Essential spectrum of a class of Riemannian manifolds

TL;DR

This paper investigates the essential spectrum of the Laplacian on certain Riemannian manifolds with specific curvature conditions, providing insights into the spectral properties of unbounded regions like horoballs and cones.

Contribution

It establishes that under particular curvature conditions, the essential spectrum contains an interval, extending spectral analysis to non-complete manifolds and specific geometric regions.

Findings

Essential spectrum contains an interval under curvature conditions

Spectrum determination for regions like horoballs and cones

Applicable to non-complete Riemannian manifolds

Abstract

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.