Contracting convex hypersurfaces by curvature

TL;DR

This paper studies curvature-driven flows of convex hypersurfaces, showing conditions under which they smoothly contract to points or develop singularities, and identifying classes of flows with guaranteed smoothing and contraction.

Contribution

It characterizes when convex hypersurfaces contract smoothly versus develop singularities under curvature flows, and introduces conditions ensuring smooth contraction to points.

Findings

Uniformly convex hypersurfaces remain smooth and contract to points.

Certain flows can cause convex hypersurfaces to become non-convex or develop singularities.

Conditions are provided to guarantee smooth contraction for weakly convex hypersurfaces.

Abstract

We consider compact convex hypersurfaces contracting by functions of their curvature. Under the mean curvature flow, uniformly convex smooth initial hypersurfaces evolve to remain smooth and uniformly convex, and contract to points after finite time. The same holds if the initial data is only weakly convex or non-smooth, and the limiting shape at the final time is spherical. We provide a surprisingly large family of flows for which such results fail, by a variety of mechanisms: Uniformly convex hypersurfaces may become non-convex, and smooth ones may develop curvature singularities; even where this does not occur, non-uniformly convex regions and singular parts in the initial hypersurface may persist, including flat sides, ridges of infinite curvature, or `cylindrical' regions where some of the principal curvatures vanish; such cylindrical regions may persist even if the speed is positive, and in such cases the hypersurface may even collapse to a line segment or higher-dimensional disc rather than to a point. We provide sufficient conditions for these various disasters to occur, and by avoiding these arrive at a class of flows for which arbitrary weakly convex initial hypersurfaces immediately become smooth and uniformly convex and contract to points.