A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic space

TL;DR

This paper proves the Payne-Pólya-Weinberger conjecture for the hyperbolic space, establishing bounds on the second eigenvalue of the Dirichlet Laplacian and analyzing eigenvalue ratios on geodesic balls.

Contribution

It extends the Payne-Pólya-Weinberger conjecture to hyperbolic space and studies eigenvalue ratios on geodesic balls, providing new bounds and monotonicity results.

Findings

Second eigenvalue bound for hyperbolic space domains

Validation of the Payne-Pólya-Weinberger conjecture in hyperbolic space

Decreasing ratio of first two eigenvalues with radius

Abstract

Let $\Omega$ be some domain in the hyperbolic space $\Hn$ (with $n\ge 2$) and $S_1$ the geodesic ball that has the same first Dirichlet eigenvalue as $\Omega$. We prove the Payne-P\'olya-Weinberger conjecture for $\Hn$, i.e., that the second Dirichlet eigenvalue on $\Omega$ is smaller or equal than 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.