A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic space

Abstract

Let Ω be some domain in the hyperbolic space Hn (with n≥2), and let S1 be a geodesic ball that has the same first Dirichlet eigenvalue as Ω. We prove the Payne-Polya-Weinberger (PPW) conjecture for Hn, namely, that the second Dirichlet eigenvalue on Ω is smaller than or equal to the second Dirichlet eigenvalue on S1. We also prove that the ratio of the first two eigenvalues on geodesic balls is a decreasing function of the radius