Communications in Analysis and Geometry

Volume 31 (2023)

Number 3

Differential Harnack inequalities via Concavity of the arrival time

Pages: 547 – 561

DOI: https://dx.doi.org/10.4310/CAG.2023.v31.n3.a1

Authors

Theodora Bourni (Department of Mathematics, University of Tennessee, Knoxville, Knoxville, Tenn., U.S.A.)

Mat Langford (Mathematical Sciences Institute, Australian National University, Canberra, ACT, Australia)

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.

Received 14 December 2019

Accepted 8 December 2020

Published 4 January 2024