A Volume Comparison Theorem for Asymptotically Hyperbolic Manifolds

2013/05/28 by S. Brendle, Simon Brendle, Otis Chodosh +1 · 24 citations
Mathematics · #Comparison theorem #Curvature #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry #Geometry and complex manifolds #Hyperbolic function #Hyperbolic manifold #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Metric (unit) #Physics #Pure mathematics #Scalar (mathematics) #Scalar curvature #Volume (thermodynamics) #math.DG

paper · pdf · doi:10.1007/s00220-014-2074-1

published in Communications in Mathematical Physics 332(2), 839-846 (Springer Science+Business Media)

arxiv created 2013/05/28 · openalex publication_date 2014/05/30 · arxiv updated 2015/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• On the capacity of surfaces in manifolds with nonnegative scalar curvature
• A Minkowski inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold