Geometry & Topology

Width and mean curvature flow

Tobias H Colding and William P Minicozzi II

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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 [?]).

Article information

Source
Geom. Topol., Volume 12, Number 5 (2008), 2517-2535.

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513800151

Digital Object Identifier
doi:10.2140/gt.2008.12.2517

Mathematical Reviews number (MathSciNet)
MR2460870

Zentralblatt MATH identifier
1165.53363

Subjects
Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 58E10: Applications to the theory of geodesics (problems in one independent variable)
Secondary: 53C22: Geodesics [See also 58E10]

Keywords
width sweepout min-max mean curvature flow extinction time

Citation

Colding, Tobias H; Minicozzi II, William P. Width and mean curvature flow. Geom. Topol. 12 (2008), no. 5, 2517--2535. doi:10.2140/gt.2008.12.2517. https://projecteuclid.org/euclid.gt/1513800151


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