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RESEARCH BLOG 8/20/03 Colding and Minicozzi posted a paper on singularity formation in
 

Summary: RESEARCH BLOG 8/20/03
Colding and Minicozzi posted a paper on singularity formation in
the Ricci flow (thanks to Mohan Ramachandran for pointing this out
to me). They prove that if one has a 3-manifold M which is non-
aspherical, then Ricci flow can exist for only finite time. This is stronger
than what Perelman proved in his third paper, in which he proved this
for manifolds which have virtually free fundamental group (although
Perelman claims a stronger result, that the Ricci flow with -cutoff be-
comes a connect sum of spherical space forms in finite time in this case).
Colding and Minicozzi prove this using similar methods to Perelman,
although they are technically somewhat simpler. If 3(M) = 0, then
they use area estimate of a minimax sphere to show that the solution
must have a singularity in finite time, as described in blog 7/20/03. If
2(M) = 0, then they give a more complicated estimate of the change
in area of a minimal 2-sphere which is homotopically non-trivial, and
again show that this leads to a singularity in finite time. Presumably
their technique would generalize to Ricci-flow with -cutoff, but I don't
understand this well enough yet.
In blog 7/20/03, the monotonicty formula for (g, ) I stated at the
end was incorrect, actually one obtains the same monotonicity formula

  

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

 

Collections: Mathematics