Differential Harnack inequalities via Concavity of the arrival time

TL;DR

This paper establishes a link between differential Harnack inequalities and concavity properties of arrival time functions, providing simplified proofs for key inequalities in hypersurface flows.

Contribution

It introduces a new approach connecting Harnack inequalities with concavity of arrival times, simplifying proofs for mean curvature and certain inverse-concave flows.

Findings

Proves concavity properties for a broad class of flows

Provides a short proof of Hamilton's Harnack inequality

Extends to Andrews' inequalities for alpha-inverse-concave flows

Abstract

We present a simple connection between differential Harnack inequalities for hypersurface flows and natural concavity properties of their time-of-arrival functions. We prove these concavity properties directly for a large class of flows by applying a concavity maximum principle argument to the corresponding level set flow equations. In particular, this yields a short proof of Hamilton's differential Harnack inequality for mean curvature flow and, more generally, Andrews' differential Harnack inequalities for certain "$\alpha$-inverse-concave" flows.