Spacelike Hypersurfaces in Spatially Parabolic Standard Static Spacetimes and Calabi–Bernstein-Type Problems

  • José A. S. PelegrínEmail author
  • Alfonso Romero
  • Rafael M. Rubio


Complete spacelike hypersurfaces in spatially parabolic standard static spacetimes are studied. Under natural boundedness assumptions, we show how the parabolicity of the base is inherited by any spacelike hypersurface and vice versa. Moreover, we give new uniqueness and non-existence results for complete spacelike hypersurfaces in these ambient spacetimes as well as solve new Calabi–Bernstein-type problems.


Spacelike hypersurface parabolicity standard static spacetime Calabi–Bernstein problem 

Mathematics Subject Classification

53C42 53C80 53C50 58J05 



  1. 1.
    Aledo, J.A., Romero, A., Rubio, R.M.: The existence and uniqueness of standard static splitting. Class. Quantum Gravity 32(1–9), 105004 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alías, L.J., Mastrolia, P., Rigoli, M.: Maximum Principles and Geometric Applications. Springer, Berlin (2016)CrossRefGoogle Scholar
  4. 4.
    Arms, J.M., Marsden, J.E., Moncrief, V.: The structure of the space of solutions of Einstein’s equations. II. Several Killing fields and the Einstein–Yang–Mills equations. Ann. Phys. 144, 81–106 (1982)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bak, D., Rey, S.-J.: Cosmic holography. Class. Quantum Gravity 17, L83–L89 (2000)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Besse, A.L.: Einstein Manifolds. Springer, New York (2007)zbMATHGoogle Scholar
  7. 7.
    Bousso, R.: The holographic principle. Rev. Mod. Phys. 74, 825–874 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Brasil, A., Colares, A.G.: On constant mean curvature spacelike hypersurfaces in Lorentz manifolds. Mat. Contemp. 17, 99–136 (1999)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Calabi, E.: Examples of Bernstein problems for some nonlinear equations. Proc. Symp. Pure Math. 15, 223–230 (1970)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cheng, S.Y., Yau, S.T.: Maximal space-like hypersurfaces in the Lorentz–Minkowski spaces. Ann. Math. 104, 407–419 (1976)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chiu, H.Y.: A cosmological model of our universe. Ann. Phys. 43, 1–41 (1967)CrossRefGoogle Scholar
  12. 12.
    Daftardar, V., Dadhich, N.: Gradient conformal Killing vectors and exact solutions. Gen. Relativ. Gravit. 26, 859–868 (1994)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Do Carmo, M.P.: Riemannian Geometry. Birkhauser, Basel (1992)CrossRefGoogle Scholar
  14. 14.
    Duggal, K., Sharma, R.: Symmetries of Spacetimes and Riemannian Manifolds. Springer Science and Business Media, New York (2013)zbMATHGoogle Scholar
  15. 15.
    Eardley, D., Isenberg, J., Marsden, J., Moncrief, V.: Homothetic and conformal symmetries of solutions to Einsteins equations. Commun. Math. Phys. 106, 137–158 (1996)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Grigor’yan, A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. 36, 135–249 (1999)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  18. 18.
    Kazdan, J.L.: Parabolicity and the Liouville property on complete Riemannian manifolds. Aspects Math. 10, 153–166 (1987)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Latorre, J.M., Romero, A.: Uniqueness of noncompact spacelike hypersurfaces of constant mean curvature in generalized Robertson–Walker spacetimes. Geom. Dedicata 93, 1–10 (2002)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lichnerowicz, A.: L’integration des équations de la gravitation relativiste et le problème des n corps. J. Math. Pure Appl. 23, 37–63 (1944)zbMATHGoogle Scholar
  21. 21.
    Marsden, J.E., Tipler, F.J.: Maximal hypersurfaces and foliations of constant mean curvature in General Relativity. Phys. Rep. 66, 109–139 (1980)MathSciNetCrossRefGoogle Scholar
  22. 22.
    O’Neill, B.: Semi-Riemannian geometry with applications to relativity. Academic Press, London (1983)zbMATHGoogle Scholar
  23. 23.
    Pelegrín, J.A.S., Romero, A., Rubio, R.M.: On maximal hypersurfaces in Lorentz manifolds admitting a parallel lightlike vector field. Class. Quantum Gravity 33(1–8), 055003 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Pelegrín, J.A.S., Romero, A., Rubio, R.M.: Uniqueness of complete maximal hypersurfaces in spatially open \((n+1)\)-dimensional Robertson–Walker spacetimes with flat fiber. Gen. Relativ. Gravit. 48, 1–14 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Pelegrín, J.A.S., Romero, A., Rubio, R.M.: On uniqueness of the foliation by comoving observers restspaces of a generalized Robertson–Walker spacetime. Gen. Relativ. Gravit. 49, 16 (2017)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Romero, A., Rubio, R.M., Salamanca, J.J.: Uniqueness of complete maximal hypersurfaces in spatially parabolic generalized Robertson–Walker spacetimes. Class. Quantum Gravity 30, 115007–115020 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Sachs, R.K., Wu, H.: General Relativity for Mathematicians, Graduate Texts in Mathematics. Springer, New York (1977)CrossRefGoogle Scholar
  28. 28.
    Sánchez, M.: On the geometry of static spacetimes. Nonlinear Anal. 63, 455–463 (2005)CrossRefGoogle Scholar
  29. 29.
    Sánchez, M., Senovilla, J.M.M.: A note on the uniqueness of global static decompositions. Class. Quantum Gravity 24(1–6), 6121 (2007)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Schoen, R., Yau, S.T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65, 45–76 (1979)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Tod, K.P.: Spatial metrics which are static in many ways. Gen. Relativ. Gravit. 32, 2079–2090 (2000)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departamento de Geometría y TopologíaUniversidad de GranadaGranadaSpain
  2. 2.Departamento de Matemáticas, Campus de RabanalesUniversidad de CórdobaCórdobaSpain
  3. 3.Departamento de Matemática Aplicada y EstadísticaUniversidad CEU San PabloMadridSpain

Personalised recommendations