Log-Sobolev inequalities: Different roles of Ric and Hess

Let $P_t$ be the diffusion semigroup generated by $L:=\mathrm{\Delta }+\mathrm{\nabla }V$ on a complete connected Riemannian manifold with $\operatorname {Ric}\ge-(\sigma ^2\rho_o^2+c)$forsomeconstants$\sigma, c>0$and$\rho_o$ the Riemannian distance to a fixed point. It is shown that $P_t$ is hypercontractive, or the log-Sobolev inequality holds for the associated Dirichlet form, provided $-{\mathrm{Hess}}_V\ge \delta$ holds outside of a compact set for some constant $\delta >(1+\sqrt{2})\sigma \sqrt{d-1}\mathrm{.}$ This indicates, at least in finite dimensions, that $\mathrm{Ric}$ and $-{\mathrm{Hess}}_V$ play quite different roles for the log-Sobolev inequality to hold. The supercontractivity and the ultracontractivity are also studied.