$L^1$-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 $P_t$ in $L^1(\mu )$ implies its hypercontractivity. Consequently, under this curvature condition $L^1$-convergence is a property stronger than hypercontractivity but weaker than ultracontractivity. Two examples are presented to show that in general, however, $L^1$-convergence and hypercontractivity are incomparable.