| | |
Summary: REMARKS ON PERELMAN'S PAPERS
MICHAEL T. ANDERSON
This note is informal commentary, (very informal in comparison with the Kleiner-Lott notes
[KL]), on Grisha Perelman's work [I], [II] on Ricci
ow and geometrization of 3-manifolds. The
comments concern issues or questions that either arise from my own thoughts, or in response to
those raised by others. They are also in
uenced by Grisha's lectures at Stony Brook in April, 03.
By and large, these comments do not address the details of the proofs in [I] or [II]; for [I], this
has already been carried out wonderfully by Kleiner-Lott. Instead, these notes basically just record
some of my thoughts and views on the papers at this time. It is hoped that this is of some use to
other non-experts on the Ricci
ow, partly as a guide in understanding some of the main issues. I
hope to add further remarks and discussion as time goes on. Comments and criticism are welcome.
x1. Comments on F .
The stationary points of the Ricci
ow on the space of metrics M on a given manifold are the
Ricci-
at metrics, or more generally Einstein metrics for the volume-normalized Ricci
ow. If the
(volume normalized) Ricci
ow is the gradient
ow of some functional, then the functional must
have critical points exactly the class of Einstein metrics. The only known functional on M with this
property is the Einstein-Hilbert action SEH = R
(R 2)dV or total scalar curvature functional
S tot = v (n 2)=n R
RdV: However, the Ricci
ow is not the gradient
ow of (any such) S; the
|
