We give a quick introduction to Ricci flow. Ricci flow is currently a hot topic in mathematics. Its development over the last two decades has primarily been aimed at understanding the Geometrization Conjecture. Recently Perelman has made spectacular progress on Hamilton's well-developed program aimed at proving the Geometrization and Poincare Conjectures. Our book will quickly lead the reader through much of the basic material in the subject up to some of the most recent developments.
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目录
Acknowledgments
A Detailed Guide for the Reader
Notation and Symbols
Chapter 1. Riemannian Geometry
§1. Introduction
§2. Metrics, connections,curvatures and covariant differentiation
§3. Basic formulas and identities in Riemannian geometry
§4. Exterior differential calculus and Bochner formulas
§5. Integration and Hodge theory
§6. Curvature decomposition and locally conformally flat manifolds
§7. Moving frames and the Gauss-Bonnet formula
§8. Variation of arc length,energy and area
§9. Geodesics and the exponential map
§10. Second fundamental forms of geodesic spheres
§11. Laplacian, volume and Hessian comparison theorems
§12. Proof of the comparison theorems
§13. Manifolds with nonnegative curvature
§14. Lie groups and left-invariant metrics
§15. Notes and commentary
Chapter 2. Fundamentals of the Ricci Flow Equation
§1. Geometric flows and geometrization
§2. Ricci flow and the evolution of scalar curvature
§3. The maximum principle for heat-type equations
§4. The Einstein-Hilbert functional
§5. Evolution of geometric quantities
§6. DeTurck's trick and short time existence
§7. Reaction-diffusion equation for the curvature tensor
§8. Notes and commentary
Chapter 3. Closed 3-manifolds with Positive Ricci Curvature
§1. Hamilton's 3-manifolds with positive Ricci curvature theorem
§2. The maximum principle for tensors
§3. Curvature pinching estimates
§4. Gradient bounds for the scalar curvature
§5. Curvature tends to constant
§6. Exponential convergence of the normalized flow
§7. Notes and commentary
Chapter 4. Ricci Solitons and Special Solutions
§1. Gradient Ricci solitons
§2. Gaussian and cylinder solitons
§3. Cigar steady soliton
§4. Rosenau solution
§5. An expanding soliton
§6. Bryant soliton
§7. Homogeneous solutions
§8. The isometry group
§9. Notes and commentary
Chapter 5. Isoperimetric Estimates and No Local Collapsing
§1. Sobolev and logarithmic Sobolev inequalities
§2. Evolution of the length of a geodesic
§3. Isoperimetric estimate for surfaces
§4. Perelman's no local collapsing theorem
§5. Geometric applications of no local collapsing
§6. 3-manifolds with positive Ricci curvature revisited
§7. Isoperimetric estimate for 3-dimensional Type I solutions
§8. Notes and commentary
Chapter 6. Preparation for Singularity Analysis
§1. Derivative estimates and long time existence
§2. Proof of Shi's local first and second derivative estimates
§3. Cheeger-Gromov-type compactness theorem for Ricci flow
§4. Long time existence of solutions with bounded Ricci curvature
§5. The Hamilton-Ivey curvature estimate
§6. Strong maximum principles and metric splitting
§7. Rigidity of 3-manifolds with nonnegative curvature
§8. Notes and commentary
Chapter 7. High-dimensional and Noncompact Ricci Flow
§1. Spherical space form theorem of Huisken-Margerin-Nishikawa
§2. 4-manifolds with positive curvature operator
§3. Manifolds with nonnegative curvature operator
§4. The maximum principle on noncompact manifolds
§5. Complete solutions of the Ricci flow on noncompact manifolds
§6. Notes and commentary
Chapter 8. Singularity Analysis
§1. Singularity dilations and types
§2. Point picking and types of singularity models
§3. Geometric invariants of ancient solutions
§4. Dimension reduction
§5. Notes and commentary
Chapter 9. Ancient Solutions
§1. Classification of ancient solutions on surfaces
§2. Properties of ancient solutions that relate to their type
§3. Geometry at infinity of gradient Ricci solitons
§4. Injectivity radius of steady gradient Ricci solitons
§5. Towards a classification of 3-dimensional ancient solutions
§6. Classification of 3-dimensional shrinking Ricci solitons
§7. Summary and open problems
Chapter 10. Differential Harnack Estimates
§1. Harnack estimates for the heat and Laplace equations
§2. Harnack estimate on surfaces with χ>0
§3. Linear trace and interpolated Harnack estimates on surfaces
§4. Hamilton's matrix Harnack estimate for the Ricci flow
§5. Proof of the matrix Harnack estimate
§6. Harnack and pinching estimates for linearized Ricci flow
§7. Notes and commentary
Chapter 11. Space-time Geometry
§1. Space-time solution to the Ricci flow for degenerate metrics
§2. Space-time curvature is the matrix Harnack quadratic
§3. Potentially infinite metrics and potentially infinite dimensions
§4. Renormalizing the space-time length yields the l-length
§5. Space-time DeTurck's trick and fixing the measure
§6. Notes and commentary
Appendix A. Geometric Analysis Related to Ricci Flow
§1. Compendium of inequalities
§2. Comparison theory for the heat kernel
§3. Green's function
§4. The Liouville theorem revisited
§5. Eigenvalues and eigenfunctions of the Laplacian
§6. The determinant of the Laplacian
§7. Parametrix for the heat equation
§8. Monotonicity for harmonic functions and maps
§9. Bieberbach theorem
§10. Notes and commentary
Appendix B. Analytic Techniques for Geometric Flows
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