Abstract
Let be some domain in the hyperbolic space (with ), and let be a geodesic ball that has the same first Dirichlet eigenvalue as . We prove the Payne-Pólya-Weinberger (PPW) conjecture for , namely, that the second Dirichlet eigenvalue on is smaller than or equal to the second Dirichlet eigenvalue on . We also prove that the ratio of the first two eigenvalues on geodesic balls is a decreasing function of the radius
Citation
Rafael D. Benguria. Helmut Linde. "A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic space." Duke Math. J. 140 (2) 245 - 279, 1 November 2007. https://doi.org/10.1215/S0012-7094-07-14022-5
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