In 1982, R. Hamilton introduced a nonlinear evolution equation for Riemannian metrics with the aim of finding canonical metrics on manifolds. This evolution equation is known as the Ricci flow, and it has since been used widely and with great success, most notably in Perelman's solution of the Poincaré conjecture. Furthermore, various convergence theorems have been established.
This book provides a concise introduction to the subject as well as a comprehensive account of the convergence theory for the Ricci flow. The proofs rely mostly on maximum principle arguments. Special emphasis is placed on preserved curvature conditions, such as positive isotropic curvature. One of the major consequences of this theory is the Differentiable Sphere Theorem: a compact Riemannian manifold, whose sectional curvatures all lie in the interval (1,4], is diffeomorphic to a spherical space form. This question has a long history, dating back to a seminal paper by H. E. Rauch in 1951, and it was resolved in 2007 by the author and Richard Schoen.
This text originated from graduate courses given at ETH Zürich and Stanford University, and it is directed at graduate students and researchers. The reader is assumed to be familiar with basic Riemannian geometry, but no previous knowledge of Ricci flow is required.
Readership: Graduate students and research mathematicians interested in differential geometry and topology of manifolds.
Author(s): Simon Brendle
Series: Graduate Studies in Mathematics 111
Publisher: American Mathematical Society
Year: 2010
Language: English
Pages: viii+176
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
