Modified Curvatures on Manifolds with Boundary and Applications

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.