A universal bound for the low eigenvalues of Neumann Laplacians on compact domains in a Hadamard manifold
Let \(\Omega\) be a bounded domain with smooth boundary in a simply connected \(n\) dimensional Hadamard manifold \(M\) with Ricci curvature bounded from below, i.e., \(\text{Ric}_M\geq-(n-1)k^2\). Let \(0<\mu_1\leq\mu_1\leq\dots\leq\mu_n\) be the first \(n\) non-trivial eigenvalues of the scalar Laplacian \(\Delta\) with Neumann boundary conditions. The author gives a computable constant \(C(n,\text{vol}(\Omega),k^2)\) so that \(\mu_1^{-1}+\cdots+\mu_n^{-1}\geq C(n,\text{vol}(\Omega),k^2)\).
scientific article; zbMATH DE number 1367697