Ricci flow  MA607
University of Warwick  Spring 2007
Lecturer:
Peter Topping.
Course description
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
dg/dt=2 Ric(g).
Often, this deforms an arbitrary metric to a
canonical metric. Hamilton's original application was to take
an arbitrary closed 3manifold with
positive Ricci curvature, and
show that the (renormalised) flow deforms it to a spherical space form.
In particular, a simply connected closed 3manifold
with positive Ricci curvature must be the 3sphere.
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.)
Outline of course
 Ricci flow introduction, and the strategy of the proof of the
Poincare conjecture

Ricci flow background material, and work of Hamilton.
Short time existence; evolution of some geometric quantities;
maximum principle techniques; Harnack estimates;
HamiltonIvey pinching; Strong maximum principle techniques;
derivative estimates;
compactness results;

Perelman's Llength and reduced volume.

The "weak" no local collapsing result of Perelman, and applications
to blowing up.

Kappasolutions. Structure and classification.

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 prePereleman publication, and so will only help with
elements of the course.
Recent texts elaborating on and correcting Perelman's work:

Notes on Perelman's Papers, by Bruce Kleiner and John Lott,
version of May 25, 2006.

A Complete Proof of the Poincaré and Geometrization Conjectures 
application of the HamiltonPerelman theory of the Ricci flow,
by HuaiDong Cao and XiPing Zhu, Asian Journal of Mathematics, June 2006.

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.
