vi CONTENTS 4. Applications of Hamilton's compactness theorem 142 5. Notes and commentary 148 Chapter 4. Proof of the Compactness Theorem 149 1. Outline of the proof 149 2. Approximate isometries, compactness of maps, and direct limits 150 3. Construction of good coverings by balls 158 4. The limit manifold (M^goo) 165 5. Center of mass and nonlinear averages 175 6. Notes and commentary 187 Chapter 5. Energy, Monotonicity, and Breathers 189 1. Energy, its first variation, and the gradient flow 190 2. Monotonicity of energy for the Ricci flow 197 3. Steady and expanding breather solutions revisited 203 4. Classical entropy and Perelman's energy 214 5. Notes and commentary 219 Chapter 6. Entropy and No Local Collapsing 221 1. The entropy functional W and its monotonicity 221 2. The functionals fi and v 235 3. Shrinking breathers are shrinking gradient Ricci solitons 242 4. Logarithmic Sobolev inequality 246 5. No finite time local collapsing: A proof of Hamilton's little loop conjecture 251 6. Improved version of no local collapsing and diameter control 264 7. Some further calculations related to T and W 273 8. Notes and commentary 284 Chapter 7. The Reduced Distance 285 1. The ^-length and distance for a static metric 286 2. The ^-length and the L-distance 288 3. The first variation of £-length and existence of ^-geodesies 296 4. The gradient and time-derivative of the L-distance function 306 5. The second variation formula for C and the Hessian of L 312 6. Equations and inequalities satisfied by L and £ 322 7. The ^-function on Einstein solutions and Ricci solitons 335 8. £-Jacobi fields and the /^-exponential map 345 9. Weak solution formulation 363 10. Notes and commentary 379 Chapter 8. Applications of the Reduced Distance 381 1. Reduced volume of a static metric 381 2. Reduced volume for Ricci flow 386 3. A weakened no local collapsing theorem via the monotonicity of the reduced volume 399 4. Backward limit of ancient ^-solution is a shrinker 406

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