1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

Appendix J. Solutions to Selected Exercises

Bibliography

• Chapter 17. Entropy, μ-invariant, and Finite Time Singularities · Notation and Symbols xvii
  
  
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    1. Compact finite time singularity models are shrinkers
    
    
    
    
  
  
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    2. Behavior ofμ (g, T) for T small
    
    
    
    
  
  
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    3. Existence of a minimizer for the entropy
    
    
    
    
  
  
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    4. 1-and 2-loop variation formulas related to RG flow
    
    
    
    
  
  
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    5. Notes and commentary
    
    
    
    
  
  
  
  


• Chapter 18. Geometric Tools and Point Picking Methods
  
  
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    1. Estimates for changing distances
    
    
    
    
  
  
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    2. Spatial point picking methods
    
    
    
    
  
  
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    3. Space-time point picking with restrictions
    
    
    
    
  
  
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    4. Necks in manifolds with positive sectional curvature
    
    
    
    
    
    
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      5. Localized no local collapsing theorem
      
      
      
      
    
    
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      6. Notes and commentary
      
      
      
      
    
    
    
    
  
  
  
  


• Chapter 19. Geometric Properties of /\;-Solutions
  
  
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    1. Singularity models and /\;-solutions
    
    
    
    
  
  
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    2. The fl;-noncollapsed condition
    
    
    
    
  
  
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    3. Perelman's /\;-solution on the n-sphere
    
    
    
    
    
    
    • without Harnack 4. Equivalence of 2- and 3-dimensional /\;-solutions with and
    
    
    
    
  
  
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    5. Existence of an asymptotic shrinker
    
    
    
    
  
  
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    6. The fl;-gap theorem for 3-dimensional /\;-solutions
    
    
    
    
  
  
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    7. Notes and commentary
    
    
    
    
  
  
  
  


• Chapter 20. Compactness of the Space of /\;-Solutions
  
  
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    1. ASCR and AVR of /\;-solutions
    
    
    
    
  
  
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    2. Almost /\;-solutions
    
    
    
    
    
    
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      3. The compactness of /\;-solutions
      
      
      
      
    
    
    
    
  
  
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    4. Derivative estimates and some conjectures
    
    
    
    
  
  
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    5. Notes and commentary vi CONTENTS
    
    
    
    
  
  
  
  


• Chapter 21. Perelman's Pseudolocality Theorem
  
  
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    1. Statement and interpretation of pseudolocality
    
    
    
    
  
  
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    2. Setting up the proof by contradiction and point picking
    
    
    
    
  
  
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    3. Local entropies are nontrivial near bad points
    
    
    
    
    
    
    • inequality 4. Contradicting the almost Euclidean logarithmic Sobolev
    
    
    
    
  
  
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    5. Notes and commentary
    
    
    
    
  
  
  
  


• Chapter 22. Tools Used in Proof of Pseudolocality
  
  
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    1. A point picking method
    
    
    
    
  
  
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    2. Heat kernels under Cheeger-Gromov limits
    
    
    
    
  
  
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    3. Upper bound for the local entropy JB v dμ
    
    
    
    
  
  
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    4. Logarithmic Sobolev inequality via the isoperimetric inequality
    
    
    
    
  
  
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    5. Notes and commentary
    
    
    
    
  
  
  
  


• Chapter 23. Heat Kernel for Static Metrics
  
  
  • Riemannian manifold 1. Construction of the parametrix for the heat kernel on a
  
  
  • via parametrix 2. Existence of the heat kernel on a closed Riemannian manifold
  
  
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    3. Differentiating a convolution with the parametrix
    
    
    
    
  
  
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    4. Asymptotics of the heat kernel for a static metric
    
    
    
    
  
  
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    5. Supplementary material: Elementary tools
    
    
    
    
  
  
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    6. Notes and commentary
    
    
    
    
  
  
  
  


• Chapter 24. Heat Kernel for Evolving Metrics
  
  
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    1. Heat kernel for a time-dependent metric
    
    
    
    
  
  
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    2. Existence of the heat kernel for a time-dependent metric
    
    
    
    
    
    
    • time-dependent metric 3. Aspects of the asymptotics of the heat kernel for a
    
    
    
    
  
  
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    4. Characterizing Ricci flow by the asymptotics of the heat kernel
    
    
    
    
  
  
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    5. Heat kernel on noncompact manifolds
    
    
    
    
  
  
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    6. Notes and commentary
    
    
    
    
  
  
  
  


• Chapter 25. Estimates of the Heat Equation for Evolving Metrics
  
  
  • with respect to evolving metrics 1. Mean value inequality for solutions of heat-type equations
  
  
  • heat-type equations with respect to evolving metrics 2. Li-Yau differential Harnack estimate for positive solutions of
  
  
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    3. Notes and commentary
    
    
    
    
  
  
  
  


• Chapter 26. Bounds for the Heat Kernel for Evolving Metrics
  
  
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    1. Heat kernel for an evolving metric
    
    
    
    
    
    
    • metric 2. Upper and lower bounds of the heat kernel for an evolving
      
      
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        3. Heat balls and the space-time mean value property CONTENTS vii
        
        
        
        
      
      
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        4. Distance-like functions on complete noncompact manifolds
        
        
        
        
      
      
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        5. Notes and commentary
        
        
        
        
      
      
      
      
    
    
    
    
  
  
  • Appendix G. Elementary Aspects of Metric Geometry
    
    
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      1. Metric spaces and length spaces
      
      
      
      
    
    
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      2. Aleksandrov spaces with curvature bounded from below
      
      
      
      
    
    
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      3. Notes and commentary
      
      
      
      
    
    
    
    
  
  
  
  


• Appendix H. Convex Functions on Riemannian Manifolds
  
  
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    1. Elementary aspects of convex analysis on Euclidean space
    
    
    
    
  
  
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    2. Connected locally convex subsets in Riemannian manifolds
       
       
       • manifolds 3. Generalized gradients of convex functions on Riemannian
       
       
       
       
    
    
    
    
  
  
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    4. Integral curves to gradients of concave functions
    
    
    
    
  
  
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    5. Notes and commentary
    
    
    
    
  
  
  
  


• Appendix I. Asymptotic Cones and Sharafutdinov Retraction
  
  
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    1. Sharafutdinov retraction theorem
    
    
    
    
  
  
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    2. The existence of asymptotic cones
       
       
       • within the injectivity radius 3. A monotonicity property of nonnegatively curved manifolds
       
       
       
       
    
    
    
    
  
  
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    4. Critical point theory and properties of distance spheres
    
    
    
    
  
  
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    5. Approximate Busemann-Feller theorem
    
    
    
    
  
  
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    6. Equivalence classes of rays and points at infinity
    
    
    
    
  
  
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    7. Notes and commentary
       
       
       • Index
       
       
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