High Energy Resolvent Estimates on Conformally Compact Manifolds with Variable Curvature at Infinity

TL;DR

This paper develops high energy resolvent estimates for the Laplacian on conformally compact manifolds with variable curvature, including cases with trapping, using semiclassical analysis and gluing techniques.

Contribution

It introduces a semiclassical parametrix construction for variable curvature manifolds and extends resolvent estimates to cases with hyperbolic trapping.

Findings

Established high energy resolvent estimates for non-trapping conformally compact manifolds.

Proved existence of resonance free strips of arbitrary height.

Extended estimates to manifolds with hyperbolic trapping using gluing methods.

Abstract

We construct a semiclassical parametrix for the resolvent of the Laplacian acing on functions on non-trapping conformally compact manifolds with variable sectional curvature at infinity, we use it to prove high energy resolvent estimates and to show existence of resonance free strips of arbitrary height away from the imaginary axis. We then use the results of Datchev and Vasy on gluing semiclassical resolvent estimates to extend these results to conformally compact manifolds with normal hyperbolic trapping.