Inverse scattering results for manifolds hyperbolic near infinity

TL;DR

This paper investigates inverse resonance problems for hyperbolic conformally compact manifolds, establishing compactness, finiteness, and topological results across various dimensions under curvature conditions.

Contribution

It provides new compactness and finiteness theorems for isoresonant and isophasal metrics on hyperbolic manifolds, extending understanding in inverse spectral geometry.

Findings

Compactness of isoresonant metrics in dimension two

Compactness of isophasal negatively curved metrics in dimension three

Topological finiteness theorems in dimensions four and higher

Abstract

We study the inverse resonance problem for conformally compact manifolds which are hyperbolic outside a compact set. Our results include compactness of isoresonant metrics in dimension two and of isophasal negatively curved metrics in dimension three. In dimensions four or higher we prove topological finiteness theorems under the negative curvature assumption.