Asymptotically hyperbolic manifolds with polyhomogeneous metric

TL;DR

This paper studies the resolvent and scattering matrix on asymptotically hyperbolic manifolds with polyhomogeneous metrics, revealing an essential singularity and applying the results to inverse problems for Einstein manifolds.

Contribution

It introduces analysis of the resolvent and scattering matrix for polyhomogeneous metrics and establishes an inverse result for conformally compact Einstein manifolds.

Findings

Existence of an essential singularity of the resolvent.

Development of the scattering matrix for these manifolds.

Inverse result for conformally compact Einstein manifolds.

Abstract

We analyze the resolvent and define the scattering matrix for asymptotically hyperbolic manifolds with metrics which have a polyhomogeneous expansion near the boundary, and also prove that there is always an essential singularity of the resolvent in this setting. We use this analysis to prove an inverse result for conformally compact odd dimensional Einstein manifolds.