 
Summary: RESEARCH BLOG 7/6/04
VOLUME ESTIMATES VIA RICCI FLOW
An article on Perelman's work on the geometrization conjecture was
published in Scientific American. It gives an introduction to the sub
ject, but doesn't have any new information or interviews.
In blog 8/20/03 I mentioned how one could improve upon the volume
estimates I gave in [1] using Ricci flow. I gave a talk about this at a
BIRS workshop last September. After the workshop, Nathan Dunfield
figured out how to improve on the volume estimate. Now we can show
that the minimal volume orientable hyperbolic 3manifold has volume
> .649 (dependent on Perelman's work). I also realized that there is
a simpler monotonicity argument than the one I used (which was first
discovered by Hamilton, and extends to Ricci flow with surgery, see
e.g. Anderson's notes). Given M3
closed and (Mt, gt) a solution to the
Ricci flow with surgery (where M0 = M, and the topology of Mt is
locally constant, but may change at the surgery times), the quantity
RminV 2/3
is nondecreasing with t when Rmin < 0 (it follows from
Perelman's claims that if M has a hyperbolic piece in its characteristic
