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RESEARCH BLOG 7/15/03 I've been trying to understand Perelman's work, and so far I'm about

Summary: RESEARCH BLOG 7/15/03
I've been trying to understand Perelman's work, and so far I'm about
through the first four sections of his paper [4], with a lot of help from
Kleiner and Lott's notes.
In his second paper [3], Perelman claims that in his first paper [4],
he verified that Ricci flow is a gradient flow for the first eigenvalue
of -4 + R. Indeed, the first section of the paper is called "Ricci
flow as a gradient flow." But the difficulty with this is that there is an
arbitrary function f which evolves in equation 1.1 as a backwards heat
equation (so the evolution is not exactly a gradient), and some trickery
is involved because of this to make things work (also, it is not exactly
the Ricci flow, but a flow modified by a diffeomorphism which is a
"gradient flow" in his context). There is some remarkable magic which
makes the functional monotonic with respect to Ricci flow, but it seems
that this functional was first discovered by string theorists. On p. 911
of the book "Quantum fields and strings: A course for mathematicians,
vol. 2" [1], equations 6.69 are essentially the stationary version of
what Perelman has in his paper as equations 1.1, with an extra field
B which Perelman has eliminated, but kept the "dilaton field". His
functional essentially appears in equation 6.74. It seems that Perelman


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


Collections: Mathematics