Intrinsic characterization for Lipschitz asymptotically hyperbolic metrics

TL;DR

This paper investigates intrinsic geometric conditions on complete Riemannian manifolds that guarantee they are asymptotically hyperbolic, using geodesic compactification and curvature decay to analyze regularity.

Contribution

It provides intrinsic criteria for asymptotic hyperbolicity based on geodesic compactification and curvature decay, advancing understanding beyond conformally compact metrics.

Findings

Established intrinsic conditions ensuring asymptotic hyperbolicity.

Used geodesic compactification to analyze regularity of the conformal boundary.

Presented an example demonstrating near-optimality of the assumptions.

Abstract

Conformally compact asymptotically hyperbolic metrics have been intensively studied. The goal of this note is to understand what intrinsic conditions on a complete Riemannian manifold (M,g) will ensure that g is asymptotically hyperbolic in this sense. We use the geodesic compactification by asymptotic geodesic rays to compactify M and appropriate curvature decay conditions to study the regularity of the conformal compactification. We also present an interesting example that shows our conclusion is nearly optimal for our assumptions.