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Some Remarks on Conformal Symmetries and Bartnik’s Splitting Conjecture

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Abstract

Inspired by the results in a recent paper by Galloway and Vega (Lett Math Phys 108(10):2285–2292, 2018), we investigate a number of geometric consequences of the existence of a timelike conformal Killing vector field in a globally hyperbolic space-time with compact Cauchy hypersurfaces, especially in connection with the so-called Bartnik’s splitting conjecture. In particular, we give a complementary result to the main theorem in [11].

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Notes

  1. The reader should be aware that such a completeness assumption is often included as part of the definition of ‘conformastationary (resp. stationary) space-time’ in the literature. Moreover, sometimes the timelike character is imposed only in some asymptotic sense, especially in the study of stationary black holes.

  2. A is then a closed \(C^0\) (indeed Lipschitz) hypersurface in M, see, e.g., Corollary 26, Chapter 14 in [20].

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Acknowledgements

We wish to thank Marcelo M. Cavalcanti, Alberto Enciso and Rafael Ortega for useful discussions. The authors are partially supported by the Spanish Grant MTM2016-78807-C2-2-P (MINECO and FEDER funds). The second author also wishes to acknowledge the Department of Mathematics, Universidade Federal de Santa Catarina (Brazil), for the kind hospitality while part of this research was being carried out.

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Authors and Affiliations

  1. Department of Mathematics, Universidade Federal de Santa Catarina, Florianópolis, SC, 88.040-900, Brazil

    I. P. Costa e Silva

  2. Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, Campus Teatinos, 29071, Málaga, Spain

    J. L. Flores

  3. Departamento de Matemáticas, Edificio Albert Einstein, Universidad de Córdoba, Campus de Rabanales, 14071, Córdoba, Spain

    J. Herrera

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  1. I. P. Costa e Silva
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  2. J. L. Flores
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  3. J. Herrera
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Correspondence to J. Herrera.

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Costa e Silva, I.P., Flores, J.L. & Herrera, J. Some Remarks on Conformal Symmetries and Bartnik’s Splitting Conjecture. Mediterr. J. Math. 17, 21 (2020). https://doi.org/10.1007/s00009-019-1447-2

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