Asian Journal of Mathematics

Volume 24 (2020)

Number 1

Harnack estimate for mean curvature flow on the sphere

Pages: 165 – 176



Paul Bryan (Department of Mathematics, Macquarie University, Sydney, NSW, Australia)

Mohammad N. Ivaki (Department of Mathematics, University of Toronto, Ontario, Canada)


We consider the evolution of hypersurfaces on the unit sphere $\mathbb{S}^{n+1}$ by their mean curvature. We prove a differential Harnack inequality for any weakly convex solution to the mean curvature flow. As an application, by applying an Aleksandrov reflection argument, we classify convex, ancient solutions of the mean curvature flow on the sphere.


mean curvature flow, ancient solutions, Aleksandrov reflection, Harnack estimate

2010 Mathematics Subject Classification

35K55, 53C44, 58J35

Received 9 August 2017

Accepted 10 May 2019

Published 21 August 2020