Two-sided non-collapsing curvature flows

TL;DR

This paper extends non-collapsing curvature flow results to a broader class of speed functions satisfying inverse-concavity, and demonstrates their utility in proving convergence of convex hypersurfaces to round points.

Contribution

It generalizes non-collapsing results to inverse-concave speed functions and applies this to prove convergence of convex hypersurfaces to round points.

Findings

Exterior non-collapsing holds for a larger class of speed functions.

Two-sided non-collapsing flows are established beyond mean curvature flow.

Convex hypersurfaces converge to round points under these flows.

Abstract

It was recently proved that embedded solutions of Euclidean hypersurface flows with speeds given by concave (convex), degree one homogeneous functions of the Weingarten map are interior (exterior) non-collapsing. These results were subsequently extended to hypersurface flows in the sphere and hyperbolic space. In the first part of the paper, we show that locally convex solutions are exterior non-collapsing for a larger class of speed functions than previously considered; more precisely, we show that the previous results hold when convexity of the speed function is relaxed to inverse-concavity. We note that inverse-concavity is satisfied by a large class of concave speed functions. As a consequence, we obtain a large class of two-sided non-collapsing flows, whereas previously two-sided non-collapsing was only known for the mean curvature flow. In the second part of the paper, we demonstrate the utility of two sided non-collapsing with a straightforward proof of convergence of compact, convex hypersurfaces to round points. The proof of the non-collapsing estimate is similar to the previous results mentioned, in that we show that the exterior ball curvature is a viscosity supersolution of the linearised flow. The new ingredient is the following observation: Since the function which provides an upper support in the derivation of the viscosity inequality is defined on $M\times M$ (or $TM$ in the `boundary case'), whereas the exterior ball curvature and the linearised flow equation depend only on the first factor, we are privileged with a freedom of choice in which second derivatives from the extra directions to include in the calculation. The optimal choice is closely related to the class of inverse-concave speed functions.