Width and flow of hypersurfaces by curvature functions

TL;DR

This paper establishes bounds on the extinction time for convex hypersurfaces evolving under curvature-dependent flows, generalizing previous results and introducing the concept of width to analyze the flow's behavior.

Contribution

It extends the concept of width to curvature flows and provides new bounds on extinction time for a broader class of hypersurfaces, including 2-convex ones.

Findings

Bound on extinction time for convex hypersurfaces

Extension of width concept to curvature flows

Results applicable to 2-convex hypersurfaces

Abstract

We give a bound on the extinction time for a compact, strictly convex hypersurface in R^{n+1} evolving by a geometric flow where the velocity is given in terms of the curvature. This result generalizes a theorem of Colding and Minicozzi for mean curvature flow solutions to a wider class of flows studied by Ben Andrews. In the proof, we use the concept of the width of a hypersurface, introduced by Colding and Minicozzi. We also extend the result to 2-convex hypersurfaces, using the 2-width.