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2008 Width and mean curvature flow
Tobias H Colding, William P Minicozzi II
Geom. Topol. 12(5): 2517-2535 (2008). DOI: 10.2140/gt.2008.12.2517

Abstract

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 [?]).

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Tobias H Colding. William P Minicozzi II. "Width and mean curvature flow." Geom. Topol. 12 (5) 2517 - 2535, 2008. https://doi.org/10.2140/gt.2008.12.2517

Information

Received: 20 June 2007; Accepted: 10 October 2008; Published: 2008
First available in Project Euclid: 20 December 2017

zbMATH: 1165.53363
MathSciNet: MR2460870
Digital Object Identifier: 10.2140/gt.2008.12.2517

Subjects:
Primary: 53C44 , 58E10
Secondary: 53C22

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

Rights: Copyright © 2008 Mathematical Sciences Publishers

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