Width and mean curvature flow
Width and mean curvature flow
复制标题
宽度和平均曲率流量
DOI：
10.2140/gt.2008.12.2517
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发表时间：
2007
期刊：
影响因子：
--
通讯作者：
W. Minicozzi
中科院分区：
文献类型：
--
作者：
T. Colding;W. Minicozzi
给定同伦球面上的黎曼度规，用以点曲线为起点和终点的连续单参数闭曲线族将其扫除。以连续的方式拉紧洗井，在保持洗井的同时尽可能拉紧每条曲线。我们展示:
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).