Gradient Estimates for Neumann Semigroups on Manifolds with Boundary under Unbounded Curvature Conditions

Abstract

This paper establishes Bismut-type formulas and gradient estimates for Feynman--Kac semigroups on Riemannian manifolds with boundary, under geometric conditions formulated in terms of Ricci curvature ${\mathrm{Ric}}_Z\ge K$ and second fundamental form $\mathrm{II}\ge \ddot{I}ƒ$ for potentially unbounded functions $K$ and $\ddot{I}ƒ$. We then apply these formulas to derive pointwise gradient estimates for the Neumann semigroup under variable, possibly unbounded, lower curvature bounds. Both convex and non-convex boundary cases are treated. In the non-convex case, the boundary contribution is controlled by a conformal change of metric and an exponential estimate for the boundary local time.