Stability of the Cheng–Yau gradient estimate

Qi S. Zhang

Vol. 225 (2006), No. 2, 379–398

DOI: 10.2140/pjm.2006.225.379

Abstract

We prove that the Cheng–Yau gradient estimate on positive harmonic functions on manifolds with nonnegative Ricci curvature is globally stable under certain perturbations of the metric. In some cases, one only needs the condition Ricci$\left(x\right)\ge -ϵ∕\left(1+d{\left(x\right)}^{2+\delta }\right)$, with $\delta >0$ and $ϵ>0$ sufficiently small.

Keywords

harmonic functions, log gradient bound, stablility

Mathematical Subject Classification 2000

Primary: 58J05, 58J35

Milestones

Received: 9 September 2004

Accepted: 21 November 2004

Published: 1 June 2006