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A Volume Comparison Theorem for Asymptotically Hyperbolic Manifolds

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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.

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, 450 Serra Mall, Bldg 380, Stanford, CA, 94305, USA

    Simon Brendle & Otis Chodosh

Authors
  1. Simon Brendle
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  2. Otis Chodosh
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Corresponding author

Correspondence to Simon Brendle.

Additional information

Communicated by P. T. Chruściel

The first author was supported in part by the National Science Foundation under grant DMS-1201924. The second author was supported in part by a National Science Foundation Graduate Research Fellowship DGE-1147470.

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Brendle, S., Chodosh, O. A Volume Comparison Theorem for Asymptotically Hyperbolic Manifolds. Commun. Math. Phys. 332, 839–846 (2014). https://doi.org/10.1007/s00220-014-2074-1

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  • DOI: https://doi.org/10.1007/s00220-014-2074-1

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