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RESEARCH BLOG 10/6/03 I've been trying to understand more of Perelman's work, but I find it
 

Summary: RESEARCH BLOG 10/6/03
I've been trying to understand more of Perelman's work, but I find it
difficult going. There are new notes on Perelman's papers by Yu Ding,
but I haven't had a chance to look at them yet. I've also been trying
to prove the conjectures made in blog 3/4/03 and blog 5/19/03 , and
I've made some progress in the special case of warped products, using
some of Perelman's results on the nature of singularities. This gives
me hope that the conjectures hold in general, since one might expect
that the metric is very close to a warped product metric near to where
a singularity is occurring.
There is a somewhat paradoxical picture of how a type II singularity
occurs for Ricci flow on a 3-manifold. Recall that a type II singularity
is a solution to the Ricci flow on a compact 3-manifold M, gt, 0
t < T, such that sup |Rm(gt)|(T - t) = (which basically means
that the sectional curvatures are blowing up faster than the inverse
of time to singularity). Some Ricci flow experts call this a "slowly-
forming" singularity, which sounds somewhat paradoxical, since the
curvature is blowing up much faster than for a type I singularity. But
this terminology may be explained by Hamilton's and Perelman's result
that one may extract a scaled limit of this solution to obtain a stable

  

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

 

Collections: Mathematics