vi CONTENTS
1. Weak maximum principles for scalar heat-type equations 140
2. Mollifying distance functions on Riemannian manifolds 158
3. Weak maximum principle for parabolic systems 170
4. Strong maximum principle for parabolic systems 180
5. Applications to the curvature operator under the Ricci flow 192
6. Notes and commentary 195
Chapter 13. Qualitative Behavior of Classes of Solutions 197
1. Curvature conditions that are not preserved 197
2. Real analyticity in the space variables for solutions of the Ricci
flow 210
Chapter 14. Local Derivative of Curvature Estimates 227
1. Introduction—fine versus coarse estimates 228
2. A quick review of the global derivative estimates 232
3. Shi's local derivative estimates 235
4. Modified Shi's local derivative estimates assuming bounds on
some derivatives of curvatures of the initial metrics 244
5. Some applications of the local derivative estimates 251
6. Local heat equation and local Ricci flow 253
7. Notes and commentary 258
Chapter 15. Differential Harnack Estimates of LYH-type 259
1. Deriving the Harnack expression using Ricci solitons 259
2. Statement of the matrix Harnack estimate 263
3. Proofs: getting started with surfaces 265
4. Proof of the matrix Harnack estimate 268
5. A variant on Hamilton's proof of the matrix Harnack estimate 287
6. Ricci solitons and ancient solutions attaining i?
ma x
293
7. Applications of Harnack estimates 300
8. Notes and commentary 303
Chapter 16. Perelman's Differential Harnack Estimate 305
1. Entropy and differential Harnack estimates for the heat equation 306
2. Properties of the heat kernel and linear entropy formula on
complete manifolds 314
3. Differential Harnack estimate and characterizing
IRn
by linear
entropy 325
4. Perelman's differential Harnack estimate 335
5. Notes and commentary 355
Appendix D. An Overview of Aspects of Ricci Flow 357
1. Existence, uniqueness, convergence, and curvature evolution 357
2. The rotationally symmetric neckpinch 360
3. Curvature pinching, derivative, and Harnack estimates 365
4. Perelman's energy, entropy, and associated invariants 369
5. Compactness, no local collapsing, and singularity models 373
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