A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic space (Q2461386)

Let \(H^n\) be the \(n\)-dimensional hyperbolic space of constant negative sectional curvature \(-\rho^2\). Let \(\Omega\) be an open bounded domain in \(H^n\). Let \(\lambda_i(\Omega)\) be the \(i^{th}\) Dirichlet eigenvalue of \(\Omega\). The authors establish the analogue of the Payne-Pólya-Weinberger inequality for hyperbolic space: Theorem 1.1. Suppose that \(S\) is a geodesic ball in \(H^n\) such that \(\lambda_1(\Omega)=\lambda_1(S)\). Then \(\lambda_2(\Omega)\leq\lambda_2(S)\) with equality if and only if \(\Omega\) is a geodesic ball. Let \(S_\theta\) be the geodesic ball of radius \(\theta\) in \(H^n\). The authors establish the following monotonicity result. Theorem 1.2. The ratio \(\lambda_2(S_\theta)/\lambda_1(S_\theta)\) is a strictly decreasing function of \(\theta\).