Heat kernel estimate for the Laplace-Beltrami operator under Bakry-Émery Ricci curvature condition and applications

Abstract

We establish a Gaussian upper bound of the heat kernel for the Laplace-Beltrami operator on complete Riemannian manifolds with Bakry-Émery Ricci curvature bounded below. As applications, we first prove an L^1-Liouville property for non-negative subharmonic functions when the potential function of the Bakry-Émery Ricci curvature tensor is of at most quadratic growth. Then we derive lower bounds of the eigenvalues of the Laplace-Beltrami operator on closed manifolds. An upper bound of the bottom spectrum is also obtained.

arXiv comment33 pages. arXiv admin note: text overlap with arXiv:1401.6155 by other authors