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 L1-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.
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