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RESEARCH BLOG 7/6/04 VOLUME ESTIMATES VIA RICCI FLOW
 

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 3-manifold 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 non-decreasing with t when Rmin < 0 (it follows from
Perelman's claims that if M has a hyperbolic piece in its characteristic

  

Source: Agol, Ian - Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago

 

Collections: Mathematics