05-387 Rafael D. Benguria, Helmut Linde
A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic space (320K, Postscript) Nov 11, 05
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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.

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