On the Li-Yau-Hamilton Harnack inequalities on Ricci flow and super Ricci flow

In this paper, we prove the Li-Yau-Hamilton differential Harnack inequality and the Harnack inequality for positive solutions to the heat equation associated with the Witten Laplacian on weighted compact or completemanifolds with time dependent complete Riemannian metrics and potentials. In particular, we prove the Li-Yau-Hamilton differential Harnack inequality and the Harnack inequality for positive solutions to the heat equation associated with theLaplace-Beltrami operatoron compact or complete Riemannian manifolds with the Ricci flow or the backward Ricci flow.