The Essential Spectrum of the Laplacian on Manifolds with Ends

Abstract

Let V be a noncompact complete Riemannian manifold. We find a geometric condition which assures that the essential spectrum of the Laplacian on V contains a half-line, by means of fiber bundle structures and the asymptotic behavior of mean curvatures on the ends of V , and give lower bounds of the essential spectrum. Our criteria can be applied to locally symmetric spaces of finite volume and manifolds of infinite volume canonically obtained from manifolds with corners.