Harnack Inequalities on Manifolds with Boundary and Applications

TL;DR

This paper establishes the equivalence between certain Harnack inequalities, boundary convexity, and curvature conditions on Riemannian manifolds with boundary, and extends these results to derive the HWI inequality.

Contribution

It proves dimension-free Harnack inequalities are equivalent to boundary convexity and curvature bounds, and extends these results to non-convex boundaries and HWI inequalities.

Findings

Harnack inequalities characterize boundary convexity and curvature conditions.

Equivalence between heat kernel inequalities and geometric conditions.

Extension of results to manifolds with non-convex boundary.

Abstract

On a large class of Riemannian manifolds with boundary, some dimension-free Harnack inequalities for the Neumann semigroup is proved to be equivalent to the convexity of the boundary and a curvature condition. In particular, for $p_t(x,y)$ the Neumann heat kernel w.r.t. a volume type measure $\mu$ and for $K$ a constant, the curvature condition $\Ric-\nn Z\ge K$ together with the convexity of the boundary is equivalent to the heat kernel entropy inequality $$\int_M p_t(x,z)\log \ff{p_t(x,z)}{p_t(y,z)} \mu(\d z)\le \ff{K\rr(x,y)^2}{2(\e^{2Kt}-1)}, t>0, x,y\in M,$$ where $\rr$ is the Riemannian distance. The main result is partly extended to manifolds with non-convex boundary and applied to derive the HWI inequality.