A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic space

Abstract

Let $\Omega$ be some domain in the hyperbolic space ${\mathbb{H}}^n$ (with $n\ge 2$), and let $S_1$ be a geodesic ball that has the same first Dirichlet eigenvalue as $\Omega$. We prove the Payne-Pólya-Weinberger (PPW) conjecture for ${\mathbb{H}}^n$, namely, that the second Dirichlet eigenvalue on $\Omega$ is smaller than or equal to the second Dirichlet eigenvalue on $S_1$. We also prove that the ratio of the first two eigenvalues on geodesic balls is a decreasing function of the radius