ESTIMATES FOR SUMS OF EIGENVALUES FOR DOMAINS IN HOMOGENEOUS SPACES

摘要

Let Omega subset of or equal to M be a bounded open subset of a homogeneous Riemannian manifold, and let sigma(k) = lambda(1) + ... + lambda(k) be the sum of the first k eigenvalues of the Dirichlet Laplacian on Omega, and similarly <(sigma) over tilde (k)> = <(lambda) over tilde (1)> ... + <(lambda) over tilde (k)> for the Neumann Laplacian. We give bounds for <(sigma) over tilde (k)> and sigma(k) generalizing results of Li-Yau and Kroger in the case M = R ''. We prove a ''generic theorem'' which in the case of compact M says sigma(k) greater than or equal to p(Omega) Sigma(k/p(Omega)) greater than or equal to <(sigma)over tilde (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 nontompact M the statement is sigma(k) greater than or equal to 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 ill the other direction of the same form with error terms. The same generic theorems hold for Laplacian on p-forms, and for subelliptic Laplacians on subRiemannian manifolds. To give life to such generic theorems it is necessary to compute the C function for a variety of examples. For Euclidean n-space, Sigma(x) = (n/(n + 2)) C(n)x(1 + 2) where C-n is the Weyl constant, so our generic result includes the known results. We discuss the computation of C for spheres, hyperbolic spaces, noncompact symmetric spaces, and the Heisenberg groups. (C) 1996 Academic Press, Inc. [References: 22]