Abstract
Given a Riemannian metric on a homotopy -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” . 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 –spheres evolving by the Ricci flow (see also Perelman [?]).
Citation
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
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