Semigroup Properties for the Second Fundamental Form

TL;DR

This paper explores the properties of the second fundamental form in relation to the Neumann semigroup on compact Riemannian manifolds with boundary, establishing inequalities and generalizing known results to boundary cases.

Contribution

It introduces new equivalences involving curvature bounds, gradient, and Poincaré inequalities for manifolds with boundary, emphasizing the role of the second fundamental form.

Findings

Generalizes known inequalities to manifolds with boundary

Clarifies the role of the second fundamental form in semigroup analysis

Studies the Lévý-Gromov isoperimetric inequality on manifolds with boundary

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 Poincar\'e/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 L\'evy-Gromov isoperimetric inequality is also studied on manifolds with boundary.