Volume 12, issue 5 (2008)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 28
Issue 3, 1005–1499
Issue 2, 497–1003
Issue 1, 1–496

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
Other MSP Journals
Width and mean curvature flow

Tobias H Colding and William P Minicozzi II

Geometry & Topology 12 (2008) 2517–2535

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 [?]).

width, sweepout, min-max, mean curvature flow, extinction time
Mathematical Subject Classification 2000
Primary: 53C44, 58E10
Secondary: 53C22
Received: 20 June 2007
Accepted: 10 October 2008
Published: 6 November 2008
Proposed: Ben Chow
Seconded: Colin Rourke, Martin Bridson
Tobias H Colding
Department of Mathematics, MIT
77 Massachusetts Avenue
Cambridge, MA 02139-4307, USA
Courant Institute of Mathematical Sciences
251 Mercer Street, New York, NY 10012
William P Minicozzi II
Department of Mathematics
Johns Hopkins University
3400 N Charles St
Baltimore, MD 21218