On the discreteness of the spectrum of the laplacian on noncompact riemannian manifolds

Abstract

Necessary and sufficient conditions for the discreteness of the Laplacian on a noncompact Riemannian manifold M are estab- lished in terms of the isocapacitary function of M. The relevant capacity takes a different form according to whether M has fi- nite or infinite volume. Conditions involving the more standard isoperimetric function of M can also be derived, but they are only sufficient in general, as we demonstrate by concrete examples. © 2011 J. Differential Geometry.