Conformal compactification of asymptotically locally hyperbolic metrics II: Weakly ALH metrics

TL;DR

This paper investigates the intrinsic characterization of conformally compact asymptotically hyperbolic metrics, demonstrating how curvature decay influences regularity and developing harmonic coordinates to improve previous results.

Contribution

It establishes the relationship between curvature decay and regularity of compactified metrics, removing previous decay assumptions on covariant derivatives, and introduces harmonic coordinates satisfying Neumann conditions.

Findings

Curvature decay controls H"older regularity of the compactified metric.

Harmonic coordinates with Neumann conditions are constructed at infinity.

Previous results are extended without decay assumptions on covariant derivatives.

Abstract

In this paper we pursue the work initiated in \cite{Bahuaud, BahuaudGicquaud}: study the extent to which conformally compact asymptotically hyperbolic metrics can be characterized intrinsically. We show how the decay rate of the sectional curvature to -1 controls the H\"older regularity of the compactified metric. To this end, we construct harmonic coordinates that satisfy some Neumann-type condition at infinity. Combined with a new integration argument, this permits us to recover to a large extent our previous result without any decay assumption on the covariant derivatives of the Riemann tensor. We believe that our result is optimal.