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Width and mean curvature
flow
Tobias H Colding and William P Minicozzi II |
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Geometry & Topology 12 (2008) 2517–2535
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Abstract |
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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 [?]). |
Keywords
width, sweepout, min-max, mean curvature flow, extinction
time
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Mathematical Subject Classification 2000
Primary: 53C44, 58E10
Secondary: 53C22
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References |
Publication
Received: 20 June 2007
Accepted: 10 October 2008
Published: 6 November 2008
Proposed: Ben Chow
Seconded: Colin Rourke, Martin Bridson
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Authors |
| Tobias H Colding | |
| Department of Mathematics, MIT 77 Massachusetts Avenue Cambridge, MA 02139-4307, USA and Courant Institute of Mathematical Sciences 251 Mercer Street, New York, NY 10012 USA |
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| William P Minicozzi II | |
| Department of Mathematics Johns Hopkins University 3400 N Charles St Baltimore, MD 21218 USA |
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