A differential Harnack inequality for noncompact evolving hypersurfaces

TL;DR

This paper establishes a differential Harnack inequality for noncompact convex hypersurfaces evolving under curvature-dependent flows, extending previous results to a broader class of noncompact solutions with specific geometric conditions.

Contribution

It generalizes Andrews' inequality from compact to noncompact hypersurfaces under curvature flows with new assumptions on speed and gradient estimates.

Findings

Proves a differential Harnack inequality for noncompact convex hypersurfaces.

Extends previous compact hypersurface results to noncompact cases.

Applicable to ancient solutions arising from singularity blow-ups.

Abstract

We prove a differential Harnack inequality for noncompact convex hypersurfaces flowing with normal speed equal to a symmetric function of their principal curvatures. This extends a result of Andrews for compact hypersurfaces. We assume that the speed of motion is one-homogeneous, uniformly elliptic, and suitably 'uniformly' inverse-concave as a function of the principal curvatures. In addition, we assume the hypersurfaces satisfy pointwise scaling-invariant gradient estimates for the second fundamental form. For many natural flows all of these hypotheses are met by any ancient solution which arises as a blow-up of a singularity.