Width and mean curvature flow
| dc.creator | Colding, Tobias H. | |
| dc.creator | Minicozzi II, William P. | |
| dc.date | 2007-05-25 | |
| dc.date | 2007-06-13 | |
| dc.date.accessioned | 2026-07-07T08:05:09Z | |
| dc.date.available | 2026-07-07T08:05:09Z | |
| dc.description | 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. Finding closed geodesics on the 2-sphere by using sweepouts goes back to Birkhoff in 1917. 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. [CM1], [CM2] 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). | |
| dc.description | Includes applications to flows by powers of mean curvature | |
| dc.identifier | https://arxiv.org/abs/0705.3827 | |
| dc.identifier | http://arxiv.org/abs/0705.3827 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/130190 | |
| dc.subject | Differential Geometry | |
| dc.subject | Analysis of PDEs | |
| dc.subject | Geometric Topology | |
| dc.title | Width and mean curvature flow | |
| dc.type | text |