A volume comparison theorem for asymptotically hyperbolic 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.

References in corpus (2)

• On The Capacity of Surfaces in Manifolds with Nonnegative Scalar Curvature
• A Minkowski inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold

Cited by in corpus (11)

• Effective versions of the positive mass theorem
• General Bounds on Holographic Complexity
• Large isoperimetric regions in asymptotically hyperbolic manifolds
• On perturbations of the Schwarzschild anti-de Sitter spaces of positive mass
• The isoperimetric inequality on asymptotically flat manifolds with nonnegative scalar curvature
• The Penrose inequality for asymptotically locally hyperbolic spaces with nonpositive mass
• Exhaustion of isoperimetric regions in asymptotically hyperbolic manifolds with scalar curvature $R\ge -6$
• Characterization of large isoperimetric regions in asymptotically hyperbolic initial data
• Volume comparison of conformally compact manifolds with scalar curvature $R\ge -n\left(n-1\right)$
• Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds with dimension $3$
• Curvature at the infinity of asymptotically flat Einstein manifold