Contents
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
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iii
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