Semigroup Properties for the Second Fundamental Form

Abstract

Let $M$ be a compact Riemannian manifold with boundary $\pp M$ and $L= \DD+Z$ 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 $\pp M$. 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.

17 pages

References in corpus (2)

• Harnack Inequalities for Functional SDEs with Multiplicative Noise and Applications
• Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex manifolds

Cited by in corpus (9)

• How to hear the corners of a drum
• Gradient Estimate on the Neumann Semigroup and Applications
• Reflecting Diffusion Process on Time-Inhomogeneous Manifolds with Boundary
• Functional Inequalities for Weighted Gamma Distribution on the Space of Finite Measures
• Gradient Estimates on Dirichlet Eigenfunctions
• Modified Curvatures on Manifolds with Boundary and Applications
• Gradient Estimates and Applications for Neumann Semigroup on Narrow Strip
• Equivalent Harnack and Gradient Inequalities for Pointwise Curvature Lower Bound
• Harnack Inequalities on Manifolds with Boundary and Applications