# Summer School 2005

*Clay Mathematics Institute 2005 Summer School. June 20 – July 15 at the Mathematical Sciences Research Institute (MSRI) Berkeley, California*

## Overview

The Clay Mathematics Institute will hold its 2005 summer school at the Mathematical Sciences Research Institute (MSRI) in Berkeley, California.

Designed for graduate students and mathematicians within five years of their Ph.D., the program is organized around Ricci Flow and the Geometrization of 3–manifolds, particularly, the recent work of Perelman.

The school will consist of three weeks of foundational courses and one week of mini-courses focusing on more advanced topics and applications.

Perelman’s work builds on earlier work of Thurston and Hamilton in a deep and original way. The aim of the school is to provide a comprehensive introduction to these exciting areas as well as the recent developments due to Perelman.

Topics covered will include an introduction to Geometrization (3–dimensional geometries, prime decomposition of 3–manifolds, incompressible tori, Thurston’s geometrization conjecture on 3–manifolds), Ricci Flow (both geometric and analytic aspects), Minimal Surfaces and various fundamental results in topology and differential geometry used in the work of Perelman.

We will also have a course dedicated to Perelman’s work on general Ricci Flow (Entropy functional of Perelman and its local form, Non-collapsing theorem, Perelman’s reduced volume and applications), as well as a course that outlines some more advanced results and applications in 3–dimensions (analysis of large curvature part of Ricci flow solutions, Ricci flow with surgery, basic properties of solutions with surgery, long time behavior of solutions, applications to geometrization).

## Organizers

Gang Tian, John Lott, John Morgan, Bennett Chow, Tobias Colding, Jim Carlson, David Ellwood, Hugo Rossi

## Lecturers

Jeff Cheeger, Bennett Chow, Tobias Colding, Richard Hamilton, Bruce Kleiner, John Lott, John Morgan, Lei Ni, Gang Tian, and others.

## 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 R^{2}, 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 R
^{2}, 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]

## Advanced Courses

# References

### Background on Ricci flow

[1] “The Ricci flow: an introduction” by B. Chow and D. Knopf, Mathematical Surveys and Monographs 110, American Mathematical Society, Providence, RI, 2004.

[2] “The formation of singularities in the Ricci flow”, by R. Hamilton, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), Internat. Press, Cambridge, MA, p. 7-136 (1995)

References for Ben Chow’s 15 lectures

### Background on the Geometrization Conjecture

[3] “Recent progress on the Poincare conjecture and the classification of 3-manifolds” by J. Morgan, Bull. Amer. Math. Soc. 42, p. 57-78 (2005)

### Background on Differential Geometry :

[4] “Riemannian geometry” by P. Petersen, Graduate Texts in Mathematics 171, Springer-Verlag, New York, 1998.

[5] “A course in metric geometry” by D. Burago, Y. Burago and S. Ivanov, Graduate Studies in Mathematics 33, American Mathematical Society, Providence, RI, 2001.

[6] “Comparison theorems in Riemannian geometry” by J. Cheeger and D. Ebin, North-Holland Mathematical Library, Vol. 9. North-Holland Publishing Co., 1975.

### Perelman’s Work on Ricci Flow

[7] “The entropy formula for the Ricci flow and its geometric applications”, by G. Perelman

[8] “Ricci flow with surgery on three-manifolds”, by G. Perelman

[9] “Notes on Perelman’s papers”, by B. Kleiner and J. Lott

### Minimal Surfaces

[10] “Minimal submanifolds“, by T. Colding and W. Minicozzi

[11] Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman, Tobias H. Colding; William P. Minicozzi II Journal: J. Amer. Math. Soc. 18 (2005), 561-569.

[12] Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, Grisha Perelman, July 17, 2003, revised June 12, 2005

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