Summary: Let $M$ be a compact Riemannian manifold with boundary $\partial M$ and $L= \delta+Z$ for a $C^1$-vector field $Z$ on $M$. Several equivalent statements, including the gradient and Poincaré/log-Sobolev type inequalities of the Neumann semigroup generated by $L$, are presented for lower bound conditions on the curvature of $L$ and the second fundamental form of $\partial M$. The main result not only generalizes the corresponding known ones on manifolds without boundary, but also clarifies the role of the second fundamental form in the analysis of the Neumann semigroup. Moreover, the Lévy-Gromov isoperimetric inequality is also studied on manifolds with boundary.

60J60, 58G32

second fundamental form, gradient estimate, Neumann semigroup, log-Sobolev inequality, Poincaré inequality