Estimates for the first non-zero eigenvalue of the Hodge Laplacian acting on differential forms defined on a Riemannian manifold

Abstract

In this article, M is a convex domain of a regular geodesic ball of a smooth Riemannian manifold (N, g). We prove that, when the radius of the ball is small enough, M contains an unique ellipsoid of maximal volume. That is a generalization of John’s theorem to Riemannian manifolds. Then, we use these results to obtain an upper bound and a lower bound for the first non-zero eigenvalue of the Hodge Laplacian acting on differential p-forms defined on M.