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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 [?]).
This is an expository article with complete proofs intended for a general nonspecialist audience. The results are two-fold. First, we discuss a geometric invariant, that we call the width, of a manifold and show how it can be realized as the sum of areas of minimal –spheres. For instance, when is a homotopy –sphere, the width is loosely speaking the area of the smallest –sphere needed to ‘pull over’ . Second, we use this to conclude that Hamilton’s Ricci flow becomes extinct in finite time on any homotopy –sphere.