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-& Eacute;mery 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. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.