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∈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.