seminar: Width and mean curvature flow
speaker: Leili Shahriyari (Department of Mathematics, Johns Hopkins University)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.
timetable:
Wed 20 May, 18:00 - 19:00, Sala Conferenze Centro De Giorgi
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