Ricci flow - MA607
University of Warwick - Spring 2007
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
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.
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
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;
Perelman's L-length and reduced volume.
The "weak" no local collapsing result of Perelman, and applications
to blowing up.
Kappa-solutions. Structure and classification.
Perelman's Canonical Neighbourhood Theorem. Understanding the structure
of Ricci flows near points of large curvature.
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.
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
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
LMS lecture notes series vol 325, CUP (2006).
To help with the Riemannian geometry prerequisites:
Riemannian Manifolds: An Introduction to Curvature
(Graduate Texts in Mathematics)
John M. Lee
Gallot, Hulin, Lafontaine
Arthur L. Besse
To help with the PDE theory (as indicated during the course):
Partial differential equations
L. C. Evans
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
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:
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 Hamilton-Perelman theory of the Ricci flow,
by Huai-Dong Cao and Xi-Ping 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:
This site includes links to many relevant papers and sets of notes.