A volume comparison theorem for asymptotically hyperbolic manifolds

TL;DR

This paper introduces a renormalized volume concept for asymptotically hyperbolic manifolds and proves a sharp volume comparison theorem, establishing conditions for equality with Anti-deSitter-Schwarzschild metrics.

Contribution

It defines a new notion of renormalized volume and proves a sharp comparison theorem for scalar curvature bounds in asymptotically hyperbolic manifolds.

Findings

Established a sharp volume comparison theorem

Proved the inequality is strict unless the metric matches Anti-deSitter-Schwarzschild

Defined a new renormalized volume concept for these manifolds

Abstract

We define a notion of renormalized volume of an asymptotically hyperbolic manifold. Moreover, we prove a sharp volume comparison theorem for metrics with scalar curvature at least -6. Finally, we show that the inequality is strict unless the metric is isometric to one of the Anti-deSitter-Schwarzschild metrics.