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Deforming the Scalar Curvature of the De Sitter–Schwarzschild Space

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Abstract

Building upon the work of Brendle, Marques, and Neves on the construction of counterexamples to Min-Oo’s conjecture, we exhibit deformations of the de Sitter–Schwarzschild space of dimension \(n\ge 3\) satisfying the dominant energy condition and agreeing with the standard metric along the event and cosmological horizons, which remain totally geodesic. Our results hold for generalized Kottler–de Sitter–Schwarzschild spaces whose cross-sections are compact rank one symmetric spaces and indicate that there exists no analogue of the rigidity statement of the Penrose inequality in the case of positive cosmological constant. As an application, we construct solutions of Einstein field equations satisfying the dominant energy condition and being asymptotic to (or agreeing with) the de Sitter–Schwarzschild space–time both at the event horizon and at spatial infinity.

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Acknowledgements

The authors thank L. Ambrozio, F. Marques, and I. Nunes for valuable discussions. Also, they would like to thank an anonymous referee for suggestions which helped substantially improve the presentation. L. L. de Lima and J. F. Montenegro were partially supported by CNPq/Brazil research grants. C. T. Cruz was supported by a CNPq Scholarship.

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Authors and Affiliations

  1. Instituto de Matemática, Universidade Federal de Alagoas (UFAL), Maceió, AL, 570272-900, Brazil

    Cícero Tiarlos Cruz

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

    Levi Lopes de Lima & José Fabio Montenegro

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  1. Cícero Tiarlos Cruz
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  2. Levi Lopes de Lima
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  3. José Fabio Montenegro
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Correspondence to Levi Lopes de Lima.

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Cruz, C.T., de Lima, L.L. & Montenegro, J.F. Deforming the Scalar Curvature of the De Sitter–Schwarzschild Space. J Geom Anal 28, 473–491 (2018). https://doi.org/10.1007/s12220-017-9829-9

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  • DOI: https://doi.org/10.1007/s12220-017-9829-9

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