Moment Spectrum and First Dirichlet Eigenvalue of Geodesic Balls in Riemannian Manifolds

Abstract

In this work, given a geodesic ball of a Riemannian manifold with radius less than the injectivity radius of its center, we prove our estimates for some geometric invariants defined on the ball. The invariants that we will study are the mean exit time function, the torsional rigidity, the Poisson hierarchy, the moment spectrum and the first eigenvalue of the Laplacian for the Dirichlet problem. To find our estimates we will compare these geometric invariants with those defined in the corresponding geodesic balls of certain rotationally symmetric model spaces. In particular, to make our comparisons, we must either construct the rotationally symmetric model spaces from the area function of the geodesic spheres of the original Riemannian manifold, or we must assume bounds between the mean curvatures of the geodesic spheres of the manifold and their corresponding on the rotationally symmetric model spaces.