Conformal compactification of asymptotically locally hyperbolic metrics

TL;DR

This paper investigates intrinsic characterizations of conformally compact asymptotically hyperbolic metrics, establishing regularity results based on curvature decay and extending these to Einstein metrics for comprehensive control of geometric derivatives.

Contribution

It proves that curvature decay conditions imply regularity of conformal compactifications and extends these results to Einstein metrics for full derivative control.

Findings

Curvature decay implies Hölder regularity of compactification.

In Einstein case, curvature decay controls all covariant derivatives of the Weyl tensor.

Results strengthen understanding of geometric regularity in asymptotically hyperbolic spaces.

Abstract

In this paper we study the extent to which conformally compact asymptotically hyperbolic metrics may be characterized intrinsically. Building on the work of the first author, we prove that decay of sectional curvature to -1 and decay of covariant derivatives of curvature outside an appropriate compact set yield H\"older regularity for a conformal compactification of the metric. In the Einstein case, we prove that the estimate on the sectional curvature implies the control of all covariant derivatives of the Weyl tensor, permitting us to strengthen our result.