The Rigidity and Stability of Gradient Estimates

Abstract

In this note, we obtain the rigidity of the sharp Cheng-Yau gradient estimate for positive harmonic functions on surfaces with non-negative Gaussian curvature, the rigidity of the sharp Li-Yau gradient estimate for positive solutions to heat equations and the related estimates for Dirichlet Green’s functions on Riemannian manifolds with non-negative Ricci curvature. Moreover, we also obtain the corresponding stability results.