Convexity estimates for hypersurfaces moving by concave curvature functions
Differential Geometry
2020-07-16 v1
Abstract
We study fully nonlinear geometric flows that deform strictly -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 -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 -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.
Cite
@article{arxiv.2007.07791,
title = {Convexity estimates for hypersurfaces moving by concave curvature functions},
author = {Stephen Lynch},
journal= {arXiv preprint arXiv:2007.07791},
year = {2020}
}