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