Ricci flow and the sphere theorem

Preface

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
§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