Harnack inequalities for evolving hypersurfaces on the sphere

TL;DR

This paper establishes Harnack inequalities for hypersurfaces evolving on the sphere under curvature flows driven by power functions, with stronger results when the curvature is mean curvature.

Contribution

It introduces new Harnack inequalities for hypersurface flows on the sphere driven by convex curvature functions raised to a power, including improved inequalities for mean curvature.

Findings

Harnack inequalities hold for flows with curvature functions to the power p, 0<p≤1

Stronger inequalities are obtained when the curvature is the mean curvature

Results extend the understanding of curvature flow behavior on the sphere

Abstract

We prove Harnack inequalities for hypersurfaces flowing on the unit sphere by $p$-powers of a strictly monotone, 1-homogeneous, convex, curvature function $f$, $0<p\leq 1.$ If $f$ is the mean curvature, we obtain stronger Harnack inequalities.