iv CONTENTS

Notes and commentary 103

Chapter 5. The Ricci flow on surfaces 105

1. The effect of a conformal change of metric 106

2. Evolution of the curvature 109

3. How Ricci solitons help us estimate the curvature from above 111

4. Uniqueness of Ricci solitons 116

5. Convergence when x

(M2)

0 120

6. Convergence when \

(M2)

= 0 123

7. Strategy for the case that \

(M2

0) 128

8. Surface entropy 133

9. Uniform upper bounds for R and |Vi2| 137

10. Differential Harnack estimates of LYH type 143

11. Convergence when R (•, 0) 0 148

12. A lower bound for the injectivity radius 149

13. The case that R (•, 0) changes sign 153

14. Monotonicity of the isoperimetric constant 156

15. An alternative strategy for the case \

(M2

0) 165

Notes and commentary 171

Chapter 6. Three-manifolds of positive Ricci curvature 173

1. The evolution of curvature under the Ricci flow 174

2. Uhlenbeck's trick 180

3. The structure of the curvature evolution equation 183

4. Reduction to the associated ODE system 187

5. Local pinching estimates 189

6. The gradient estimate for the scalar curvature 194

7. Higher derivative estimates and long-time existence 200

8. Finite-time blowup 209

9. Properties of the normalized Ricci flow 212

10. Exponential convergence 218

Notes and commentary 221

Chapter 7. Derivative estimates 223

1. Global estimates and their consequences 223

2. Proving the global estimates 226

3. The Compactness Theorem 231

Notes and commentary 232

Chapter 8. Singularities and the limits of their dilations 233

1. Classifying maximal solutions 233

2. Singularity models 235

3. Parabolic dilations 237

4. Dilations of finite-time singularities 240

5. Dilations of infinite-time singularities 246

6. Taking limits backwards in time 250