Hamilton's Harnack inequality and the $W$-entropy formula on complete Riemannian manifolds

Abstract: In this paper, we prove Hamilton's Harnack inequality and the gradient estimates of the logarithmic heat kernel for the Witten Laplacian on complete Riemainnian manifolds. As applications, we prove the $W$-entropy formula for the Witten Laplacian on complete Riemannian manifolds, and prove a family of logarithmic Sobolev inequalities on complete Riemannian manifolds with natural geometric condition.