 
Summary: RESEARCH BLOG 11/12/03
Last week, I decided to take a break from thinking about Ricci flow.
I was working hard on computing the variation of the Willmore energy
under Ricci flow, and I came up with a computation, but I couldn't see
how to make use of it as a maximum principle, to show that a Willmore
energy minimizer is increasing with Ricci flow (as conjectured in blog
10/17/03). Also, I wasn't able to prove the conjecture on backwards
propagation of minimal surfaces under Ricci flow, as conjectured in
blog 3/4/03. The main reason is that for an unstable minimal surface
, 2
A()/r2
could be > 0 (that is, the second variation of area
in the normal direction may be positive, even though there is some
variation which has negative 2nd variation), so I wasn't able to make
the heuristic I gave precise. Right now, I don't have any ideas on how
to approach these conjectures.
In blog 10/28/03, I worked out an improvement of the BishopGromov
comparison theorem in the case of negatively curved 3manifolds. I
stated that I thought the inequality should hold more generally. But
the difficulty is that the exponential map may become orientation re
