Width and mean curvature flow

Colding, Tobias H¹ ; Minicozzi II, William P²

¹ Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA, and, Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA
² Department of Mathematics, Johns Hopkins University, 3400 N Charles St, Baltimore, MD 21218, USA

Geometry & topology, Tome 12 (2008) no. 5, pp. 2517-2535.

Résumé

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 [?]).

Détail

DOI : 10.2140/gt.2008.12.2517

Keywords: width, sweepout, min-max, mean curvature flow, extinction time

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