#### Volume 12, issue 5 (2008)

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Width and mean curvature flow

### Tobias H Colding and William P Minicozzi II

Geometry & Topology 12 (2008) 2517–2535
##### 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 [?]).

##### Keywords
width, sweepout, min-max, mean curvature flow, extinction time
##### Mathematical Subject Classification 2000
Primary: 53C44, 58E10
Secondary: 53C22