L<sup>1</sup>-convergence and hypercontractivity of diffusion semigroups on manifolds
L<sup>1</sup>-convergence and hypercontractivity of diffusion semigroups on manifolds
Let $P_t$ be the Markov semigroup generated by a weighted Laplace operator on a Riemannian manifold, with $\mu $ an invariant probability measure. If the curvature associated with the generator is bounded below, then the exponential convergence of $P_t$ i