Contents
Preface ix
What Part III is about ix
Acknowledgments x
Contents of Part III of Volume Two xiii
Notation and Symbols xvii
Chapter 17. Entropy, µ-invariant, and Finite Time Singularities 1
1. Compact finite time singularity models are shrinkers 1
2. Behavior of µ (g, τ) for τ small 15
3. Existence of a minimizer for the entropy 23
4. 1- and 2-loop variation formulas related to RG flow 31
5. Notes and commentary 36
Chapter 18. Geometric Tools and Point Picking Methods 39
1. Estimates for changing distances 40
2. Spatial point picking methods 49
3. Space-time point picking with restrictions 57
4. Necks in manifolds with positive sectional curvature 62
5. Localized no local collapsing theorem 68
6. Notes and commentary 76
Chapter 19. Geometric Properties of κ-Solutions 79
1. Singularity models and κ-solutions 80
2. The κ-noncollapsed condition 85
3. Perelman’s κ-solution on the n-sphere 93
4. Equivalence of 2- and 3-dimensional κ-solutions with and
without Harnack 104
5. Existence of an asymptotic shrinker 106
6. The κ-gap theorem for 3-dimensional κ-solutions 116
7. Notes and commentary 120
Chapter 20. Compactness of the Space of κ-Solutions 123
1. ASCR and AVR of κ-solutions 124
2. Almost κ-solutions 129
3. The compactness of κ-solutions 136
4. Derivative estimates and some conjectures 149
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