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Complete spacelike hypersurfaces in generalized Robertson–Walker and the null convergence condition: Calabi–Bernstein problems

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Abstract

We study constant mean curvature spacelike hypersurfaces in generalized Robertson–Walker spacetimes \(\overline{M}= I \times _f F\) which are spatially parabolic (i.e. its fiber F is a (non-compact) complete Riemannian parabolic manifold) and satisfy the null convergence condition. In particular, we provide several rigidity results under appropriate mathematical and physical assumptions. We pay special attention to the case where the GRW spacetime is Einstein. As an application, some Calabi–Bernstein type results are given.

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References

  1. Albujer, A.L., Alías, L.J.: Spacelike hypersurfaces in the steady state space. Math. Proc. Am. Math. Soc. 137, 711–721 (2009)

    Article  MATH  Google Scholar 

  2. Albujer, A.L., Camargo, F.E.C., de Lima, H.F.: Complete spacelike hypersurfaces in a Robertson–Walker spacetime. Math. Proc. Camb. Phil. Soc. 151, 271–282 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aledo, J.A., Alías, L.J.: Curvature properties of compact spacelike hypersurfaces in de Sitter Space. Differ. Geom. Appl. 14, 137–149 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aledo, J.A., Rubio, R.M., Romero, A.: Constant mean curvature spacelike hypersurfaces in Lorentzian warped products and Calabi–Bernstein type problems Nonlinear Analysis: Theory. Methods Appl. 106, 57–69 (2014)

    MATH  Google Scholar 

  5. Alías, L.J., Colares, A.G.: Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes. Math. Proc. Camb. Phil. Soc. 143, 703–730 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Alías, L.J., Colares, A.G., de Lima, H.F.: On the rigidity of complete spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime. Bull. Braz. Math. Soc. 44, 195–217 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Alías, L.J., Mira, P.: On the Calabi–Bernstein theorem for maximal hypersurfaces in the Lorentz–Minkowski space. Publicaciones de la RSME 4, 23–55 (2003)

    Google Scholar 

  8. Alías, L.J., Romero, A., Sánchez, M.: Uniqueness of complete spacelike hypersurfaces of constant mean curvature in Generalized Robertson–Walker spacetimes. Gen. Relativ. Gravit. 27, 71–84 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  9. Alías, L.J., Romero, A., Sánchez, M.: Spacelike hypersurfaces of constant mean curvature and Calabi–Bernstein type problems. Tôhoku Math. J. 49, 337–345 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bartnik, R.: Existence of maximal surfaces in asymptotically flat spacetimes. Commun. Math. Phys. 94, 155–175 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Brill, D., Flaherty, F.: Isolated maximal surfaces in spacetime. Commun. Math. Phys. 50, 157–165 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  12. Caballero, M., Romero, A., Rubio, R.M.: Constant mean curvature spacelike hypersurfaces in Lorentzian manifolds with a timelike gradient conformal vector field. Class. Quant. Grav. 28, 145009–145022 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Calabi, E.: Examples of Bernstein problems for some nonlinear equations. Proc. Symp. Pure Math. 15, 223–230 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  14. Camargo, F.E.C., Camninha, A., de Lima, H.F., Parente, U.L.: Generalized maximum principles and the rigidity of complete spacelike hypersurfaces. Math. Proc. Camb. Phil. Soc. 153, 541–556 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cheng, S.Y., Yau, S.T.: Maximal space-like hypersurfaces in the Lorentz–Minkowski spaces. Ann. Math. 104, 407–419 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chiu, H.Y.: A cosmological model of universe. Ann. Phys. 43, 1–41 (1967)

    Article  Google Scholar 

  17. Choquet-Bruhat, Y.: Quelques proprietes des sous-varietes maximales d’un variete lorentzienne. Comptes Rend. Acad. Sci. (Paris) Serie A 281, 577–580 (1975)

  18. Galloway, G.J.: Cosmological spacetimes with \(\Lambda >0\). Advances in differential geometry and general relativity. Contemp. Math. Am. Math. Soc. Provid. RI 359, 87–101 (2004)

  19. Geroch, R.P.: Limits of spacetimes. Comm. Math. Phys. 13, 180–193 (1969)

    Article  MathSciNet  Google Scholar 

  20. de Lima, H.F., Parente, U.L.: On the geometry of maximal spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime. Ann. di Matematica Pura ed Appl. 192, 649–663 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Marsden, J.E., Tipler, F.J.: Maximal hypersurfaces and foliations of constant mean curvature in General Relativity. Phys. Rep. 66, 109–139 (1980)

    Article  MathSciNet  Google Scholar 

  22. O’Neill, B.: Semi-Riemannian geometry with applications to relativity. Academic Press (1983)

  23. Rainer, M., Schmidt, H.-J.: Inhomogeneous cosmological models with homogeneous inner hypersurface geometry. Gen. Relativ. Grav. 27, 1265–1293 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  24. Riess, A.G.: Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astrophys. J. 116, 1009–1038 (1999)

    Google Scholar 

  25. Romero, A., Rubio, R., Salamanca, J.J.: Uniqueness of complete maximal hypersurfaces in spatially parabolic generalized Robertson–Walker spacetimes. Class. Quantum Grav. 30, 115007 (2013)

  26. Sachs, R.K., Wu, H.: General relativity for mathematician. Springer, New York (1977)

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Acknowledgments

The Juan A. Aledo is partially supported by the Spanish MICINN Grant with FEDER funds MTM2013-43970-P and by the Junta de Comunidades de Castilla-La Mancha Grant PEII-2014-001-A. The Rafael M. Rubio and Juan J. Salamanca are partially supported by the Spanish MICINN Grant with FEDER funds MTM2013-47828-C2-1-P.

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Authors and Affiliations

  1. Departamento de Matemáticas, E.S.I. Informática, Universidad de Castilla-La Mancha, 02071, Albacete, Spain

    Juan A. Aledo

  2. Departamento de Matemáticas, Campus de Rabanales, Universidad de Córdoba, 14071, Córdoba, Spain

    Rafael M. Rubio & Juan J. Salamanca

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  1. Juan A. Aledo
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  2. Rafael M. Rubio
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  3. Juan J. Salamanca
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Correspondence to Rafael M. Rubio.

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Aledo, J.A., Rubio, R.M. & Salamanca, J.J. Complete spacelike hypersurfaces in generalized Robertson–Walker and the null convergence condition: Calabi–Bernstein problems. RACSAM 111, 115–128 (2017). https://doi.org/10.1007/s13398-016-0277-3

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  • DOI: https://doi.org/10.1007/s13398-016-0277-3

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