Convexity estimates for hypersurfaces moving by concave curvature functions

TL;DR

This paper investigates fully nonlinear geometric flows deforming strictly k-convex hypersurfaces in Euclidean space, establishing convexity estimates that show high curvature regions are approximately convex, and extends these results to certain Riemannian backgrounds.

Contribution

It introduces convexity estimates for hypersurfaces evolving under nonlinear flows with concave curvature functions, including interpolations between mean and harmonic mean curvatures, and extends results to curved ambient spaces.

Findings

High curvature regions are approximately convex in the flow.

Convexity estimates are preserved under certain ambient curvature conditions.

The flow preserves k-convexity in Riemannian backgrounds with pinched curvature.

Abstract

We study fully nonlinear geometric flows that deform strictly $k$-convex hypersurfaces in Euclidean space with pointwise normal speed given by a concave function of the principal curvatures. Specifically, the speeds we consider are obtained by performing a nonlinear interpolation between the mean and the $k$-harmonic mean of the principal curvatures. Our main result is a convexity estimate showing that, on compact solutions, regions of high curvature are approximately convex. In contrast to the mean curvature flow, the fully nonlinear flows considered here preserve $k$-convexity in a Riemannian background, and we show that the convexity estimate carries over to this setting as long as the ambient curvature satisfies a natural pinching condition.