Estimates for Sums of Eigenvalues for Domains in Homogeneous Spaces

Let&0Hgr;⊆Mbe a bounded open subset of a homogeneous Riemannian manifold, and letσk=λ1+…+λkbe the sum of the firstkeigenvalues of the Dirichlet Laplacian on&0Hgr;, and similarlyσk=λ1+…+λkfor the Neumann Laplacian. We give bounds forσkandσkgeneralizing results of Li–Yau and Kröger in the caseM=Rn. We prove a “generic theorem” which in the case of compactMsaysσk⩾p(&0Hgr;) Σ(k/p(&0Hgr;))⩾σkwherep(&0Hgr;)=|&0Hgr;|/|M| is the relative volume of&0Hgr;andΣ(x) is the eigenvalue sum function forM(interpolated linearly for non integer values). For noncompactMthe statement isσk⩾|&0Hgr;| Σ(k/|&0Hgr;|) whereΣis a renormalized eigenvalue sum function forM(defined using the spectral resolution ofΔonM). There are also estimates in the other direction of the same form with error terms. The same generic theorems hold for Laplacian onp-forms, and for subelliptic Laplacians on subRiemannian manifolds. To give life to such generic theorems it is necessary to compute theΣfunction for a variety of examples. For Euclideann-space,Σ(x)=(n/(n+2)) Cnx1+2/nwhereCnis the Weyl constant, so our generic result includes the known results. We discuss the computation ofΣfor spheres, hyperbolic spaces, noncompact symmetric spaces, and the Heisenberg groups.