Width and mean curvature flow

Details Width and mean curvature flow Width and mean curvature flow

2007-05-25

arxiv.org/abs/0705.3827v2

Mathematics

Differential Geometry

Includes applications to flows by powers of mean curvature

Scientific paper

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).

Affiliated with

Also associated with

No associations

Rating

Width and mean curvature flow does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Width and mean curvature flow, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Width and mean curvature flow will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-680559