[CvGmt News] GFO-Pisa Seminars - Wednesday 20 May 2009

[Carlo Mantegazza]

Wednesday 20 May 2009 - Leili Shahriyari, Department of Mathematics,
Johns Hopkins University
H. 18.00 - Aula Conferenze of the Centro De Giorgi,
Collegio Puteano of the SNS


Width and Mean Curvature Flow

Abstract: We will discuss the following results of Colding and Minicozzi
(Eprint arXiv:0705.3827)

Given a Riemannian metric on a homotopy n-sphere, sweep it out by a
continuous one-parameter family of closed curves starting and ending at
point curves. Pull the sweepout tight by, in a continuous way, pulling
each curve as tight as possible yet preserving the sweepout.
We show: each curve in the tightened sweepout whose length is close to the
length of the longest curve in the sweepout must itself be close to a
closed geodesic. In particular, there are curves in the sweepout that are
close to closed geodesics. Finding closed geodesics on the 2-sphere by
using sweepouts goes back to Birkhoff in 1917. As an application, we bound
from above, by a negative constant, the rate of change of the width for a
one-parameter family of convex hypersurfaces that flows by mean curvature.
The width is loosely speaking up to a constant the square of the length of
the shortest closed curve needed to ``pull over'' M. This estimate is
sharp and leads to a sharp estimate for the extinction time.