Differential Harnack inequalities on Riemannian manifolds I: Linear heat equation

In the first part of this paper, we get new Li-Yau type gradient estimates for positive solutions of heat equation on Riemannian manifolds with Ricci(M) >= -k, k is an element of 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) >= -k, which generalizes a result of L. Ni (2004, 2006) [20,21]. 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. Published by Elsevier Inc.