Differential Harnack inequalities on Riemannian manifolds I : linear heat equation

Details Differential Harnack inequalities on Riemannian manifolds I : linear heat equation Differential Harnack inequalities on Riemannian manifolds I : linear heat equation

2009-01-24

arxiv.org/abs/0901.3849v1

Mathematics

Differential Geometry

Scientific paper

In the first part of this paper, we get new Li-Yau type gradient estimates for positive solutions of heat equation on Riemmannian manifolds with $Ricci(M)\ge -k$, $k\in \mathbb R$. As applications, several parabolic Harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with Ricci curvature bounded from below. In the second part, we establish a Perelman type Li-Yau-Hamilton differential Harnack inequality for heat kernels on manifolds with $Ricci(M)\ge -k$, which generalizes a result of L. Ni \cite{NL1,NL4}. As applications, we obtain new Harnack inequalities and heat kernel estimates on general manifolds. We also obtain various entropy monotonicity formulas for all compact Riemannian manifolds.

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