Gradient estimates and the first Neumann eigenvalue on manifolds with boundary

By studying the local time of reflecting diffusion processes, explicit gradient estimates of the Neumann heat semigroup on non-convex manifolds are derived from a recent derivative formula established by Hsu. As an application, an explicit lower bound of the first Neumann eigenvalue is presented via dimension, radius and bounds of the curvature and the second fundamental form. Finally, some new estimates are also presented for the strictly convex case.