the scalar curvature attains its maximum, is isometric to either a finite quotient of R3 , or the Bryant soliton (up to scaling). Theorem 7.42. Every three-dimensional complete gradient expanding Ricci soliton with non-negative Ricci curvature and div3 (C) = 0 on M is rotationallysymmetric. Note that, in dimension three, the Bach tensor is defined as (see equation (1.87)) Bij = Cijk,k . Hence, in this case, the condition div3 (C) = 0 is equivalent to div2 (B) = ∇i ∇j Bij = 0 and Theorems 7.41 and 7.42 improve the results [31, Corollary 1.3] and [31, Theorem 5.9], respectively. Note also that on the steady three-dimensional gradient Ricci soliton Σ2 × R the “triple divergence” of the Cotton tensor div3 (C) does not vanish identically, see Lemma 7.45. Finally, note that, as it will be clear from the proof, the scalar assumptions on the vanishing of div3 (C) and div4 (W ) in all the above theorems can be trivially relaxed to a (suitable) inequality. For instance, Theorem 7.41 holds just assuming div3 (C) ≤ 0 on M . The proof of our results relies heavily on two main ingredients. The first consists of new integrability conditions for gradient Ricci solitons that will be shown below in Proposition 7.7. The second one is an integral formula (see Theorem 7.43) which relates the squared norm |C|2 to the double divergence of the Cotton tensor C, for every gradient Ricci soliton and for a suitable family of cutoff functions (depending on the potential f ) with compact support. A careful (double) use of this identity allows us to avoid imposing any Lebesgue integrability assumptions on the curvature of the soliton in our main results.

7.4.1 A key integral formula In this section we show an integral formula that holds on every gradient Ricci soliton. Theorem 7.43. Let (M n , g), n ≥ 3, be a gradient Ricci soliton with potential function f . For every C 2 function ψ : R → R such that ψ(f ) having compact support in M , one has Z Z 1 |C|2 ψ(f ) dVg = − Ckti,it fk ψ(f ) dVg . 2 M M In particular, if n ≥ 4, this is equivalent to Z Z 1 n−2 |C|2 ψ(f ) dVg = Wlkti,lit fk ψ(f ) dVg . 2 M n−3 M

7.4. Rigidity III: vanishing conditions on the Weyl tensor

195

Proof. Let ψ satisfy the hypotheses. We multiply equation (7.19) by ψ(f ) and integrate over M . By using the soliton equation and the fact that C is totally trace-free, and integrating by parts, we obtain Z Z Z 1 2 |C| ψ(f )dVg − Ckti,i fkt ψ(f )dVg = (n − 2) Ditk,tki ψ(f )dVg 2 M M M Z = −(n − 2) Ditk,tk fi ψ 0 (f )dVg M Z =− Ckti Rkt fi ψ 0 (f )dVg M Z = Ckti fkt fi ψ 0 (f )dVg , M

where we also made use of equation (7.18). Another integration by parts and the fact that Ckti = −Ckit yield Z Z 1 2 |C| ψ(f )dVg − Ckti,i fkt ψ(f )dVg 2 M M Z Z Z =− Ckti,t fk fi ψ 0 (f )dVg − Ckti fk fit ψ 0 (f )dVg − Ckti fk fi ft ψ 00 (f )dVg M M ZM 0 =− Ckti,t fk fi ψ (f )dVg . M

Hence, renaming indices, Z Z 1 |C|2 ψ(f )dVg − Ckti,i fkt ψ(f )dVg 2 M M Z =− Ckit,i fk ft ψ 0 (f )dVg M Z
= Ckti,i fk ψ(f ) t dVg M Z Z =− Ckti,it fk ψ(f ) − Ckti,i fkt ψ(f )dVg . M

M

Simplifying, we obtain the result. The second equation in the statement follows from (1.82).  Remark 7.44. In case n = 3 the formula of Theorem 7.43 holds trivially. Indeed, it is easy to see, using formulas (7.16) and (7.18), that Cijk = Dijk ,

Rkt Ckti = Citk,tk ;

hence, thanks to the symmetries of C, one has 1 2 1 1 |C| = Ckti Dkti = Ckti (Rkt fi − Rik ft ) = Ckti Rkt fi = Citk,tk fi = −Ckti,it fk , 2 2 2

196

Chapter 7. Ricci Solitons: Selected Results

i.e., pointwise on M 3 , 1 2 |C| = −Ckti,it fk . 2

(7.59)

7.4.2 Proof of the results Shrinking Ricci solitons In this section we prove Theorem 7.38. Let (M n , g), n ≥ 4, be a complete gradient shrinking Ricci soliton with potential function f . If M is compact, then we choose ψ(f ) ≡ 1 on M in Theorem 7.43. Thus, integrating by parts, we have Z Z Z 1 2 |C| dVg = − Ckti,it fk dVg = Ckti,itk f dVg . 2 M M M By formula (1.82), Ckti,itk = −

n−2 n−2 Wjkti,jitk = − div4 (W ) ≡ 0 n−3 n−3

(7.60)

on M , by assumption. Consequently, C ≡ 0 on M . Theorem 7.38 now follows from [76, 121]. If M is complete and non-compact, then we choose ψ(f ) = e−f φ(f ), where, for any fixed s > 0, φ : R → R is a non-negative C 3 function such that φ ≡ 1 on [0, s], φ ≡ 0 on [2s, +∞), and φ0 ≤ 0 on [s, 2s]. It is well known that on every complete, non-compact gradient shrinking soliton the potential function f is proper with quadratic growth at infinity (see [35]). Then, for every s > 0, the cutoff function ψ(f ) has compact support in M . Thus, integrating by parts in the integral formula of Theorem 7.43 we obtain Z Z 1 |C|2 e−f φ(f )dVg = − Ckti,it fk e−f φ(f )dVg 2 M M Z
= Ckti,it e−f k φ(f )dVg (7.61) M Z Z =− Ckti,itk e−f φ(f )dVg − Ckti,it fk e−f φ0 (f )dVg . M

M

Now, since the function ψe = e−f φ0 (f ) is C 2 with compact support, we can apply e obtaining again the integral formula of Theorem 7.43 with ψ, Z Z Z 1 − Ckti,it fk e−f φ0 (f )dVg = − Ckti,it fk ψe dVg = |C|2 ψe dVg 2 M M ZM 1 2 −f 0 = |C| e φ (f )dVg ≤ 0, 2 M

7.4. Rigidity III: vanishing conditions on the Weyl tensor

197

since φ0 ≤ 0. Equation (7.61) yields Z Z 1 |C|2 e−f φ(f )dVg ≤ − Ckti,itk e−f φ(f )dVg = 0 2 M M by assumption and equation (7.60). Hence, C ≡ 0 on the compact set Ωs = {f ≤ s}, since φ(f ) ≥ 0 on M and φ(f ) ≡ 1 on Ωs . Then taking the limit as s → +∞ we find that C ≡ 0 on M , and the conclusion follows again from [76, 121]. Steady and expanding Ricci solitons in dimension greater than four In this section we prove Theorem 7.39 and Theorem 7.40. Let (M n , g), n ≥ 4, be a complete gradient steady or expanding Ricci soliton with potential function f . It is well known that, if M is compact, then (M n , g) is Einstein. In particular, (M n , g) has harmonic Weyl curvature. On the other hand, if M is complete and non-compact, then we proceed similarly as in the shrinking case. We let ψ(f ) = ef φ(−f ), where, for any fixed s > 0, φ : R → R is a non-negative C 3 function such that φ ≡ 1 on [0, s], φ ≡ 0 on [2s, +∞) and φ0 ≤ 0 on [s, 2s]. The assumptions of Theorem 7.39 and Theorem 7.40 imply that −f is proper, with linear or quadratic growth in the steady or expanding case, respectively (see for instance [33] and [31]). Then, for every s > 0, the cutoff function ψ(f ) has compact support in M . Integrating by parts in the integral formula of Theorem 7.43 we obtain Z Z 1 2 f |C| e φ(−f )dVg = − Ckti,it fk ef φ(−f ) 2 M M Z
=− Ckti,it ef k φ(−f )dVg (7.62) M Z Z = Ckti,itk ef φ(−f )dVg − Ckti,it fk ef φ0 (−f )dVg . M

M

Now, since the function ψe = ef φ0 (−f ) is C 2 with compact support, applying the e we get integral formula of Theorem 7.43 with ψ, Z Z e g − Ckti,it fk ef φ0 (−f )dVg = − Ckti,it fk ψdV M M Z Z 1 e g=1 = |C|2 ψdV |C|2 ef φ0 (−f )dVg ≤ 0, 2 M 2 M since φ0 ≤ 0. Equation (7.62) yields Z Z 1 |C|2 ef φ(−f )dVg ≤ Ckti,itk ef φ(−f )dVg = 0, 2 M M

(7.63)

by assumption and equation (7.60). Hence, arguing as before, C ≡ 0 on M , i.e., (M, g) has harmonic Weyl curvature.

198

Chapter 7. Ricci Solitons: Selected Results

Steady and expanding Ricci solitons in dimension three In this section we prove Theorem 7.41 and Theorem 7.42. First of all, in the expanding case, arguing exactly as before, we obtain equation (7.63). This again implies that C ≡ 0 on M , i.e., (M, g) is locally conformally flat. The rotational symmetry now follows from [33]. Finally, let (M 3 , g) be a three-dimensional complete gradient steady Ricci soliton. By B.-L. Chen [61], g must have non-negative sectional curvature. Further, Hamilton’s identity (7.6) 2 R + |∇f | = c, for some c ∈ R, and thus g has bounded curvature. From Hamilton’s strong
maximum principle (see e.g. [65]) we deduce that M 3 , g is: (i) flat, or (ii) it has strictly positive sectional curvature, or (iii) it splits as a product Σ2 × R, where Σ2 is the cigar steady soliton. In case (i) the proof is complete. If (ii) holds, then, by [33], −f is proper and has linear growth at infinity. By the same argument we used before, we conclude that C ≡ 0 on M , i.e., (M, g) is locally conformally flat, and the result follows from [33]. Let us show that case (iii) cannot occur, by proving that the steady soliton Σ2 × R does not satisfy div3 (C) ≡ 0. We recall that Hamilton’s cigar steady soliton is defined as the complete Riemannian surface (Σ2 , ge), where Σ2 = R2 , ge =

dx2 + dy 2 1 + x2 + y 2

and with the potential function is given by fe(x, y) = − log(1 + x2 + y 2 ). On (Σ2 × R, g) we adopt global coordinates s, x, y, and then the metric and the potential take the form dx2 + dy 2 g = ds2 + 1 + x2 + y 2 and f (s, x, y) = − log(1 + x2 + y 2 ). In particular, the Ricci tensor is diagonal, with Rxx = Ryy =

1 R > 0, 2

Rss = 0.

7.4. Rigidity III: vanishing conditions on the Weyl tensor

199

Moreover, Hamilton’s identity (7.6) implies that at the origin O = (0, 0, 0) one has ∇f (O) = ∇R(O) = 0. Now the conclusion of Theorem 7.41 is a consequence of the following lemma. Lemma 7.45. In the above notation, div3 (C)(O) =

1 R(O)3 6= 0. 8

Proof. By (7.16) and (7.19), C = D and 1 1 Ckti,itk = Dkti,itk = −Dkit,itk = − |D|2 −Rti Dtik,k = − |D|2 −Rij Dijk,k . (7.64) 2 2 Since ∇f (O) = 0, from the definition of D, namely, 1 1 Dijk = (fk Rij − fj Rik ) + ft (Rtk gij − Rtj gik ) − R(fk gij − fj gik ), 2 2 we have D(O) = 0. To compute the value of the last term in (7.64) at the origin O, we take the divergence of D and obtain 1 ftk Rtk gij + ft Rtk,k gij 2
− fti Rtj − ft Rtj,i − fk Rk gij − R∆f gij + Ri fj + Rfij .

Dijk,k = ∆f Rij + fk Rij,k − fjk Rik − fj Rik,k +

Evaluating this at the origin O, we get Dijk,k = ∆f Rij − fjk Rik +


1 ftk Rtk gij − fti Rtj − R∆f gij + Rfij . 2

Using the soliton equations (4.3) and (4.3), one has at the origin O 3 3 1 1 Dijk,k = − RRij + Rjk Rik − | Ric |2 gij + R2 gij , 2 2 2 2 and then taking the trace with the Ricci tensor we obtain 3 1 Rij Dijk,k = −2R| Ric |2 + Rij Rjk Rik + R3 . 2 2 Since | Ric |2 =

1 2 R 2

and Rij Rjk Rik =

1 3 R , 4

we obtain

1 Rij Dijk,k = − R3 , 8 and now thw conclusion follows from (7.64).



200

7.5

Chapter 7. Ricci Solitons: Selected Results

Rigidity IV: Weyl scalars

In this section we follow closely [51], which can be thought of as a first, preliminary step in a general research program which aims at showing that gradient Ricci solitons can be classified by finding a “generic” (in a suitable sense) [k, s]-vanishing condition on the Weyl tensor, for every k, s ∈ N, where k is the order of the covariant derivatives of Weyl and s is the type of the (covariant) tensor involved. To quote some important examples, the results recalled in the previous sections, for instance, in [75, 151], [76, 121], [34] and [55], deal, respectively, with [0, 4], [1, 3], [2, 2] and [4, 0] conditions. Obviously, the study of the case [k, 0] is harder, since it involves only a scalar condition. We study general [4, 0] conditions, defined as linear combinations of divergences of the Weyl tensor, contracted with suitable covariant derivatives of the potential function f . With this choice we improve all the previous quoted results, introducing a general Weyl scalar that has the same homogeneity under rescaling of the metric (see Section 7.5.1 for the precise definitions). For instance, we prove the following triviality result under a single Weyl scalar vanishing assumption. Proposition 7.46. For n ≥ 4 there are no non-Einstein compact Ricci solitons, provided that at least one of the following Weyl scalars vanishes: Wijkl Rik Rjl , Wijkl,i Rjl,k , Bij fi fj , Wijkl,ilk fj , or Wijkl,ilkj . In particular, in the compact setting, the proposition improves the results in [75, 151, 76, 121, 34]. More generally, Proposition 7.46 holds true assuming a generic [4, 0] vanishing condition on the Weyl tensor (see Propositions 7.56 and 7.57); in particular, we show that the Bach tensor Bij =

1 1 Wikjl,lk + Wikjl Rkl , n−3 n−2

although interesting for its connections with conformal geometry and physics, plays no special role in the classification of Ricci solitons. In fact, we prove the following Proposition 7.47. Let (M n , g) be a compact Ricci soliton of dimension n ≥ 4. If  c1 Wikjl,lk + c2 Wtikj,t fk +

 1 Wikjl Rkl Rij = 0 n−2

for some constants c1 , c2 ∈ R with c1 6= Einstein.

1 n−3

on M,

1 and c2 > − n−2 , then (M, g) is

1 Note that the Bach case corresponds to the choice c1 = n−3 and c2 = 0. Moreover, we can show two triviality results under mixed vanishing assumptions on Weyl scalars. In general dimension n ≥ 4 we can prove the following

7.5. Rigidity IV: Weyl scalars

201

Proposition 7.48. Let (M n , g) be a compact Ricci solitons of dimension n ≥ 4. If, on M , 1 Wtijk,tk Rij + c4 Wtijk,t Rik,j n−3 1 + c5 Wtijk,t Rik fj + c6 Wtijk Rik,jt + c7 Wtijk Rik ft fj + Wtijk Rtj Rik = 0, n−2

c1 Wtijk,tkji + c2 Wtijk,tkj fi + c3 Wtijk,tk fi fj +

for some ci ∈ R, i = 1, . . . , 7, with either c1 > 0 or c1 = 0 and c4 + then (M, g) is Einstein.

n−2 n−3 c6

6= 0,

In dimension four, we have Proposition 7.49. Let (M 4 , g) be a compact Ricci solitons of dimension four. If, on M , c1 Wtijk,tkji + c2 Wtijk,tkj fi + c3 Wtijk,tk fi fj + c4 Wtijk,tk Rij 1 + c5 Wtijk,t Rik,j − c3 Wtijk,t Rik fj + c6 Wtijk Rik,jt + Wtijk Rtj Rik = 0, 2 for some ci ∈ R, i = 1, . . . , 6, with 1+c2 +c4 +c5 +c6 6= 0, then (M, g) is Einstein. We recall that, for a gradient Ricci solitons, the integrability conditions relating the tensors C and D (and also W and B), see Proposition 7.7 are valid; note that if D ≡ 0, then C ≡ 0 (see [34]). Moreover, when M is compact, C ≡ 0 implies that (M, g) is Einstein (see [76, 121]). Thus, a possible strategy to obtain the classification is to provide suitable assumptions ensuring the vanishing of C. The proof of our results can be divided into three steps: 1. first, we obtain some pointwise identities for each Weyl scalar given by linear combinations of the three terms |C|2 , |D|2 and CD := Cijk Dijk , with possibly a remainder term of divergence type; 2. second, exploiting these pointwise identities, we derive integral identities with a parametric exponential weight of the type e−ωf , ω ∈ R. More precisely, we prove that, for every ω ∈ R, the weighted integral of a general Weyl scalar is given by the expression Z
α|C|2 + 2β CD + γ|D|2 e−ωf , M

with explicit coefficients α, β and γ depending on ω and the Weyl scalar itself (Section 7.5.1); 3. finally, a simple algebraic argument shows that the vanishing of a class of good Weyl scalars implies that D ≡ C ≡ 0 (Section 7.5.2). In the final Section 7.5.3, we provide some applications of the previous analysis and prove Propositions 7.46, 7.47, 7.48 and 7.49. Moreover, in Remark 7.60 we discuss possible extensions to the non-compact case.

202

Chapter 7. Ricci Solitons: Selected Results

7.5.1 Weyl scalars on a Ricci soliton We define the following ten Weyl scalars: w0,1 := Wtijk ftj fik , w1,1 w2,1 w3,1 w4,1

w0,2 := Wtijk fik ft fj , w0,4 := Wtijk fikjt , := Wtijk,t fik fj , w1,2 := Wtijk,t fikj , := Wtijk,tk fij , w2,2 := Wtijk,tk fi fj , := Wtijk,tkj fi , := Wtijk,tkji .

w0,3 := Wtijk fikj ft ,

(7.65)

Note that these are the only scalar quantities that depend linearly on the Weyl tensor and its divergences. Moreover, all these functions have the same homogene−3 ity under rescaling of the metric. Indeed, if ge = λ g, λ ∈ R, then w g wa,b . a,b = λ We can now define the notion of a general Weyl scalar as follows: Definition 7.50. A general Weyl scalar is a linear combination of the ten Weyl scalars in (7.65), i.e., a function w of the form wG = ap0 w0,p,+ aq1 w1,q + ar2 w2,r + a13 w3,1 + a14 w4,1 with ap0 , aq1 , ar2 , a13 , a14 ∈ R and p = 1, . . . , 4, q, r = 1, 2. Pointwise identities We now obtain some pointwise identities for each Weyl scalar given by linear combinations of the three terms |C|2 , |D|2 and CD := Cijk Dijk , with possibly a remainder term of divergence type. Indeed we have Lemma 7.51. Let (M n , g), n ≥ 3, be a gradient Ricci soliton with potential function f . Then the Weyl scalars defined in (7.65) satisfy the following pointwise identities: 1 2 n−4 |C| + CD + (Wtijk Rij ft )k , 2 2
n−2 = −Wtijk Rik ft fj = |D|2 − CD , 2
1 = −Wtijk Rik,j ft = CD − |C|2 , 2 n−3 = −Wtijk Rik,jt = − |C|2 − (Wtijk Rtj,i )k , 2(n − 2) n−3 = −Wtijk,t Rik fj = − CD, 2 n−3 = −Wtijk,t Rik,j = − |C|2 , 2(n − 2) n−3 = −Wtijk,tk Rij = − |C|2 − (Wtijk,t Rij )k , 2(n − 2)

w0,1 = Wtijk Rtj Rik = w0,2 w0,3 w0,4 w1,1 w1,2 w2,1

7.5. Rigidity IV: Weyl scalars

203

n−3 CD + (Wtijk,t fi fj )k , 2 n−3 = Wtijk,tkj fi = |C|2 + (Wtijk,t fi )kj , 2(n − 2) = Wtijk,tkji .

w2,2 = Wtijk,tk fi fj = w3,1 w4,1

Proof. We give only the proof of the first identity. Using the soliton equation and Proposition 7.7, we have w0,1 = Wtijk ftj fik = Wtijk Rtj Rik = Wtikj ftj Rik = (Wtikj ft Rik )j − Wjkit,j ft Rik − Wtikj ft Rik,j n−3 1 = (Wtijk ft Rij )k + ft Rik Ckit − Wtikj ft Cikj n−2 2 1 2 n−4 = |C| + CD + (Wtijk Rij ft )k . 2 2 The other identities are in a similar manner.



Integral identities We first derive integral identities with a general weight function depending on the potential f . To simplify the expressions, we omit the notation dVg . Lemma 7.52. Let (M n , g), n ≥ 3, be a gradient Ricci soliton with potential function f . For every smooth function ψ : R → R such that ψ(f ) has compact support in M , the Weyl scalars defined in (7.65) satisfy the following weighted integral identities: Z W0,1 := w0,1 ψ(f ) M Z  1 = ψ(f )|C|2 + [(n − 4)ψ(f ) + (n − 2)ψ 0 (f )]CD − (n − 2)ψ 0 (f )|D|2 , 2 Z M Z
n−2 W0,2 := w0,2 ψ(f ) = ψ(f ) |D|2 − CD , 2 M ZM Z
1 W0,3 := w0,3 ψ(f ) = ψ(f ) CD − |C|2 , 2   ZM ZM  1 n−3 0 2 0 W0,4 := w0,4 ψ(f ) = ψ (f ) − ψ(f ) |C| − ψ (f ) CD , 2 M n−2 ZM Z n−3 W1,1 := w1,1 ψ(f ) = − ψ(f ) CD, 2 M M Z Z n−3 W1,2 := w1,2 ψ(f ) = − ψ(f )|C|2 , 2(n − 2) M M  Z Z  n−3 1 2 0 W2,1 := w2,1 ψ(f ) = − ψ(f )|C| + ψ (f )CD , 2 n−2 M M

204

Chapter 7. Ricci Solitons: Selected Results Z n−3 ψ(f ) CD, 2 M M Z Z n−3 := w3,1 ψ(f ) = ψ(f )|C|2 , 2(n − 2) M M Z Z n−3 := w4,1 ψ(f ) = − ψ 0 (f )|C|2 . 2(n − 2) M M

W2,2 := W3,1 W4,1

Z

w2,2 ψ(f ) =

Proof. Using Lemma 7.51, integrating by parts and using the definitions of C and D, we obtain Z Z (Wtijk Rij ft )k ψ(f ) = − Wtijk Rij ft fk ψ 0 (f ) M M Z n−2 =− (Dijk − Cijk )Dijk ψ 0 (f ) 2 M Z
n−2 = CD − |D|2 ψ 0 (f ) , 2 M Z

Z Z (Wtijk Rtj,i )k ψ(f ) = − Wtijk Rtj,i fk ψ 0 (f ) = − Wtijk Rik,j ft ψ 0 (f ) M M M Z Z
1 1 0 = (Dijk − Cijk )Cijk ψ (f ) = CD − |C|2 ψ 0 (f ), 2 M 2 M

and Z

Z

Wtijk,t Rij fk ψ 0 (f ) M Z Z n−3 n−3 = Cijk Rij fk ψ 0 (f ) = CD ψ 0 (f ) . n−2 M 2 M

(Wtijk,t Rij )k ψ(f ) = − M

Moreover, thanks to the symmetries of the Weyl tensor, Z Z (Wtijk,t fi fj )k ψ(f ) = − Wtijk,t fi fj fk ψ 0 (f ) = 0, M Z Z M (Wtijk,t fi )kj ψ(f ) = Wtijk,t fi [ψ(f )]jk = 0 . M

M

Finally, Z

Z Wtijk,tkji ψ(f ) = −

M

ZM =− M

Z

Wtijk,t [ψ(f )]ijk Wtijk,t [ψ 0 (f )fij + ψ 00 (f )fi fj ]k

h Wtijk,t ψ 00 (f )fij fk + ψ 0 (f )fijk + ψ 000 (f )fi fj fk M i + ψ 00 (f )fik fj + ψ 00 (f )fi fkj

=−

7.5. Rigidity IV: Weyl scalars Z

Wtijk,t [ψ 0 (f )fijk ] = M Z n−3 =− |C|2 ψ 0 (f ) . 2(n − 2) M

205 Z

=−

Wtijk,t Rij,k ψ 0 (f )

M



In particular, it follows that are only six integrals independent. Indeed, a simple computation shows that Corollary 7.53. The following identities hold: 1 n−2 W1,1 + W1,2 ; n−2 n−3 n−4 1 n−2 = W1,2 + W2,1 − W4,1 ; n−3 n−3 n−3 = −W1,1 ; = −W1,2 .

W0,3 = − W0,4 W2,2 W3,1

Since (M n , g) is compact, we choose ψ(f ) := e−ωf , with ω ∈ R, and obtain Corollary 7.54. Let (M n , g), n ≥ 3, be a compact gradient Ricci soliton with potential function f . Then, for every ω ∈ R, the Weyl scalars defined in (7.65) satisfy the following weighted integral identities: Z W0,1 = w0,1 e−ωf M Z  2 1 = |C| + [(n − 4) − (n − 2)ω]CD + (n − 2)ω|D|2 e−ωf , 2 Z M Z
n−2 W0,2 = w0,2 e−ωf = |D|2 − CD e−ωf , 2 M ZM Z n − 3 −ωf W1,1 = w1,1 e =− CD e−ωf , 2 M M Z Z n−3 W1,2 = w1,2 e−ωf = − |C|2 e−ωf , 2(n − 2) M M  Z Z  1 n−3 −ωf W2,1 = w2,1 e =− |C|2 − ω CD e−ωf , 2 n−2 ZM ZM n − 3 W4,1 = w4,1 e−ωf = ω|C|2 e−ωf . 2(n − 2) M M Note that, in the case n = 4 and ω = 0, the first integral identities appeared in [36].

7.5.2 Main results In view of Corollary 7.53, to compute the weighted integral of a general Weyl R scalar wG , i.e., M wG e−ωf , it is sufficient to consider a linear combination of the

206

Chapter 7. Ricci Solitons: Selected Results

six independent integrals W0,1 , W0,2 , W1,1 , W1,2 , W2,1 , W4,1 . Thus, letting   A := A10 , A20 , A11 , A21 , A12 , A14 ∈ R6 we can define WG (A, ω) := A10 W0,1 + A20 W0,2 + A11 W1,1 + A21 W1,2 + A12 W2,1 + A14 W4,1 (7.66) and correspondingly, at the pointwise level, wG (A) := A10 w0,1 + A20 w0,2 + A11 w1,1 + A21 w1,2 + A12 w2,1 + A14 w4,1 .

(7.67)

Based on Corollary 7.54, a long but straightforward algebraic computation establishes the validity of the following Corollary 7.55. Let (M n , g), n ≥ 3, be a compact gradient Ricci soliton with potential function f . Then, for every ω ∈ R, Z
WG (A, ω) = α|C|2 + 2β CD + γ|D|2 e−ωf , M

with  
1 1 n−3 2 1 1 α = α(A, ω) := A − A + A2 − ωA4 2 0 n−2 1 n 1 β = β(A, ω) := (n − 4)A10 − (n − 2)A20 − (n − 3)A11 4  o − ω (n − 2)A10 − (n − 3)A12 γ = γ(A, ω) :=


n−2 2 A0 + ωA10 . 2

We now need the following definition. We say that (A, ω) ∈ R6 × R is Cdegenerate or D-degenerate if α(A, ω) 6= 0

and β(A, ω) = γ(A, ω) = 0

or γ(A, ω) 6= 0 and α(A, ω) = β(A, ω) = 0, respectively. From Corollary 7.55 we immediately deduce the following proposition justifying the terminology. Proposition 7.56. Let (M, g) be a compact shrinking Ricci soliton of dimension n ≥ 4 with wG (A) = 0 for some A ∈ R6 . If there exists ω ∈ R such that (A, ω) is C-degenerate or D-degenerate, then C ≡ 0 or D ≡ 0, respectively.

7.5. Rigidity IV: Weyl scalars

207

Let ∆ = ∆(A, ω) := αγ − β 2 . A (long) computation shows that ∆ = δ2 ω 2 + 2δ1 ω + δ0 , where δ2 = δ2 (A) :=

1h − (n − 2)2 (A10 )2 + 2(n − 3)A10 ((n − 2)A12 + 2A14 ) 16 i

− (n − 3)2 (A12 )2 ,  1n 1 2 δ1 = δ1 (A) := (A0 ) (n − 2)2 + A10 n(5A11 + 5A12 16 + 4A20 − 2A21 ) + n2 (−(A11 + A12 + A20 )) − 6A11 − 6A12 
o − 4A20 + 6A21 + (n − 3) A11 A12 (n − 3) + A12 A20 (n − 2) + 2A14 A20 , and   1n − (A10 )2 (n − 4)2 + 2A10 A11 (n − 4)(n − 3) + A20 (n − 2)2 16 − (A11 )2 (n − 3)2 − 2A11 A20 (n − 3)(n − 2)  o − A20 4A12 (n − 3) + A20 (n − 2)2 + 4A21 (n − 3) .

δ0 = δ0 (A) :=

We now define the following subsets of R6 :  Ω+ := A ∈ R6 : δ2 (A) > 0 ,  Ω0 := A ∈ R6 : δ2 (A) = 0 and δ1 (A) 6= 0 ∪  ∪ A ∈ R6 : δ2 (A) = δ1 (A) = 0 and δ0 (A) > 0 ,  Ω− := A ∈ R6 : δ2 (A) < 0 and δ12 (A) − δ2 (A)δ0 (A) > 0 , and  Ωd = A ∈ R6 : δ12 (A) − δ2 (A)δ0 (A) > 0 ⊃ Ω− . We can now state our main triviality result. Proposition 7.57. Let (M, g) be a compact shrinking Ricci soliton of dimension n ≥ 4. If wG (A) = 0 for some A ∈ Ω+ ∪ Ω0 ∪ Ω− ∪ Ωd , then C ≡ D ≡ 0. Remark 7.58. Note that if D ≡ 0 then C ≡ 0 (see [34]). Moreover when M is compact, C ≡ 0 implies (M, g) Einstein (see [76]). To prove Proposition 7.57 we first need the following

208

Chapter 7. Ricci Solitons: Selected Results

Lemma 7.59. Let (M, g) be a compact shrinking Ricci soliton of dimension n ≥ 4. If WG (A, ω ¯ ) = 0 for some (A, ω ¯ ) ∈ R6 × R such that ∆(A, ω ¯ ) > 0, then C ≡ D ≡ 0. ¯ = ∆(A, ω Proof. We set α ¯ = α(A, ω ¯ ), β¯ = β(A, ω ¯ ), γ¯ = γ(A, ω ¯ ) and ∆ ¯ ). Since ¯ ∆ > 0m we have α ¯ 6= 0, and from WG (A, ω ¯ ) = 0 we deduce that Z
0= α ¯ |C|2 + 2β¯ CD + γ¯ |D|2 e−¯ωf M ! 2 Z ¯ β¯ ∆ 2 = α ¯ C + D + |D| e−¯ωf , α ¯ α ¯ M which implies that D ≡ 0 and C ≡ 0.



Proof of Proposition 7.57. Let (M, g) be a compact shrinking Ricci soliton of dimension n ≥ 4 satisfying wG (A) = 0, with A ∈ Ω+ ∪ Ω0 ∪ Ω− . In particular, WG (A, ω) = 0 for all ω ∈ R. 1. If A ∈ Ω+ , then δ2 (A) > 0 and thus ∆(A, ω) > 0 for ω sufficiently large; 2. if A ∈ Ω0 , then δ2 (A) = 0 and we have two possibilities: if δ1 (A) 6= 0, then ∆(A, ω) > 0 for |ω| sufficiently large with ωδ1 (A) > 0. If δ1 (A) = 0, then ∆(A, ω) = δ0 (A) > 0; 3. if A ∈ Ω− , then δ2 (A) < 0 and δ12 (A) − δ2 (A)δ0 (A) > 0; 4. if A ∈ Ωd , then δ12 (A) − δ2 (A)δ0 (A) > 0 and, clearly, there exists ω ∈ R such that ∆(A, ω) > 0. In any case, there exists ω ∈ R such that ∆(A, ω) > 0 and the conclusion follows from Lemma 7.59. 

7.5.3 Special cases In this final section we highlight some special cases in which Propositions 7.56 and 7.57 are applicable. Single Weyl scalars We consider here five general Weyl scalars wG given by a single term, in order to extend all the known results (at least in the compact case) discussed in the Introduction. We summarize them in the following tables: A (1, 0, 0, 0, 0, 0) (0, 0, 0, 1, 0, 0)

W ∗ Ric ∗ Ric div(W ) ∗ ∇ Ric B ∗ Ric B(∇f, ∇f ) div4 (W )

 



1 1 n−2 , 0, 0, 0, n−3 , 0

 1 1 0, − n−2 , n−3 , 0, 0, 0 (0, 0, 0, 0, 0, 1)

α

β

1 2 n−3 − 2(n−2)

(n−4)−ω(n−2) 4

0

γ ω n−2 2 0

0

(n−4) 4(n−2)

ω 2

0

0

− 12

n−3 ω 2(n−2)

0

0

7.5. Rigidity IV: Weyl scalars

W ∗ Ric ∗ Ric div(W ) ∗ ∇ Ric

−

209

δ2 (n−2)2 16

δ1 (n−2)2 16

−

δ0 (n−4)2 16

0

0

B ∗ Ric

0

0

0  2 n−4 − 4(n−2)

B(∇f, ∇f ) div4 (W )

0 0

0 0

0 0

A ∈ Ω− C-deg. D-deg. (n=4)

D-deg. C-deg.

Here ∗ denotes a suitable contraction, according to the definition of the Weyl scalars given in Lemma 7.51. By Propositions 7.56 and 7.57, we can deduce that every compact shrinking solitons of dimension n ≥ 4 for which one of the five Weyl scalars in the first column of the previous tables vanishes must be Einstein. In the case B ∗ Ric we get the result only in dimension n = 4. By Corollary 7.53, we include also the condition Wijkl,ilk fj = 0. Note that, in particular, if B(∇f, ∇f ) = 0 or div4 (W ) = 0, we recover the results in [34] or [55] (see Theorem 7.38), respectively; on the other hand, the first three cases provide new conditions ensuring the classification. This proves Proposition 7.46. Modified Bach tensors We prove Proposition 7.47. Let (M n , g) be a compact Ricci solitons of dimension n ≥ 4 such that   1 c1 Wikjl,lk + c2 Wtikj,t fk + Wikjl Rkl Rij = 0 on M, n−2 1 and c2 > − n−2 . Equivalently, one has wG (A) ≡   1 A= , 0, −c2 , 0, c1 , 0 . n−2 A computation shows that in this case we have

for some c1 , c2 ∈ R with c1 6= 0 with

1 n−3

α

β

1−(n−3)c1 2(n−2)

1 4

n

n−4 n−2

γ

o + (n − 3)c2 − ω[1 − (n − 3)c1 ]

ω 2

and δ2  2 (n−3)c1 −1 − 4

δ1 −(n−3)2 c1 c2 +(n−3)(c2 −c1 )+1 16

δ0  2 (n−4)+(n−2)(n−3)c2 − 4(n−2)

In particular, a straightforward computation yields 2

δ12 − δ0 δ2 =

(n − 3)[(n − 2)c2 + 1][(n − 3)c1 − 1] > 0. 64(n − 2)2

Thus A ∈ Ω− and Proposition 7.47 follows from Proposition 7.57.

210

Chapter 7. Ricci Solitons: Selected Results

Mixed Weyl scalars We prove first Proposition 7.48. Let (M n , g) be a compact Ricci soliton of dimension n ≥ 4 such that 1 Wtijk,tk Rij + c4 Wtijk,t Rik,j n−3 1 + c5 Wtijk,t Rik fj + c6 Wtijk Rik,jt + c7 Wtijk Rik ft fj + Wtijk Rtj Rik = 0, n−2

c1 Wtijk,tkji + c2 Wtijk,tkj fi + c3 Wtijk,tk fi fj +

for some ci ∈ R, i = 1, . . . , 7, with either c1 > 0 or c1 = 0 and c4 + From Corollary 7.53, we get wG (A) ≡ 0 with   1 c6 n−2 1 A= , −c7 , − c5 , − c6 − c4 , , c1 . n−2 n−2 n−3 n−3 A computation shows that δ2 =

n−2 n−3 c6

6= 0.

n−3 c1 . 4(n − 2)

If c1 > 0, then A ∈ Ω+ and Proposition 7.48 follows from Proposition 7.57. On the other hand, if c1 = 0 and c4 + n−2 n−3 c6 6= 0, then δ2 = 0 and   n−3 n−2 δ1 = c4 + c6 6= 0. 4(n − 2) n−3 Thus A ∈ Ω0 and Proposition 7.48 follows again from Proposition 7.57. Now let us prove Proposition 7.49. Let (M 4 , g) be a compact Ricci soliton of dimension four such that c1 Wtijk,tkji + c2 Wtijk,tkj fi + c3 Wtijk,tk fi fj + c4 Wtijk,tk Rij 1 + c5 Wtijk,t Rik,j − c3 Wtijk,t Rik fj + c6 Wtijk Rik,jt + Wtijk Rtj Rik = 0, 2 for some ci ∈ R, i = 1, . . . , 6, with 1 + c2 + c4 + c5 + c6 6= 0. Using Corollary 7.53 we see that wG (A) ≡ 0, with   1 A= , 0, 0, −c2 − c5 , −c4 − c6 , c1 . 2 A computation shows that α 1+c1 ω+c2 +c4 +c5 +c6 4

and

β − 1+c85 +c6 ω

γ ω 2

7.5. Rigidity IV: Weyl scalars δ2 (1+c5 +c6 )2 − 32

211

+

c1 8

δ1 1+c2 +c4 +c5 +c6 16

δ0 0

In particular, δ12 − δ0 δ2 = δ12 > 0 and A ∈ Ωd . Now Proposition 7.49 follows from Proposition 7.57. Remark 7.60. We explicitely note that, if (M, g) is a complete non-compact gradient shrinking Ricci soliton with bounded curvature, then Proposition 7.57 applyes. In fact, by Shi’s estimates (see e.g. [65]), |∇k Riem| is bounded for every k ∈ N, implying that |C| and |D are bounded. Moreover, it is well known that the potential function f grows quadratically at infinity, while the volume of geodesic balls is at most Euclidean (see [35]). Hence, all the integration by parts in the proof of Corollary 7.54 can be justified using standard cutoff functions as soon as ω can be chosen to be positive. For instance, this is the case when A ∈ Ω+ . Thus, using the classification of non-compact shrinkers with D = 0 (see [34]), one has Proposition 7.61. Let (M n , g) be a non-compact Ricci solitons of dimension n ≥ 4. If, on M , 1 Wtijk,tk Rij + c4 Wtijk,t Rik,j n−3 1 + c5 Wtijk,t Rik fj + c6 Wtijk Rik,jt + c7 Wtijk Rik ft fj + Wtijk Rtj Rik = 0, n−2

c1 Wtijk,tkji + c2 Wtijk,tkj fi + c3 Wtijk,tk fi fj +

for some ci ∈ R, i = 1, . . . , 7, with c1 > 0, then (M, g) is isometric to a finite quotient of N n−1 × R, where N n−1 is Einstein and R is the Gaussian shrinking soliton.

Chapter 8

Existence Results of Canonical Metrics on Four-Manifolds In this chapter, following the recent work [56], we study the existence of minimizers in a conformal class for the quadratic functional D(g) given by the (rescaled) L2 norm of the divergence of the Weyl tensor. In general, the problem is degenerate elliptic due to possible vanishing of the Weyl tensor. In order to overcome this issue, we minimize the functional in the conformal class determined by the reference metric constructed by Aubin, with nowhere vanishing Weyl tensor (see Theorem 3.10 in Chapter 3).

8.1

A new variational problem: Weak harmonic Weyl metrics

As we recalled in the Introduction, and also in Chapter 5, given a closed smooth manifold M it is natural to study canonical Riemannian metrics g on M as critical points of certain functionals defined on the space of metrics. Perhaps the most famous of such functionals is the Einstein-Hilbert action Z n−2 S(g) := Volg (M )− n Rg dVg . M

All stationary points of S(g) are Einstein metrics; while the existence of Einstein metrics as critical points of S(g) is not guaranteed (for instance, in dimension four, the topological restriction (5.3) of Chapter 5 holds), the constrained version of the problem (i.e., the so-called Yamabe problem) always admits a solution. More precisely, Yamabe, Aubin, Trudinger, and Schoen (see [111]) showed that the Yamabe invariant Y(M, [g]) := inf S(e g) g e∈[g]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 G. Catino, P. Mastrolia, A Perspective on Canonical Riemannian Metrics, Progress in Mathematics 336, https://doi.org/10.1007/978-3-030-57185-6_8

213

214

Chapter 8. Existence Results of Canonical Metrics on Four-Manifolds

is always attained in the conformal class [g]. Moreover, every critical point of the normalized functional in the conformal class has constant scalar curvature. In the last decades several curvature conditions generalizing Einstein metrics have been investigated by many authors, with important examples arising as critical points of functionals which are quadratic in the curvatures (see the discussions in Chapter 4 and 5). In general, the associated Euler-Lagrange equation is of the fourth order in the metric, hence obtaining a satisfactory existence theory can be challenging. We know that an important class of metrics generalizing the Einstein condition is given by the harmonic Weyl metrics, i.e., the metrics with divergence-free Weyl tensor, δg Wg = 0 (see again [11], [68] and Chapter 6). In fact, it is well known that all Einstein metrics have harmonic Weyl tensor and that, on four-dimensional closed manifolds, there are topological obstructions to the existence of harmonic Weyl metrics with constant scalar curvature, i.e. metrics with harmonic curvature (see [19, 70]). From now on, we assume that M 4 is a four-dimensional closed smooth manifold; observe that all harmonic Weyl metrics are critical points of the quadratic scale-invariant functional Z 1 2 D(g) := Volg (M ) |δg Wg |2g dVg , M

while in general the converse does not hold. Note that conformal variations give rise to a second-order Euler-Lagrange equation, since the transformation law of δW (see (2.86)) is given by δgeWge = δg W + Wg (∇g u, ·, ·, ·) for every conformal metric ge = e2u g ∈ [g]. Thus, in the same spirit as the Yamabe problem, it seems natural to define the conformal invariant D(M, [g]) := inf D(e g) . g e∈[g]

In this chapter, following the recent paper [56], we study the existence of minimizers in the conformal class for the functional g 7→ D(g). In general, the problem is degenerate elliptic due to possible vanishing of the Weyl tensor. To overcome this issue, we minimize the functional in the conformal class determined by a reference metric constructed by Aubin, with nowhere vanishing Weyl tensor (see Theorem 3.10). On the contrary, for the Yamabe problem the existence of minimizers is guaranteed in every conformal class. Besides the aforementioned variational point of view, there is another geometric motivation for studying constrained critical points of g 7→ D(g). Indeed, it was proved by Derdzinski [68] that, on four-manifolds, harmonic Weyl metrics satisfy the Weitzenb¨ock equation (6.5), that is 1 1 ∆|W |2 = |∇W |2 + R|W |2 − 3 Wijkl Wijpq Wklpq . 2 2

(8.1)

8.1. A new variational problem: Weak harmonic Weyl metrics

215

On the other hand, Chang, Gursky and Yang [60] showed that, on every closed four-manifold (M 4 , g), the following integral formula holds (see Corollary 6.2):  Z  1 2 2 2 |∇W | − 4|δW | + R|W | − 3 Wijkl Wijpq Wklpq dVg = 0 . (8.2) 2 M A simple consequence is that, on a closed four-manifold M 4 , δg Wg = 0

⇐⇒

equation (8.1) holds on (M 4 , g) .

In Section 8.2.1 we show that a metric is critical in the conformal class for the functional g 7→ D(g) if and only if it satisfies the Weitzenb¨ock equation 1 1 ∆|W |2 = |∇W |2 + R|W |2 − 3 Wijkl Wijpq Wklpq − 8|δW |2 2 2Z 4 + |δW |2 dVg . Vol(M ) M

(8.3)

Note that this equation reduces to (8.1) if δW = 0. Hence we are led to the following Definition 8.1. Let M 4 be a closed four-dimensional manifold. A Riemannian metric g on M 4 is a weak harmonic Weyl metric if the Weitzenb¨ock equation (8.3) holds on (M 4 , g). Clearly, harmonic Weyl metrics (and Einstein metrics) are weak harmonic Weyl metrics. Let us emphasize that integrating equation (8.3) we obtain the identity (8.2) and this gives no a priori obstructions to the existence of weak harmonic Weyl metrics, contrary to what happens with (8.1). Our first main result is the following Theorem 8.2. On every closed four-dimensional manifold there exists a weak harmonic Weyl metric. Remarks. 1. Aubin [5] proved that every closed Riemannian manifold admits a metric of constant negative scalar curvature (see also Section 2.2.3). Besides this fact, to the best of our knowledge, Theorem 8.2 is the only existence result of a canonical metric, which generalizes the Einstein condition, on every fourdimensional Riemannian manifold, without any topological obstructions. 2. To be more precise, the metric in Theorem 8.2 is constructed as follows: first, thanks to the result of Aubin contained in Theorem 3.10, on any given four-dimensional manifold M 4 we can choose a reference metric g0 such that |Wg0 |g0 > 0. Then, we prove that on (M 4 , g0 ) the infimum D(M, [g0 ]) is attained by a conformal metric g ∈ [g0 ], which is a weak harmonic Weyl metric. Moreover, we show that every critical point in the conformal class [g0 ] is necessarily a minimum point.

216

Chapter 8. Existence Results of Canonical Metrics on Four-Manifolds

3. From both the geometric and the analytic points of view, it would be interesting to understand which conformal classes of metrics contain a weak harmonic Weyl representative. 4. We can also consider the (anti-)self-dual functional Z 1 ± 2 D (g) := Volg (M ) |δg Wg± |2g dVg , M

and define weak half harmonic Weyl metrics as its critical points in the conformal class. In particular, we can prove that, given a closed four-manifold (M 4 , g0 ) with |Wg±0 |g0 > 0, there exists a weak half harmonic Weyl metric ge ∈ [g0 ]. However, we do not know if the aforementioned result of Aubin can be extended to the (anti-)self-dual Weyl tensor W ± . In order to prove this theorem, we endow a closed four-manifolds M 4 with the metric g0 constructed by Aubin and we consider the functional D(v) := D(v −2 g0 ) Z  12 Z −4 = v dV M



M

 1 2 2 2 2 2 |W | |∇v| + |δW | v − (v)s Wsijk Wpijk,p dV , 4

where all the geometric quantities are referred to g0 and the function v belongs to the convex cone   Z H(M ) := u ∈ H 1 (M ) : u > 0 a.e. and u−4 dV < ∞ . M

The condition |W | > 0 is crucial, as it ensures the uniform ellipticity of the problem. A variational argument, combined with some spectral analysis and maximum principles, shows that u 7→ D(u) admits a minimum point v in H(M ). Consequently, v is a (weak) solution of the Euler-Lagrange equation
1 − div(|W |2 ∇v) + |δW |2 + div(Wsijk Wpijk,p ) v = D(v) 4

Z v M

−4

−3/2 dV

1 , v5

which is a uniformly elliptic semilinear equation with singular non-linearity. Here, again, all the geometric quantities are referred to g0 . Hence, by standard elliptic regularity theory, v ∈ C ∞ (M ) and D(M, [g0 ]) =

min

0 0 a.e. and u−4 dV < ∞ M

and define D :=

inf

D(u) .

u∈H(M )

By standard elliptic theory (see e.g. [82]), there exists a smooth, positive, first eigenfunction ϕ1 of L, i.e., a solution of Lϕ1 = λ1 ϕ1 . Note that R(ϕ1 ) = λ1 . We have the following weak maximum principle:

8.3. Existence of minimizers

221

Lemma 8.4. Let λ1 > 0. Under the previous assumptions, if u ∈ H 1 (M ) satisfies Lu ≥ 0 in the weak sense, then u ≥ 0 a.e. on M . Moreover, using Lemma 8.4, one can prove the following strong maximum principle. Lemma 8.5. Let λ1 > 0. Under the previous assumptions, if u ∈ H 1 (M ) satisfies Lu ≥ 0 in the weak sense, then either u = 0 a.e. on M , or ess inf M u > 0. We have a two-sided estimate on D in terms of λ1 Lemma 8.6. Under the previous assumptions, R 3

Vol(M ) 2 λ1 ≤ D ≤ R

M M

ϕ21 dV

ϕ−4 1 dV


12 λ1 .

Proof. By the Jensen inequality, for every u ∈ H(M ), 1 ≤ 2 dV u M

Z

R

u−4 dV

 12

3

Vol(M )− 2 .

M

Consequently, 3

λ1 ≤ R(u) ≤ D(u) Vol(M )− 2 and the first inequality follows. Moreover, for every u ∈ H(M ), R u2 dV D(u) = R(u) R M
1 , −4 dV 2 u M and so

R D ≤ D(ϕ1 ) = R(ϕ1 ) R

M M

ϕ21 dV

ϕ−4 1 dV

R
12 = R

M M

ϕ21 dV

ϕ−4 1 dV


12 λ1 .



Consequently, by Lemma 8.4 and Lemma 8.5, maximum principles hold whenever D > 0, and D = 0 if and only if λ1 = 0.

8.3.3 Existence In this subsection we prove that the functional u 7→ D(u) admits a minimum v in H(M ), and that v satisfies the associated Euler-Lagrange equation and is smooth. Lemma 8.7. Suppose that u ∈ H(M ) and that D(u) = D > 0. Then Lu ≥ 0 in the weak sense, i.e., Z {ah∇u, ∇ϕi + cuϕ} dV ≥ 0 for any ϕ ∈ C 1 (M ), ϕ ≥ 0 . M

222

Chapter 8. Existence Results of Canonical Metrics on Four-Manifolds

Proof. By contradiction, suppose that there exists ϕ ∈ C 1 (M ), ϕ ≥ 0, such that Z {ah∇u, ∇ϕi + cuϕ} dV < 0 . M 1

For v ∈ H (M ), define Z



Q(v) :=

a|∇v|2 + cv 2 dV .

M

Note that 2 Q(v) ≥ λ1 kvkL 2 .

Take any t ∈ R with |t| small enough. We have that D(u + tϕ) − D(u)  21 Z Z  12 = (u + tϕ)−4 dV Q(u + tϕ) − u−4 dV Q(u) M M "Z  1 Z 1 # 2

(u + tϕ)−4 dV

= M

Z

2

Q(u + tϕ)

M −4

+

u−4 dV

−

u

 12 [Q(u + tϕ) − Q(u)] .

dV

M

Furthermore, Q(u + tϕ) ≥ 0 , 1

1 (u + tϕ)−4 dV 2 − u−4 dV 2 for any t > 0 , and Q(u + tϕ) − Q(u) = Q0 (u)[ϕ]t + o(t) where Q0 (u)[ϕ] =

as t → 0,

Z {ah∇u, ∇ϕi + cuϕ}dV > 0 . M

Thus, for t > 0 sufficiently small, Z

−4

Q(u + tϕ) − Q(u) ≤

u

 12 dV

{Q0 (u)[ϕ]t + o(t)} < 0,

M

and so D(u + tϕ) < D(u) with u + tϕ > 0 a.e., u + tϕ ∈ H(M ). This is a contradiction, since D(u) = D .



8.3. Existence of minimizers

223

Corollary 8.8. Suppose that u ∈ H(M ) and D(u) = D > 0. Then ess inf M u > 0. Proof. The assertion follows from Lemmas 8.5 and 8.7.



∞

Theorem 8.9. There exists v ∈ C (M ), v > 0, such that b. D(v) = D Moreover, v satisfies Lu = D

Z v

−4

− 23 dV

v −5 .

M

Proof. First we suppose that D = 0. By Lemma 8.6, λ1 = 0. Moreover, R R ϕ2 dV λ1 M ϕ21 dV M 1 D(ϕ1 ) =
12 R(ϕ1 ) =
12 = 0 = D . ϕ−4 ϕ−4 1 dV 1 dV b = 0. Hence Since ϕ1 ∈ C ∞ (M ), also D b = 0. D(ϕ1 ) = D = D From now on we suppose that D > 0. Let {vn }n∈N ⊂ H(M ) be a sequence of functions such that D(vn ) → D. Since the R functional D is scale-invariant, without loss of generality we can assume that M vn−4 dV = 1. Since D > 0, in view of Lemma 8.6, we have that λ1 > 0. Moreover, Z Z 2 2 {a|∇vn | + cvn }dV ≥ λ1 vn2 dV . M

M

Clearly, for any n ∈ N sufficiently large, 0 < D ≤ D(vn ) ≤ D + 1 . Hence, Z λ1 M

vn2 dV

Z

{a|∇vn |2 + cvn2 }dV Z
1 = vn−4 dV 2 {a|∇vn |2 + cvn2 }dV ≤

M

M

= D(vn ) ≤ D + 1 . Therefore, the sequence {vn } is bounded in L2 (M ). Moreover, for n ∈ N sufficiently large, Z Z Z min a |∇vn |2 dV ≤ a|∇vn |2 dV = D(vn ) − cvn2 dV M

M

M

≤ D(vn ) + kckL∞ kvn k2L2 D+1 ≤ D + 1 + kckL∞ . λ1

M

(8.16)

224

Chapter 8. Existence Results of Canonical Metrics on Four-Manifolds

Therefore, the sequence {∇vn } is bounded in L2 (M ), and {vn } is bounded in H 1 (M ). Consequently, there exist a subsequence of {vn }, which we still denote by {vn }, and a function v ∈ H 1 (M ) such that vn −→ v

in H 1 (M ) ,

vn −→ v

in L2 (M ) ,

vn −→ v

a.e. in M .

n→∞

n→∞

n→∞

Therefore, vn−4 −→ v −4 n→∞

a.e. in M ;

1 here we have assumed that vn , v : M → [0, +∞] and ∞ = 0, 10 = ∞. By Fatou’s lemma, Z Z Z −4 −4 v dV = lim inf vn dV ≤ lim inf vn−4 dV = 1 . M n→∞

M

Thus

Z

n→∞

M

v −4 dV < +∞ .

M

This implies that v >R 0 a.e. in M . In fact, if {v = 0} in a set of positive measure, −4 since v ≥ 0, v ∈ H 1 (M ), v > 0 R we−4get M v dV = ∞, a contradiction. Hence 1 a.e. in M , M v dV ≤ 1. Using the fact that vn −→ v in H (M ) and vn −→ v in L2 (M ) as n → ∞, we can infer that Z
12 −4 D ≤ D(v) = v dV {a|∇v|2 + cv 2 }dV M Z ≤ {a|∇v|2 + cv 2 }dV M Z ≤ lim inf {a|∇vn |2 + cvn2 }dV = lim inf D(vn ) = D . n→∞

n→∞

M

So, D(v) = D > 0 . From Lemma 8.5 it follows that ess inf v > 0. Take any ϕ ∈ C 1 (M ). Since D(v) = D, we get d [D(v + tϕ)] = 0. dt t=0 Consequently, for any ϕ ∈ C 1 (M ), we have Z M

{ah∇v, ∇ϕi + cvϕ}dV = D

Z v M

−4

− 32 Z dV M

ϕ dV . v5

8.3. Existence of minimizers

225

Thus, Lu = D

Z v

−4

− 32

v −5 =: f

dV

weakly on M .

M

Since ess inf v > 0, we have that f ∈ L∞ (M ). Therefore, by standard elliptic regularity theory, v ∈ C ∞ (M ) and v > 0 on M . We can therefore infer that b ≤ D(v) = D ≤ D. b D Hence, D(v) = D . This completes the proof.



b so Remark 8.10. The proof of Theorem 8.9 also shows that D = D, b Lv = D

Z v

−4

− 23 dV

v −5

on M .

(8.17)

M

8.3.4 Uniqueness Observe that equation (8.17) is scale-invariant, in the sense that if u1 solves (8.17), then u2 := βu1 , with β ∈ R+ , satisfies b Lu2 = D

Z

− 23

u−4 2 dV

u−5 2

on M .

M

Therefore, uniqueness for equation (8.17) does not hold. However, we have the following result. Theorem 8.11. Suppose that u1 and u2 are solutions of equation (8.17) and that u1 > 0, u2 > 0 on M . Then there exists a β ∈ R+ such that u1 = βu2

on M .

Proof. Let Z

u−4 1 dV

µ :=

 14

Z ,

γ :=

M

u−4 2 dV

M

So, the functions ψ := µu1 ,

w := γu2

satisfy Z M

ψ −4 dV =

Z

w−4 dV = 1 .

M

Then b −5 Lψ = Dψ

on M ,

 14 .

226

Chapter 8. Existence Results of Canonical Metrics on Four-Manifolds b −5 Lw = Dw

on M .

We choose α > 0 such that ψ − αw ≥ 0 and min{ψ − αw} = 0 . M

Since M is compact, the continuous function ψ − αw admits a minimum point b > 0. We have that x0 ∈ M , so that ψ(x0 ) = αw(x0 ) . First assume that D b −5 − αw−5 ) L(ψ − αw) = D(ψ

in M .

In particular, at x0 we obtain 6 b b −5 (x0 ) − αw−5 (x0 )) = D(1 − α ) . 0 ≥ D(ψ α5 w5 (x0 )

This yields α ≥ 1, and so, ψ ≥ αw ≥ w

in M .

By repeating the same argument with the roles of ψ and w interchanged, we get w≥ψ

in M .

Hence µu1 = ψ = w = γu2

in M .

Thus, we obtain the thesis with β = µγ . b = 0. Since ψ − αw ≥ 0, if we take m > kckL∞ , then Now, assume that D L(ψ − αw) + m(ψ − αw) ≥ 0

on M,

and minM {ψ − αw} = 0 . Observe that λ1 (L + m Id) = λ1 (L) + m = m > 0. Thus, by Lemma 8.5 applied to the operator L + m Id for the function ψ − αw, we obtain ψ = αw. Therefore, u1 = The proof is now complete.

γ αu2 µ

in M . 

b = 0 is equivalent to the Remark 8.12. The proof of Theorem 8.11 in the case D proof that λ1 (L) is simple.

8.3. Existence of minimizers

227

Consider equation Z Lu = D(u)

u

−4

− 32 dV

u−5

on M .

(8.18)

M

Observe that D(u) = D(βu) for any β ∈ R+ . Furthermore, equation (8.18) is scale-invariant as before, so uniqueness for equation (8.18) fails. However, we have the following result. Theorem 8.13. Suppose that u1 and u2 are solutions to equation (8.18), and that u1 > 0,u2 > 0 on M . Then there exists a β ∈ R+ such that u1 = βu2

on M .

Proof. First assume that D(u1 ) > 0, D(u2 ) > 0. Let Z

u−4 1 dV

µ :=

 41

Z D(u1 ),

γ :=

M

u−4 2 dV

 14 D(u2 ).

M

So, the functions ψ := µu1 ,

w := γu2

satisfy Lψ = ψ −5

on M ,

−5

on M .

Lw = w

b > 0. In the Hence the conclusion follows as in the proof of Theorem 8.11, when D case D(u1 ) = D(u2 ) = 0, too, the assertion is obtained by the same arguments as b = 0. in the proof of Theorem 8.11, when D We claim that the case D(u1 ) > 0 and D(u2 ) = 0 cannot happen. Indeed, by contradiction, assume that D(u1 ) > 0 and D(u2 ) = 0. Define ψ := µu1 ,

w := u2 .

Choose α > 0 such that ψ − αw ≥ 0 and min{ψ − αw} = 0 . M

Since M is compact, we can find a minimum point x0 ∈ M of the continuous function ψ − αw, so that ψ(x0 ) = αw(x0 ) . We have that L(ψ − αw) = ψ −5

in M .

In particular, at x0 we obtain 0 ≥ ψ −5 (x0 ) > 0 . This is a contradiction. The proof is now complete.



228

Chapter 8. Existence Results of Canonical Metrics on Four-Manifolds

Corollary 8.14. Every critical point of the functional u 7→ D(u), defined on H(M ), is a minimum point. Proof. Let w be a critical point of the functional u 7→ D(u). Recall that this is equivalent to requiring that D0 (w) = 0, i.e., w is a solution of equation (8.18). By Theorem 8.9, there exists a minimum point v of D(u), which is a solution of equation (8.17). By Theorem 8.13 with u1 = w and u2 = v, we can infer that w = βv, for some β > 0. Then b. D(w) = D(βv) = D(v) = D This proves the claim.



8.3.5 Further results For an arbitrary β > 0, consider the equation Lu = βu−5

on M .

(8.19)

Let l := min ϕ1 ,

l := max ϕ1 .

M

M

Proposition 8.15. Assume that λ1 > 0 and β > 0. Then there exists a solution u ∈ C ∞ (M ) of equation (8.19) such that l l



β λ1

 16 ≤u≤

l l



β λ1

 16 on M .

Moreover, if v > 0 is any other solution of equation (8.19), then v = u on M . Proof. Define u := αϕ1 ,

u := αϕ1 ,

where α, α are positive constants to be chosen below. It is readily seen that if 1 1 α ≤ β 6 /(lλ16 ), then u is a subsolution of equation (8.19), that is Lu ≤ βu−5

on M .

In fact, Lu = λ1 αϕ1 ≤ βu−5 = βα−5 ϕ−5 1 1

1 6

on M , 1

1

provided that α ≤ β 6 /(lλ1 ). It is similarly seen that if α ≥ β 6 /(lλ16 ), then u is a supersolution of equation (8.19), that is Lu ≥ β u−5

on M .

Clearly, 0 < α ≤ α. Define Lu := − div (a∇u) .

8.3. Existence of minimizers

229

Hence equation (8.18) is equivalent to equation Lu = f (u) on M ,

(8.20)

where f (u) := −cu + u−5 . We have shown that u is a subsolution of equation (8.20), while u is a supersolution of equation (8.20). Moreover, 0 < α l ≤ u ≤ u ≤ αl

on M ,

and f ∈ C 1 ([α l, αl]). Hence, by the standard method of sub- and supersolutions, we can infer that there exists a weak solution to equation (8.20), and hence to equation (8.19), satisfying u ≤ u ≤ u in M . By standard regularity theory it follows that u ∈ C ∞ (M ). Moreover, arguing as b > 0 we can infer that if v > 0 is any other in the proof of Theorem 8.11 when D solution of equation (8.19), then v = u. This completes the proof.  b > 0. Let v be a solution of equation (8.17). Then Proposition 8.16. Suppose that D l l

b D λ1

! 61

Z ≤v M

v −4 dV

 14

l ≤ l

b D λ1

! 16 in M .

Proof. Let v be a solution of equation (8.17). Then v is also a solution of equation
− 3 R b (8.19) with β = D v −4 dV 2 . It remains to apply Proposition 8.15.  M

List of Symbols (U, ϕ)

local chart, page 1

δij

suggestive way of writing the Kronecker symbol, page 2

δij

Kronecker symbol, page 2

Λ2 (U )

space of skew-symmetric 2-forms on the open set U , page 7

L∇f g

Lie derivative of the metric g in the direction of ∇f , page 17

X(U )

set of smooth vector fields on the open set U , page 3

∇ω

covariant derivative of the 1-form ω, page 4

∇T

covariant derivative of the tensor field T , page 4

∇u

gradient of the function u, page 5

∇X

covariant derivative of the vector field X, page 3

∇

connection induced by the Levi-Civita connection forms, page 3

∇Y ω

covariant derivative of ω in the direction of Y , page 4

∇Y X

covariant derivative of X in the direction of Y , page 3

ωik

covariant derivative of the coefficient ωi , page 4

div X

divergence of the vector field X, page 4

Tor

torsion tensor, page 6

Tr

trace, page 4

⊗

tensor product, page 3

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 G. Catino, P. Mastrolia, A Perspective on Canonical Riemannian Metrics, Progress in Mathematics 336, https://doi.org/10.1007/978-3-030-57185-6

231

232

LIST OF SYMBOLS Kulkarni-Nomizu product, page 22

? (M, g)

Riemannian manifold with metric g, page 1

˚ Ric

traceless Ricci tensor, page 14

Ric

Ricci tensor, page 10

Riem

(0, 4)-version of the Riemann curvature tensor, page 8

θji

Levi-Civita connection forms, page 2

Θij

curvature forms, page 6

{θi }

(local) orthonormal coframe, page 2

{ei }

(local) orthonormal frame, page 2

A

Schouten tensor, page 22

B

Bach tensor, page 24

C

Cotton tensor, page 23

dθ

exterior differential of the 1-form θ, page 6

du

differential of the function u, page 5

gij

(local) components of the metric, page 2

K(u ∧ v), Kp (Π),

M

K(u ∧ v) sectional curvature of the plane Π ⊂ Tp M spanned by u and v, page 11

Kp (Π)

sectional curvature of the 2-plane Π ⊂ Tp M , page 11 (1, 3)-version of the Riemann curvature tensor, page 7

R S,

M

M

S, R

scalar curvature (of the manifold M ), page 10

Sec(p)

infimum of the sectional curvature at the point p, page 11

Tp∗ M

cotangent space at a point p ∈ M , page 4

Tp M

tangent space at a point p ∈ M , page 4

ui

local components of the differential du, page 5

W

Weyl tensor, page 21

Xki

covariant derivative of the coefficient X i , page 3

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Index f -Einstein tensor, 80 f -Laplacian, 161 f -Schouten tensor, 78 f -Yamabe metric, 73 f -curvatures, 74 f -harmonic curvature metric, 73 f -locally symmetric Einstein metric, 73 f -locally symmetric metric, 73 f -space form, 73 f -structure trivial, xv, 74 f -structures, 73 X-Laplacian, 172 X-structures, 103 (local) components of the metric, 2 “cathedral” of f -structures, 74 “fake” second Bianchi identity for the Weyl tensor, 31 anti-self-dual two-forms, 58 anti-self-dual part of the Weyl tensor, 58 Aubin’s deformation of the metric, 45 Bach tensor, 24 Bakry-Emery Ricci tensor, xiv, 72 Betti numbers, 65 Bianchi identities, 8 Bochner formula for functions, 18 Bochner-Weizenb¨ ock formula for functions, 18

canonical metrics and curvature conditions, 71 CC approach algebraic, xii analytic, xii Christoffel symbols, 14 classification of HCf spaces, 83 of LSf and LSEf spaces, 82 of f -space forms, 81 Codazzi tensor, 24 commutation relations for the covariant derivatives of a smooth function, 28 for the covariant derivatives of a vector field, 29 for the covariant derivatives of the Ricci tensor, 30 for the covariant derivatives of the Schouten tensor, 30 for the second and third covariant derivatives of the Riemann curvature tensor, 29 for the second and third covariant derivatives of the Weyl tensor, 31 compatibility of the connection with the metric in coordinates, 35 compatibility of the covariant derivative with the metric, 5 components of the Hessian, 17

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 G. Catino, P. Mastrolia, A Perspective on Canonical Riemannian Metrics, Progress in Mathematics 336, https://doi.org/10.1007/978-3-030-57185-6

243

244 components of the Riemann curvature tensor, 7 conformal vector field, 90 conformal class of a metric, 111 conformal deformation of a metric, 19 connection dual, 3 induced by the Levi-Civita connection forms, 3 Levi-Civita, 6 coordinate functions, 2 cotangent space at a point p, 4 Cotton tensor, 23 and covariant derivative of the Schouten tensor, 23 covariant derivative, 2 of a 1-form, 4 of a function, 5 of a tensor field, 4 of a vector field, 3 curvature forms, 6 curvature tensor, 7 de Rham decomposition theorem, xiii, 72 decomposition of the curvature in dimension four, 58 decomposition of the curvature tensor in three orthogonal parts, 24 using the Ricci tensor, 21 using the Schouten tensor, 22 using the traceless Ricci tensor, 24 Derdzinski basis, 58 differential of a function, 5 divergence of the Riemann curvature tensor, 52 of a vector field, 4

Index of the Bach tensor, 25 of the Weyl tensor, 23 divergences of the Cotton tensor, 33 dual frame, 2 eigenspaces of the Hodge operator, 58 Einstein manifold, 13 Einstein summation convention, 1 Einstein tensor, 79 Einstein-Hilbert action, 109 Euler characteristic, 65 expanding Ricci soliton, xv, 73 exterior differential of a 1-form, 6 first Bianchi identity, 8 for the Cotton tensor, 33 for the Weyl tensor, 21 fundamental theorem of Riemannian geometry, 6 Gaussian curvature, 13 generic Ricci soliton, 172 geometric quantity, 33 geometric meaning of the first structure equations, 6 of the skew-symmetry of the LeviCivita connection forms, 5 gradient of a function, 5 of a functional, 110 of a geometric functional, 39 Ricci soliton, 73 Gram-Schmidt orthonormalization process, 2 half harmonic Weyl metric, 67 half-conformally flat metric, 58

Index harmonic curvature, 52 Weyl curvature, 52 harmonic coordinates, 164 Hessian of a function, 17 Hitchin-Thorpe inequality, 111 Hodge Laplacian, xiii Hodge operator, 58 inequality of Thorpe and Hitchin, 111 Yamabe-Sobolev, 111 integrability conditions for gradient Ricci solitons, 166 Jacobi formula, 34 Kato inequality, 67 Kazdan-Warner identity, 90 Kronecker symbol, 2 Kulkarni-Nomizu product, 22 Laplacian of a function, 18 of a tensor (rough), 18 of the squared norm of a (covariant) tensor, 18 Levi-Civita connection, 6 Levi-Civita connection forms, 2 Lie bracket, 5 Lie derivative of the metric, 17 linear connection, 2 local chart, 1 locally conformally flat, 57 locally symmetric Einstein metric, 72 locally symmetric metric, 72 lowering an index, 8 metric f -Yamabe, 73 f -locally symmetric, 73 f -locally symmetric Einstein, 73 half-conformally flat, 58

245 locally conformally flat, 57 locally symmetric, 72 locally symmetric Einstein, 72 with f -harmonic curvature, 73 with parallel Bakry-Emery Ricci tensor, 74 with parallel Ricci curvature, 72 Yamabe, 72 musical isomorphism, 5 obstructions to the existence of Einstein metrics, 112 orthonormal coframe, 2 frame, 2 parallel metric, 16 parallelism of the metric, 5 Perelman’s functionals, 75 Poincar´e conjecture, xiv, 72 polarization formula, 12 potential, xiv, 72 of a Ricci soliton, 159 quaternionic structure on Tx M , 59 regularity of gradient Ricci solitons, 164 Ricci flow, xiv, 72, 163 Ricci soliton, xiv, 72, 159 expanding, 159 generic, 172 shrinking, 159 steady, 159 trivial, 159 Ricci soliton structure, 159 Ricci tensor, 10 Riemann curvature tensor (0, 4)-version, 8 (1, 3)-version, 7

246 rough Laplacian of a tensor, 18 scalar curvature, 10 Schouten tensor, 22 Schur’s identity, 13 Schur’s theorem, 13 second Bianchi identity, 10 second elementary symmetric function, 120 sectional curvature, 11 constant, 12 self-dual two-forms, 58 self-dual part of the Weyl tensor, 58 sharp map, 5 Shi estimates, 211 shrinking Ricci soliton, xv, 73 shrinking Ricci solitons with positive sectional curvature, 167 signature, 65 soliton equation, 159 space form, 71 steady Ricci soliton, xv, 73 structure equations first, 2 second, 6 symmetries of the curvature tensor, 8 tangent space at a point p, 4 tensor f -Einstein, 80 f -Schouten, 78 Einstein, 79 tensor field, 4 torsion, 6 torsion tensor, 6 trace, 4 traceless Ricci tensor, 14

Index trivial f -structure, xv, 74 variation of a geometric functional, 39 of a geometric quantity, 33 of the Christoffel symbols, 35 of the covariant derivative of the Ricci tensor, 38 of the covariant derivative of the Riemann tensor, 37 of the differential of the scalar curvature, 38 of the gradient of a function, 38 of the Hessian of a function, 38 of the inverse of the metric tensor, 34 of the Laplacian of a function, 39 of the Ricci tensor, 36 of the Riemann curvature tensor ((0, 4)-version), 35 of the scalar curvature, 36 of the volume form, 34 of the Weyl tensor, 37 variation of the Bach tensor under a conformal deformation of the metric, 44 variation of the Christoffel symbols under Aubin’s deformation of the metric, 46 variation of the Cotton tensor under a conformal deformation of the metric, 43 variation of the covariant derivative of a vector field under a conformal deformation of the metric, 44 variation of the covariant derivative of the Schouten tensor under a conformal deformation of the metric, 42

Index variation of the divergence of a vector field under a conformal deformation of the metric, 45 variation of the gradient of a function under a conformal deformation of the metric, 41 under Aubin’s deformation of the metric, 46 variation of the Hessian of a function under a conformal deformation of the metric, 41 under Aubin’s deformation of the metric, 46 variation of the inverse of the metric under Aubin’s deformation of the metric, 45 variation of the Laplacian of a function under a conformal deformation of the metric, 41 variation of the Lie derivative of the metric under a conformal deformation of the metric, 44 variation of the Ricci tensor under Aubin’s deformation of the metric, 48 variation of the Riemann curvature tensor under Aubin’s deformation of the metric, 47 variation of the scalar curvature under Aubin’s deformation of the metric, 48 variation of the Schouten tensor under a conformal deformation of the metric, 42 variation of the second covariant derivative of the Schouten tensor under a conformal deformation of the metric, 42

247 variation of the Weyl tensor under a conformal deformation of the metric, 43 under Aubin’s deformation of the metric, 48 vector field conformal, 90 Weyl scalars mixed, 210 Weyl tensor, 21 Yamabe invariant, 111 metric, 111 Yamabe metric, 71 Yamabe-Sobolev inequality, 111 Yang-Mills connections, xiii functional, xiii Theory, xiii, 72