Estimates for sums of eigenvalues for domains in homogeneous spaces (Q1911521)

The main purpose of this paper is to estimate the sums of eigenvalues for domains in homogeneous Riemannian spaces. Let \(\Omega \subseteq M\) be a bounded open subset of a homogeneous Riemannian manifold \(M\). Let \(\sigma_k = \lambda_1 + \cdots + \lambda_k\) be the sum of the first \(k\) eigenvalues of the Dirichlet Laplacian on \(\Omega\), and similarly \(\widetilde{\sigma}_k = \widetilde{\lambda}_1 + \cdots + \widetilde {\lambda}_k\) for the Neumann Laplacian. The author gives bounds for \(\sigma\) and \(\widetilde{\sigma}\) generalizing results of Li-Yau and Kröger in the case \(M = \mathbb{R}^n\). He has proved a ``generic theorem'' which in the case of compact \(M\) says \(\sigma_k \geq p(\Omega) \Sigma (k/p(\Omega) \geq \widetilde{\sigma}_k\) where \(p(\Omega) = |\Omega|/|M|\) is the relative volume of \(\Omega\) and \(\Sigma(x)\) is the eigenvalue sum function for \(M\) (interpolated linearly for non integer values). For noncompact \(M\) the statement is \(\sigma_k \geq |\Omega|\Sigma (k / |\Omega|)\) where \(\Sigma\) is a renormalized eigenvalue sum function for \(M\) (defined using the spectral resolution of \(\Delta\) on \(M\)). There are also estimates in the other direction of the same form with error terms. The same generic theorems hold for Laplacians on \(p\)-forms, and for subelliptic Laplacians on sub-Riemannian manifolds. The author discusses the computation of \(\Sigma\) for spheres, hyperbolic spaces, noncompact symmetric spaces, and the Heisenberg groups.