Ricci Flow and the Sphere Theorem (cid:27)(cid:15)(cid:22)(cid:28)(cid:16)(cid:4)!(cid:23)(cid:8)(cid:16)"(cid:26)(cid:8) #(cid:23)(cid:6)"’(cid:6)(cid:25)(cid:8)(cid:4)(cid:27)(cid:25)’"(cid:15)(cid:8)(cid:9)(cid:4) (cid:15)(cid:16)(cid:4)(cid:24)(cid:6)(cid:25)(cid:17)(cid:8)(cid:22)(cid:6)(cid:25)(cid:15)(cid:12)(cid:9) ((cid:28)(cid:26)’(cid:22)(cid:8)(cid:4)(cid:1)(cid:1)(cid:1) (cid:21)(cid:22)(cid:8)(cid:23)(cid:15)(cid:12)(cid:6)(cid:16)(cid:4)(cid:24)(cid:6)(cid:25)(cid:17)(cid:8)(cid:22)(cid:6)(cid:25)(cid:15)(cid:12)(cid:6)(cid:26)(cid:4)(cid:27)(cid:28)(cid:12)(cid:15)(cid:8)(cid:25)(cid:29) Ricci Flow and the Sphere Theorem Simon Brendle Graduate Studies in Mathematics Volume 111 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Steven G. Krantz Rafe Mazzeo Martin Scharlemann 2000 Mathematics Subject Classification. Primary 53C20, 53C21, 53C44. For additional informationand updates on this book, visit www.ams.org/bookpages/gsm-111 Library of Congress Cataloging-in-Publication Data Brendle,Simon,1981– Ricciflowandthespheretheorem/SimonBrendle. p.cm. —(Graduatestudiesinmathematics;v.111) Includesbibliographicalreferencesandindex. ISBN978-0-8218-4938-5(alk.paper) 1.Ricciflow. 2.Sphere. I.Title. QA377.3B74 2010 516.3(cid:1)62—dc22 2009037261 Copying and reprinting. Individual readers of this publication, and nonprofit libraries actingforthem,arepermittedtomakefairuseofthematerial,suchastocopyachapterforuse in teaching or research. Permission is granted to quote brief passages from this publication in reviews,providedthecustomaryacknowledgmentofthesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication is permitted only under license from the American Mathematical Society. Requests for such permissionshouldbeaddressedtotheAcquisitionsDepartment,AmericanMathematicalSociety, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by [email protected]. (cid:1)c 2010bytheauthor. PrintedintheUnitedStatesofAmerica. (cid:1)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 151413121110 Contents Preface v Chapter 1. A survey of sphere theorems in geometry 1 §1.1. Riemannian geometry background 1 §1.2. The Topological Sphere Theorem 6 §1.3. The Diameter Sphere Theorem 7 §1.4. The Sphere Theorem of Micallef and Moore 9 §1.5. Exotic Spheres and the Differentiable Sphere Theorem 13 Chapter 2. Hamilton’s Ricci flow 15 §2.1. Definition and special solutions 15 §2.2. Short-time existence and uniqueness 17 §2.3. Evolution of the Riemann curvature tensor 21 §2.4. Evolution of the Ricci and scalar curvature 28 Chapter 3. Interior estimates 31 §3.1. Estimates for the derivatives of the curvature tensor 31 §3.2. Derivative estimates for tensors 33 §3.3. Curvature blow-up at finite-time singularities 36 Chapter 4. Ricci flow on S2 37 §4.1. Gradient Ricci solitons on S2 37 §4.2. Monotonicity of Hamilton’s entropy functional 39 §4.3. Convergence to a constant curvature metric 45 Chapter 5. Pointwise curvature estimates 49 §5.1. Introduction 49 iii (cid:160)iv Contents §5.2. The tangent and normal cone to a convex set 49 §5.3. Hamilton’s maximum principle for the Ricci flow 53 §5.4. Hamilton’s convergence criterion for the Ricci flow 58 Chapter 6. Curvature pinching in dimension 3 67 §6.1. Three-manifolds with positive Ricci curvature 67 §6.2. The curvature estimate of Hamilton and Ivey 70 Chapter 7. Preserved curvature conditions in higher dimensions 73 §7.1. Introduction 73 §7.2. Nonnegative isotropic curvature 74 §7.3. Proof of Proposition 7.4 77 §7.4. The cone C˜ 87 §7.5. The cone Cˆ 90 §7.6. An invariant set which lies between C˜ and Cˆ 93 §7.7. An overview of various curvature conditions 100 Chapter 8. Convergence results in higher dimensions 101 §8.1. An algebraic identity for curvature tensors 101 §8.2. Constructing a family of invariant cones 106 §8.3. Proof of the Differentiable Sphere Theorem 112 §8.4. An improved convergence theorem 117 Chapter 9. Rigidity results 121 §9.1. Introduction 121 §9.2. Berger’s classification of holonomy groups 121 §9.3. A version of the strict maximum principle 123 §9.4. Three-manifolds with nonnegative Ricci curvature 126 §9.5. Manifolds with nonnegative isotropic curvature 129 §9.6. K¨ahler-Einstein and quaternionic-K¨ahler manifolds 135 §9.7. A generalization of a theorem of Tachibana 146 §9.8. Classification results 149 Appendix A. Convergence of evolving metrics 155 Appendix B. Results from complex linear algebra 159 Problems 163 Bibliography 169 Index 175 Preface In this book, we study the evolution of Riemannian metrics under the Ricci flow. This evolution equation was introduced in a seminal paper by R. Hamilton [44], following earlier work of Eells and Sampson [33] on the harmonic map heat flow. Using the Ricci flow, Hamilton proved that ev- ery compact three-manifold with positive Ricci curvature is diffeomorphic to a spherical space form. The Ricci flow has since been used to resolve longstandingopenquestionsinRiemanniangeometryandthree-dimensional topology. In this text, we focus on the convergence theory for the Ricci flow in higher dimensions and its application to the Differentiable Sphere Theo- rem. The results we describe have all appeared in research articles. How- ever, we have made an effort to simplify various arguments and streamline the exposition. InChapter1,wegiveasurveyofvariousspheretheoremsinRiemannian geometry (see also [22]). We first describe the Topological Sphere Theorem of Berger and Klingenberg. We then discuss various generalizations of that theorem,suchastheDiameterSphereTheoremofGroveandShiohama[42] and the Sphere Theorem of Micallef and Moore [60]. These results rely on the variational theory for geodesics and harmonic maps, respectively. We will discuss the main ideas involved in the proof; however, this material will not be used in later chapters. Finally, we state the Differentiable Sphere Theorem obtained by the author and R. Schoen [20]. In Chapter 2, we state the definition of the Ricci flow and describe the short-time existence and uniqueness theory. We then study how the Riemann curvature tensor changes when the metric evolves under the Ricci flow. This evolution equation will be the basis for all the a priori estimates established in later chapters. v vi Preface In Chapter 3, we describe Shi’s estimates for the covariant derivatives of the curvature tensor. As an application, we show that the Ricci flow cannotdevelopasingularityinfinitetimeunlessthecurvatureisunbounded. Moreover, we establish interior estimates for solutions of linear parabolic equations. These estimates play an important role in Sections 4.3 and 5.4. In Chapter 4, we consider the Ricci flow on S2. In Section 4.1, we show thatanygradientRiccisolitononS2 hasconstantcurvature. Wethenstudy solutionstotheRicciflowonS2 withpositivescalarcurvature. Atheoremof Hamilton[46] asserts that such a solution converges to a constant curvature metric after rescaling. A key ingredient in the proof is the monotonicity of Hamilton’s entropy functional. This monotonicity formula will be discussed in Section 4.2. Alternative proofs of this theorem can be found in [4], [6], [48], or [82]. The arguments in [4] and [48] are based on a careful study of the isoperimetric profile, while the proofs in [6] and [82] employ PDE techniques. In Chapter 5, we describe Hamilton’s maximum principle for the Ricci flow and discuss the notion of a pinching set. We then describe a general convergence criterion for the Ricci flow. This criterion was discovered by Hamilton [45] and plays an important role in the study of Ricci flow. InChapter6,weexplainhowHamilton’sclassificationofthree-manifolds with positive Ricci curvature follows from the general theory developed in Chapter5. Wethendescribeanimportantcurvatureestimate,duetoHamil- ton and Ivey. This inequality holds for any solution to the Ricci flow in dimension 3. In Chapter 7, we describe various curvature conditions which are pre- servedbytheRicciflow. Wefirstprovethatnonnegativeisotropiccurvature is preserved by the Ricci flow in all dimensions. This curvature condition originated in Micallef and Moore’s work on the Morse index of harmonic two-spheres and plays a central role in this book. We then consider the conditionthat M×R has nonnegative isotropiccurvature. This conditionis stronger than nonnegative isotropic curvature, and is also preserved by the Ricciflow. Continuinginthisfashion,weconsidertheconditionthatM×R2 has nonnegative isotropic curvature, and the condition that M ×S2(1) has nonnegative isotropic curvature. (Here, S2(1) denotes a two-dimensional sphere of constant curvature 1.) We show that these conditions are pre- served by the Ricci flow as well. InChapter8,wepresenttheproofoftheDifferentiableSphereTheorem. More generally, we show that every compact Riemannian manifold M with the property that M ×R has positive isotropic curvature is diffeomorphic to a spherical space form. This theorem is the main result of Chapter 8. It Preface vii(cid:160) can be viewed as a generalization of Hamilton’s work in dimension 3 and was originally proved in [17]. In Chapter 9, we prove various rigidity theorems. In particular, we clas- sify all compact Riemannian manifolds M with the property that M ×R has nonnegative isotropic curvature. Moreover, we show that any Einstein manifoldwithnonnegativeisotropiccurvatureisnecessarilylocallysymmet- ric. This generalizes classical results due to Berger [10], [11] and Tachibana [84]. In order to handle the borderline case, we employ a variant of Bony’s strict maximum principle for degenerate elliptic equations. The material presented in Chapters 2–9 is largely, though not fully, self- contained. In Section 2.2, we employ the existence and uniqueness theory forparabolicsystems. InSection4.2, we usetheconvergencetheoryforRie- mannian manifolds developed by Cheeger and Gromov. Finally, in Chapter 9, we use Berger’s classification of holonomy groups, as well as some basic facts about K¨ahler and quaternionic-K¨ahler manifolds. There are some importantaspectsof Ricciflowwhich are not mentioned inthisbook. Forexample,wedonotdiscussHamilton’sdifferentialHarnack inequality (cf. [47], [49]) or Perelman’s crucial monotonicity formulae (see [68], [69]). A detailed exposition of Perelman’s entropy functional can be found in [63] or [85]. A generalization of Hamilton’s Harnack inequality is described in [18] (see also [24]). This book grew out of a Nachdiplom course given at ETH Zu¨rich. It is a pleasure to thank the Department of Mathematics at ETH Zu¨rich for its hospitality. I am especially grateful to Professor Michael Struwe and Professor Tristan Rivi`ere for many inspiring discussions. Without their en- couragement, this book would never have been written. Finally, I thank Professor Camillo De Lellis for valuable comments on an earlier version of this manuscript. Chapter 1 A survey of sphere theorems in geometry 1.1. Riemannian geometry background Let M be a smooth manifold of dimension n, and let g be a Riemannian metric on M. The Levi-Civita connection is defined by 2g(D Y,Z) = X(g(Y,Z))+Y(g(X,Z))−Z(g(X,Y)) X +g([X,Y],Z)−g([X,Z],Y)−g([Y,Z],X) for all vector fields X,Y,Z. The connection D is torsion-free and metric- compatible; that is, D Y −D X =[X,Y] X Y and X(g(Y,Z))= g(D Y,Z)+g(Y,D Z) X X for all vector fields X,Y,Z. The Riemann curvature tensor of (M,g) is defined by (cid:1) (cid:2) g D D Z −D D Z −D Z,W = −R(X,Y,Z,W). X Y Y X [X,Y] Hence, if we write D2 Z = D D Z −D Z, then we obtain X,Y X Y DXY D2 Z −D2 Z =D D Z −D D Z −D Z X,Y Y,X X Y Y X [X,Y] (cid:3)n =− R(X,Y,Z,e )e . k k k=1 The Levi-Civita connection on (M,g) induces a connection on tensor bundles. For example, if S is a (0,4)-tensor, then the covariant derivative 1