Revisiting Li-Yau type inequalities on Riemannian manifolds

Abstract

In this paper, we first present a refined Hamilton's gradient estimate for the heat equation which is firstly obtained by R.S. Hamilton (Comm. Anal. Geom., 1993). New Harnack inequalities and new bounds of the associated heat kernels are obtained. Inspired by Yau's work (Comm. Anal. Geom., 1994), we obtain a generalized Li-Yau gradient estimate for the linear heat equation, which generalizes some known results and generates new gradient estimates.