Essential spectrum of a class of Riemannian manifolds

In this paper we consider a family of Riemannian manifolds, not necessarily complete, with curvature conditions in a neighborhood of a ray. Under these conditions we obtain that the essential spectrum of the Laplacian contains an interval. The results presented in this paper allow to determine the spectrum of the Laplace operator on unlimited regions of space forms, such as horoball in hyperbolic space and cones in Euclidean space.