Width and mean curvature flow

TL;DR

This paper investigates the relationship between sweepouts of homotopy spheres, closed geodesics, and the evolution of convex hypersurfaces under mean curvature flow, providing bounds on the width's rate of change and implications for extinction time.

Contribution

It introduces a method to tighten sweepouts to find curves close to geodesics and establishes bounds on the width's rate of change during mean curvature flow.

Findings

Curves close to the maximum length in a sweepout are near closed geodesics.

The width decreases at a rate bounded by a negative constant during mean curvature flow.

The bounds are sharp and relate to extinction time estimates.

Abstract

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).