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
§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  ............................................... 87
§7.5 The cone Ĉ ............................................... 90
§7.6 An invariant set which lies between and Ĉ .............. 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 Kähler-Einstein and quaternionic-Kähler 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
| 
