Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds

TL;DR

This paper investigates the meromorphic extension of the Laplacian's resolvent on asymptotically hyperbolic manifolds, revealing that even metrics allow a meromorphic extension with finite rank poles, while non-even metrics may have essential singularities.

Contribution

It establishes a precise criterion linking the geometric property of the metric being 'even' to the meromorphic extendability of the resolvent.

Findings

Resolvent extends meromorphically with finite rank poles for even metrics.

Non-even metrics can exhibit essential singularities in the resolvent.

Characterization of the resolvent's singularities based on metric properties.

Abstract

We show that the resolvent of the Laplacian on asymptotically hyperbolic spaces extends meromorphically with finite rank poles to the complex plane if and only if the metric is `even' (in a sense). If it is not even, there exist some cases where the resolvent has an essential singularity in the non-physical sheet.