1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

CONTENTS vii

Appendix C. Glossary

Bibliography

• Chapter 1. Ricci Solitons Contents of Part I of Volume Two· xvii
  
  
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    1. General solitons and their canonical forms
    
    
    
    
  
  
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    2. Differentiating the.soliton equation - local and global analysis
    
    
    
    
  
  
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    3. Warped products and 2-dimensional solitons
    
    
    
    
  
  
  • · 4. Constructing the Bryant steady soliton
    
    
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      5. Rotationally symmetric expanding solitons
      
      
      
      
    
    
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      6. Homogeneous expanding solitons
      
      
      
      
    
    
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      7. When breathers and solitons are Einstein
      
      
      
      
    
    
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      8. Perelman's energy and entropy in relation to Ricci solitons
      
      
      
      
    
    
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      9. Buscher duality transformation of warped product solit.ons.
      
      
      
      
    
    
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      10. Summary of results and open problems on :i=ucci solitons
      
      
      
      
    
    
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      11. Notes and commentary
      
      
      
      
    
    
    
    
  
  
  
  


• Chapter 2. Kahler-Ricci Flow and Kahler-Ricci Solitons
  
  
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    1. Introduction to Kahler manifolds
    
    
    
    
  
  
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    2. Connection, curvature, and covariant differentiation
    
    
    
    
  
  
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    3. Existence of Kahler-Einstein metrics
    
    
    
    
  
  
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    4. Introduction to the Kahler-Ricci flow
    
    
    
    
  
  
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    5. Existence and convergence of the Kahler-Ricci flow '
    
    
    
    
  
  
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    6. Survey of some results for the Kahler-Ricci flow.
    
    
    
    
  
  
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    7. Examples of Kahler-Ricci solitons
    
    
    
    
  
  
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    8. Kahler-Ricci flow with nonnegative bisectional curvature
    
    
    
    
  
  
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    9. Matrix differential Harnack estimate for the Kahler-Ricci flow
    
    
    
    
  
  
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    10. Linear and interpolated differential Harnack estimates
    
    
    
    
  
  
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    11. Notes and commentary
    
    
    
    
  
  
  
  


• Chapter 3. The Compactness Theorem for Ricci Flow
  
  
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    1. Introduction and statements of the compactness theorems
    
    
    
    
  
  
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    2. Convergence at all times from convergence at one time
    
    
    
    
  
  
  • · 3. Extensions of Hamilton's compactness theorem
    
    
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      5. Notes and commentary
      
      
      
      
    
    
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• Chapter 4. Proof of the Compactness Theorem
  
  
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    1. Outline of the proof
    
    
    
    
  
  
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    2. Approximate isometries, compactness of maps, and direct limits
    
    
    
    
  
  
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    3. Construction of good coverings by balls
    
    
    
    
  
  
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    4. The limit manifold (M~, g 00 )
    
    
    
    
  
  
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    5. Center of mass and nonlinear averages
    
    
    
    
  
  
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    6. Notes and commentary
       
       
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         5. Notes and commentary
         
         
         
         
       
       
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• Chapter 6. Entropy and No Local Collapsing
  
  
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    1. The entropy functional W and its monotonicity
    
    
    
    
  
  
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    2. The functionalsμ and v
    
    
    
    
  
  
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    3. Shrinking breathers are shrinking gradient Ricci solitons
    
    
    
    
  
  
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    4. Logarithmic Sobolev inequality
       
       
       • conjecture 5. No finite time local collapsing: A proof of Hamilton's little loop
       
       
       
       
    
    
    
    
  
  
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    6. Improved version of no local collapsing and diameter control
    
    
    
    
  
  
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    7. Some further calculations related to :F and W
    
    
    
    
  
  
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    8. Notes and commentary
    
    
    
    
  
  
  
  


• Chapter 7. The Reduced Distance
  
  
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    1. The £-length and distance for a static metric
    
    
    
    
  
  
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    2. The £-length and the £-distance
    
    
    
    
  
  
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    3. The first variation of £-length and existence of £-geodesics
    
    
    
    
  
  
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    4. The gradient and time-derivative of the £-distance function
    
    
    
    
  
  
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    5. The second variation formula for C and the Hessian of L
    
    
    
    
  
  
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    6. Equations and inequalities satisfied by L and f ·
    
    
    
    
  
  
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    7. The £-function on Einstein solutions and Ricci solitons
    
    
    
    
  
  
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    8. £-Jacobi fields and the £-exponential map
    
    
    
    
  
  
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    9. Weak solution formulation
    
    
    
    
  
  
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    10. Notes and commentary
    
    
    
    
  
  
  
  


• Chapter 8. Applications of the Reduced Distance.
  
  
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    1. Reduced volume of a static metric
    
    
    
    
  
  
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    2. Reduced volume for Ricci fl.ow
    
    
    
    
    
    
    • the reduced volume 3. A weakened no local collapsing theorem via the monotonicity of
    
    
    
    
  
  
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    4. Backward limit of ancient K-solution is a shrinker
       
       
       • dimensions 5. Perelman's Riemannian formalism in potentially infinite
       
       
       
       
    
    
    
    
    
    
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      6. Notes and commentary
      
      
      
      
    
    
    
    
  
  
  
  


• Chapter 9. Basic Topology of 3-Manifolds
  
  
  • l. Essential 2-spheres and irreducible 3-manifolds
  
  
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    2. Incompressible surfaces and the geometrization conjecture
    
    
    
    
  
  
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    3. Decomposition theorems and the Ricci fl.ow
    
    
    
    
  
  
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    4. Notes and commentary
    
    
    
    
  
  
  
  


• Appendix A. Basic Ricci Flow Theory
  
  
  • l. Riemannian geometry
  
  
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    2. Basic Ricci fl.ow
    
    
    
    
  
  
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    3. Basic singularity theory for Ricci fl.ow
    
    
    
    
  
  
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    4. More Ricci fl.ow theory and ancient solutions
    
    
    
    
  
  
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    5. Classical singularity theory
    
    
    
    
  
  
  
  


• Appendix B. Other Aspects of Ricci Flow and Related Flows
  
  
  • l. Convergence to Ricci solitons
  
  
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    2. The mean curvature fl.ow
    
    
    
    
  
  
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    3. The cross curvature fl.ow
    
    
    
    
  
  
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    4. Notes and commentary
       
       
       • Index
       
       
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