Gradient estimates for a nonlinear Lichnerowicz equation under general geometric flow on complete noncompact manifolds

Liang Zhao and Shouwen Fang

Vol. 285 (2016), No. 1, 243–256

DOI: 10.2140/pjm.2016.285.243

Abstract

We study gradient estimates for positive solutions to the nonlinear parabolic equation

$$\frac{\partial u}{\partial t}=\Delta u+c{u}^{-\alpha }$$

under general geometric flow on complete noncompact manifolds, where $\alpha ,c$ are two real constants and $\alpha >0$. As an application, we give the corresponding Harnack inequality.

Keywords

gradient estimates, geometric flow, Harnack inequality

Mathematical Subject Classification 2010

Primary: 58J05

Secondary: 58J35

Milestones

Received: 16 September 2015

Revised: 20 April 2016

Accepted: 22 May 2016

Published: 27 September 2016