Modified Curvatures on Manifolds with Boundary and Applications

TL;DR

This paper introduces new curvature operators on manifolds with boundary to analyze reflecting diffusion processes, leading to gradient estimates and inequalities for the Neumann semigroup.

Contribution

It proposes novel curvature operators combining Bakry-Emery curvature and second fundamental form for manifolds with boundary.

Findings

Established gradient estimates for Neumann semigroup

Derived log-Harnack inequalities

Proved Poincaré and log-Sobolev inequalities

Abstract

To study the reflecting diffusion processes on manifolds with boundary, some new curvature operators are introduced by using the Bakry-Emery curvature and the second fundamental form. As applications, the gradient estimates, log-Harnack inequality and Poincar\'e/log-Sobolev inequalities are investigated for the Neumann semigroup on manifolds with boundary.