Gradient estimates of Poisson equations on Riemannian manifolds and applications

For the reflected diffusion generated by L=Δ−∇V⋅∇ on a connected and complete Riemannian manifold M with empty or convex boundary, we establish some sharp estimates of supx∈M|∇G|(x) of the Poisson equation −LG=g in terms of the dimension, the diameter and the lower bound of curvature. Applications to transportation-information inequality, to Cheeger's isoperimetric inequality and to Gaussian concentration inequality are given. Several examples are provided.