A volume comparison theorem for asymptotically hyperbolic manifolds (Q466042)

First, a notion of normalized volume of an asymptotically hyperbolic manifold of dimension three is defined. Then, motivated by a volume comparison theorem for three-dimensional manifolds with scalar curvature at least 6, due to \textit{H. Bray} [The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature. Stanford, CA: Stanford University (PhD Thesis) (1997)], a sharp volume comparison theorem for metrics with scalar curvature at least \(-6\) is proved. In the end, it is shown that the inequality is strict unless the metric is isometric to one of the anti-deSitter-Schwarzschild metrics.