Absence of eigenvalue at the bottom of the continuous spectrum on asymptotically hyperbolic manifolds

TL;DR

This paper proves that on certain asymptotically hyperbolic manifolds, the bottom of the continuous spectrum of the Laplace-Beltrami operator is not an eigenvalue, using a localized approach near infinity.

Contribution

It introduces a method that relies solely on properties near infinity, avoiding global assumptions on topology or curvature, unlike previous studies.

Findings

Bottom of the continuous spectrum is not an eigenvalue on the studied manifolds.

Method relies only on asymptotic properties, not global geometric assumptions.

Applicable to a class of asymptotically hyperbolic manifolds.

Abstract

For a class of asymptotically hyperbolic manifolds, we show that the bottom of the continuous spectrum of the Laplace-Beltrami operator is not an eigenvalue. Our approach only uses properties of the operator near infinity and, in particular, does not require any global assumptions on the topology or the curvature, unlike previous papers on the same topic.