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Width and mean curvature flow

Tobias H Colding and William P Minicozzi II

Geometry & Topology 12 (2008) 2517–2535

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.

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; cf our papers [??] where a similar bound for the rate of change for the two dimensional width is shown for homotopy 3–spheres evolving by the Ricci flow (see also Perelman [?]).

width, sweepout, min-max, mean curvature flow, extinction time
Mathematical Subject Classification 2000
Primary: 53C44, 58E10
Secondary: 53C22
Received: 20 June 2007
Accepted: 10 October 2008
Published: 6 November 2008
Proposed: Ben Chow
Seconded: Colin Rourke, Martin Bridson
Tobias H Colding
Department of Mathematics, MIT
77 Massachusetts Avenue
Cambridge, MA 02139-4307, USA
Courant Institute of Mathematical Sciences
251 Mercer Street, New York, NY 10012
William P Minicozzi II
Department of Mathematics
Johns Hopkins University
3400 N Charles St
Baltimore, MD 21218