Abstract
An (n + 1)-dimensional Generalized Robertson–Walker (GRW) spacetime such that the universal Riemannian covering of the fiber is parabolic (thus so is the fiber) is said to be spatially parabolic. This class of spacetimes allows to model open relativistic universes which extend to the spatially closed GRW spacetimes from the viewpoint of the geometric-analysis of the fiber and which are not incompatible with certain cosmological principle. We explain here a new technique for the study of non-compact complete spacelike hypersurfaces in such spacetimes. Thus, a complete spacelike hypersurface in a spatially parabolic GRW spacetime inherits the parabolicity, whenever some boundedness assumptions on the restriction of the warping function to the spacelike hypersurface and on the hyperbolic angle between the unit normal vector field and a certain timelike vector field are assumed. Conversely, the existence of a simply connected parabolic spacelike hypersurface, under the previous assumptions, in a GRW spacetime also leads to its spatial parabolicity. Then, all the complete maximal hypersurfaces in a spatially parabolic GRW spacetime are determined in several cases, extending known uniqueness results. Finally, all the entire solutions of the maximal hypersurface equation on a parabolic Riemannian manifold are found in several cases, solving new Calabi–Bernstein problems.
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Notes
- 1.
This definition simplifies the one given in [16] where each GRW spacetime considered was explicitly assumed with parabolic universal Riemannian covering of its fiber.
References
Alías, L.J., Montiel, S.: Uniqueness of spacelike hypersurfaces with constant mean curvature in generalized Robertson–Walker spacetimes. Differential Geometry, Valencia, 2001 World Sci. Publ., River Edge, pp. 59–69 (2002)
Alías, L.J., Romero, A., Sánchez, M.: Uniqueness of complete spacelike hypersurfaces of constant mean curvature in Generalized Robertson–Walker spacetimes. Gen. Relat. Gravit. 27, 71–84 (1995)
Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian Geometry, 2nd edn. Pure and Applied Mathematics, vol. 202. Marcel Dekker, New York (1996)
Bousso, R.: The holographic principle. Rev. Mod. Phys. 74, 825–874 (2002)
Caballero, M., Romero, A., Rubio, R.M.: Constant mean curvature spacelike hypersurfaces in Lorentzian manifolds with a timelike gradient conformal vector field. Classical Quant. Grav. 28, 145009–145022 (2011)
Chavel, I.: Eigenvalues in Riemannian Geometry. Pure and Applied Mathematics, vol. 115. Academic, New York (1984)
Chiu, H.Y.: A cosmological model for our universe. Ann. Phys. 43, 1–41 (1967)
Grenne, R.E., Wu, H.: Function theory on manifolds which possess a pole. Lecture Notes Series in Mathematics, vol. 699. Springer, New York (1979)
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)
Kanai, M.: Rough isometries and the parabolicity of Riemannian manifolds. J. Math. Soc. Jpn. 38, 227–238 (1986)
Kazdan, J.K.: Parabolicity and the Liouville property on complete Riemannian manifolds. In: Tromba, A.J. (ed.) Aspects of Mathematics, vol. E10, pp. 153–166. Vieweg and Sohn, Bonn (1987)
Li, P.: Curvature and function theory on Riemannian manifolds. Surveys in Differential Geometry, vol. II, pp. 375–432. International Press, Somerville (2000)
Nishikawa, S.: On maximal spacelike hypersurfaces in a Lorentzian manifold. Nagoya Math. J. 95, 117–124 (1984)
O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic, New York (1983)
Romero, A., Rubio, R.M.: On the mean curvature of spacelike surfaces in certain three-dimensional Robertson–Walker spacetimes and Calabi–Bernstein’s type problems. Ann. Glob. Anal. Geom. 37, 21–31 (2010)
Romero, A., Rubio, R.M., Salamanca, J.J.: Uniqueness of complete maximal hypersurfaces in spatially parabolic generalized Robertson–Walker spacetimes. Class. Quant. Grav. 30, 115007(1–13) (2013)
Romero, A., Rubio, R.M., Salamanca, J.J.: Parabolicity of spacelike hypersurfaces in generalized Robertson–Walker spacetimes. Applications to uniqueness results. Int. J. Geom. Methods Mod. Phys. 10, 1360014(1–8) (2013)
Royden, H.: Harmonic functions on open Riemann surfaces. Trans. Am. Math. Soc. 73, 40–94 (1952)
Sachs, R.K., Wu, H.: General Relativity for Mathematicians. Graduate Texts in Mathematics, vol. 48. Springer, New York (1977)
Acknowledgements
Supported by the Spanish MICINN Grant with FEDER funds MTM2010-18099, the Junta de Andalucía Regional Grant with FEDER funds P09-FQM-4496, National Institute for Mathematical Sciences, Daejeon, Korea, and Grassmannian Research Group of the Dep. of Mathematics of the Kyungpook National University, Daegu, Korea. The author would like also to express his sincere thanks to Prof. Y.J. Suh and Dr. Hyunjin Lee.
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Romero, A. (2014). A New Technique for the Study of Complete Maximal Hypersurfaces in Certain Open Generalized Robertson–Walker Spacetimes. In: Suh, Y.J., Berndt, J., Ohnita, Y., Kim, B.H., Lee, H. (eds) Real and Complex Submanifolds. Springer Proceedings in Mathematics & Statistics, vol 106. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55215-4_3
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