Lecturer: Peter Topping.

We will take a look at the Ricci flow -- introduced in 1982 by Hamilton, following work of Eells and Sampson -- which deforms a Riemannian metric g in terms of its Ricci curvature Ric(g) according to the PDE

Often, this deforms an arbitrary metric to a canonical metric. Hamilton's original application was to take an arbitrary closed 3-manifold with positive Ricci curvature, and show that the (renormalised) flow deforms it to a spherical space form. In particular, a simply connected closed 3-manifold with positive Ricci curvature must be the 3-sphere.

In the twenty years following its introduction, the Ricci flow was steadily developed, largely by Hamilton and his school, partly with a view to proving Thurston's geometrization conjecture (which includes the Poincaré conjecture). Starting a few years ago, Perelman released a series of papers which culminated with a claim of the Poincaré conjecture using Hamilton's approach.

This course will cover some of the techniques along these lines which are required to give such a proof. More precisely, we hope to cover many of the key ideas used to study singularity development in smooth flows. An international workshop at the end of the course will include an advanced lecture by Bruce Kleiner (Yale) on the 'surgery' argument required to complete the proof. (Tuesday 27 March 2007.)

1. Ricci flow introduction, and the strategy of the proof of the Poincare conjecture
2. Ricci flow background material, and work of Hamilton. Short time existence; evolution of some geometric quantities; maximum principle techniques; Harnack estimates; Hamilton-Ivey pinching; Strong maximum principle techniques; derivative estimates; compactness results;
3. Perelman's L-length and reduced volume.
4. The "weak" no local collapsing result of Perelman, and applications to blowing up.
5. Kappa-solutions. Structure and classification.
6. Perelman's Canonical Neighbourhood Theorem. Understanding the structure of Ricci flows near points of large curvature.
   
   
   Prerequisites
   
   The absolute prerequisite is a knowledge of Riemannian geometry -- at least "Differential geometry" MA4C0 from term 1. There are many other ingredients required to understand the course fully - in particular a knowledge of PDE theory - but we will try to lighten these requirements as much as possible. It should be possible to follow the course concurrently with "Advanced PDE" MA4A2.
   
   Lecture times
   
   Tuesday 11:00, B3.02
   Wednesday 10:00, B3.02
   Thursday MOVED TO: 12:00, B3.01

   All lectures in the Mathematics Institute.

   First lecture: Tuesday 9th January 2007
   

   
   
   Bibliography
   
   

   There will be some overlap with my previous lecture notes (although this course should be heavily adapted to the techniques required for the Poincaré conjecture):

   Lectures on the Ricci flow
   Peter Topping
   LMS lecture notes series vol 325, CUP (2006).
   http://www.maths.warwick.ac.uk/~topping/RFnotes.html

   To help with the Riemannian geometry prerequisites:

   Riemannian Manifolds: An Introduction to Curvature (Graduate Texts in Mathematics)
   John M. Lee
   Springer

   Riemannian geometry
   Gallot, Hulin, Lafontaine
   springer

   Einstein manifolds
   Arthur L. Besse
   Springer

   To help with the PDE theory (as indicated during the course):

   Partial differential equations
   L. C. Evans
   AMS

   If you have never done a first course in PDE theory (eg our third year undergrad course "Theory of PDE") then you must look up the basic theory of the heat equation in any basic PDE book (or chapter 2 of Evans' book above). A minimum requirement is to digest the "maximum principle" for this equation. You must understand the basics of solving the equation (forwards in time) with given initial conditions.

   Other books we'll refer to:

   Collected papers on Ricci flow
   Eds: Cao, Chu, Chow, Yau
   International press

   This collects together some of the main papers which have been written on Ricci flow, with corrections in footnotes. This is a pre-Pereleman publication, and so will only help with elements of the course.

   Recent texts elaborating on and correcting Perelman's work:

7. Notes on Perelman's Papers, by Bruce Kleiner and John Lott, version of May 25, 2006.
8. A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow, by Huai-Dong Cao and Xi-Ping Zhu, Asian Journal of Mathematics, June 2006.
9. Ricci Flow and the Poincaré Conjecture, by John Morgan and Gang Tian, July 25, 2006.

Useful link - to Ricci flow surveys, and other commentary on Perelman's work: http://www.math.lsa.umich.edu/research/ricciflow/perelman.html

This site includes links to many relevant papers and sets of notes.