Methods and Applications of Analysis

Volume 27 (2020)

Number 4

Special Issue for the 60th Birthday of John Urbas: Part I

Guest Editors: Neil Trudinger and Xu-Jia Wang (Australian National University)

Li–Yau gradient estimates for curvature flows in positively curved manifolds

Pages: 341 – 358



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

Heiko Kröner (Fakultät für Mathematik, Universität Duisburg-Essen, Germany)

Julian Scheuer (Cardiff University, School of Mathematics, Cardiff, Wales, United Kingdom)


We prove differential Harnack inequalities for flows of strictly convex hypersurfaces by powers $p, 0 \lt p \lt 1$, of the mean curvature in Einstein manifolds with a positive lower bound on the sectional curvature. We assume that this lower bound is sufficiently large compared to the derivatives of the curvature tensor of the ambient space and that the mean curvature of the initial hypersurface is sufficiently large compared to the ambient geometry. We also obtain some new Harnack inequalities for more general curvature flows in the sphere, as well as a monotonicity estimate for the mean curvature flow in non-negatively curved, locally symmetric spaces.


curvature flow, Harnack inequality, pinching

2010 Mathematics Subject Classification

53C21, 53C44

This work was supported by the “Deutsche Forschungsgemeinschaft” (DFG, German research foundation) within the research grant “Harnack inequalities for curvature flows and applications”, number SCHE 1879/1-1; and by the ARC within the research grant “Analysis of fully non-linear geometric problems and differential equations”, number DE180100110.

Received 24 December 2019

Accepted 2 April 2020

Published 24 September 2021