  We establish a Gaussian upper bound of the heat kernel for the
Laplace-Beltrami operator on complete Riemannian manifolds with Bakry-\'Emery
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-\'Emery 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.
