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The Mass of an Asymptotically Hyperbolic Manifold with a Non-compact Boundary

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Abstract

We define a mass-type invariant for asymptotically hyperbolic manifolds with a non-compact boundary which are modelled at infinity on the hyperbolic half-space and prove a sharp positive mass inequality in the spin case under suitable dominant energy conditions. As an application, we show that any such manifold which is Einstein and either has a totally geodesic boundary or is conformally compact and has a mean convex boundary is isometric to the hyperbolic half-space.

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Acknowledgements

Part of this work was carried out during the first author’s visit to Princeton University in the academic year 2018–2019. He would like to express his deep gratitude to Prof. Fernando Marques and the Mathematics Department. The authors thank the anonymous referee for comments that improved the final version of this paper.

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Authors and Affiliations

  1. Instituto de Matemática, Campus do Gragoatá, Universidade Federal Fluminense (UFF), Rua Prof. Marcos Waldemar de Freitas s/n bloco H, Niterói, RJ, 24210-201, Brazil

    Sérgio Almaraz

  2. Departamento de Matemática, Campus do Pici, Universidade Federal do Ceará (UFC), Av. Humberto Monte, s/n, Bloco 914, Fortaleza, CE, 60455-760, Brazil

    Levi Lopes de Lima

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  1. Sérgio Almaraz
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  2. Levi Lopes de Lima
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Correspondence to Sérgio Almaraz.

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Communicated by Mihalis Dafermos.

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S. Almaraz has been partially supported by CNPq/Brazil Grant 309007/2016-0 and CAPES/Brazil Grant 88881.169802/2018-01, and L. de Lima has been partially supported by CNPq/Brazil Grant 312485/2018-2. Both authors have been partially supported by FUNCAP/CNPq/PRONEX Grant 00068.01.00/15.

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Almaraz, S., de Lima, L.L. The Mass of an Asymptotically Hyperbolic Manifold with a Non-compact Boundary. Ann. Henri Poincaré 21, 3727–3756 (2020). https://doi.org/10.1007/s00023-020-00954-w

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