Heat kernel bounds and Cheng-Yau type estimate for the Laplace-Beltrami operator with Bakry-Émery Ricci curvature lower bound

On complete Riemannian manifolds with Bakry-Émery Ricci curvature bounded below, we first derive a parabolic Harnack inequality for positive solutions of the heat equation and Gaussian upper and lower bounds of the heat kernel for the Laplace-Beltrami operator. As applications of the heat kernel estimates, an L¹-Liouville theorem for non-negative subharmonic functions and lower bounds of the Dirichlet eigenvalues are shown. Finally, we prove Cheng-Yau type local gradient estimates for positive harmonic functions and Dirichlet and Neumann eigenfunctions.