Summer School Program 2005
Ricci Flow, 3-Manifolds and Geometry
June 20 — July 16
at MSRI
General Information
The program will run Monday through Friday, 09:15 to 17:15. Breakfast will be provided at Clark Kerr College for all student participants, and lunch will be organized at MSRI for all participants Monday through Friday.
There will be an opening reception on Tuesday, June 21, and a closing banquet on Thursday, July 14, 2005. Evening meals will not be organized by the school, but all participants will receive a per diem to supplement the cost of weekend and evening meals.
Main Courses
Minimal Surfaces
Tobias Colding
Perelman's work on Ricci Flow I
Bruce Kleiner & John Lott
This course concerns Perelman's works on general Ricci flow. The
topics include: Entropy functional of Perelman and its local form,
Noncollapsing theorem, Perelman's reduced volume and applications,
Kappa-ancient solutions and their classification in 3-dimensions.
References [7], [9]
Perelman's work on Ricci Flow II
Bruce Kleiner, John Lott & Gang Tian
The emphasis of this course is Perelman's works on Ricci flow in
3-dimensions and geometrization of 3-manifolds. The topics include:
Analysis of large curvature part of Ricci flow solutions, Ricci flow with
surgery, basic properties of solutions with surgery,
long time behavior of solutions with surgery, applications to geometrization.
References [7], [8], [9]
Ricci Flow I
Bennett Chow
Hamilton's 3-manifolds with positive Ricci curvature theorem: background and basic techniques used in its proof - linearization of the Ricci tensor, short time existence, basic evolution equations, maximum principles, curvature pinching estimates, convergence criteria.
Student's guide to Ricci Flow I, II, III
Exercises on Riemannian Geometry. Exercises on Ricci Flow. From chapters 1, 2, 3, 4 of Hamilton's Ricci Flow, by Bennett Chow, Peng Lu, Lei Ni, to be published by Science Press, China.
Selected solutions to Exercises on Riemannian Geometry and Ricci Flow, I
Schedule and notes in PDF form.
- 1. Connections, curvatures, and variation formulas
- 2. The Ricci flow equation and associated equations
- 3. Heat equations and maximum principles
- 4. Short time existence and curvature estimates
- 5. Convergence of Ricci flow for closed 3-manifolds with positive Ricci curvature
- Supplement A. Divergence theorem and integration by parts
- Additional notes for Lecture 5: Proof of Hamilton's 3-manifold theorem using compactnes
Background and reading: [1], [2], References
Ricci Flow II
Bennett Chow
Special solutions: Ricci solitons and homogeneous solutions - gradient Ricci solitons and basic associated formulas, examples: cigar soliton, expanding soliton on R2, Bryant soliton, Rosenau solution, homogeneous solutions in dimension 3.
Schedule
- 6. Gradient Ricci solitons, related monotonicity on surfaces and the Kazdan-Warner identity
- 7. The cigar soliton, the Rosenau solution, and moving frames calculations
Supplement for lecture 7. A canonical form for gradient Ricci solitons - 8. Expanding soliton on R2, the 3-dimensional Bryant soliton, and no closed 3-dimensional shrinkers
- 9. Basic formulas associated to the gradient Ricci soliton equation
Supplement for lecture 9. Classification of ancient solutions on surfaces - 10. Homogeneous solutions in dimension 3.
Supplement: Survey of some previous work on the 2-d case of Ricci flow
Supplement: Ricci flow of left-invariant metrics on 3-dimensional unimodular Lie groups
Background and reading: [1], [2], References
Ricci Flow III
Bennett Chow
Analytic and geometric techniques: more maximum principle and monotonicity — Li-Yau Harnack estimate for the heat equation, Hamilton's Harnack estimates for the Ricci flow, consequences for eternal solutions, Shi's local and global derivative estimates, Hamilton-Ivey estimate and its consequences.
Schedule
- 11. The Li-Yau differential Harnack estimate for the heat equation
- 12. Hamilton's trace
Harnack estimate for the Ricci flow on surfaces and its
consequences
Supplement: Curvature estimates for the Ricci flow on 3-manifolds with positive Ricci curvature via the maximum principle for system - 13. More gradient Ricci solitons, Hamilton's matrix Harnack estimate for the Ricci flow and eternal solutions
- 14. Shi's local and global derivative estimates
- 15. The Hamilton-Ivey 3-dimensional curvature pinching estimate and some consequences
Background and reading: [1], [2], References
Topics in Geometry and Topology I,
II
John Morgan & Jeff Cheeger
Geometrization (3 lectures) : The eight basic 3-dimensional geometries, prime decomposition of 3-manifolds, incompressible tori, Thurston's geometrization conjecture on 3-manifolds, graph manifolds.
Fundamental results in differential geometry which are used in Perelman's work:
Compactness theorems in Riemannian geometry [4, Chapter 10], [5, Chapter 7]
Compactness theorems for Ricci Flow [1, Chapter 7.3]
Structure of manifolds with nonnegative curvature [4, Chapter 11.4] [6, Chapter 8]
Basics of Alexandrov spaces [5, Chapter 4]
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