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