Rafael D. Benguria, Helmut Linde
A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic space
(320K, Postscript)
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.