Abstract
The aim of this article is to study the uniqueness of a complete spacelike hypersurface \({\sum^{n}}\) immersed with constant mean curvature H in a spatially closed generalized Robertson–Walker spacetime \({\overline{M}^{n+1} = -I {\times_{f}} {M^{n}}}\), whose Riemannian fiber \({M^n}\) has positive curvature. Supposing that the warping function f is such that −log f is convex and \({H{f^{\prime}} \leq 0}\) along \({\sum^{n}}\), we show that \({\sum^{n}}\) must be isometric to a totally geodesic slice of \({\overline{M}^{n+1}}\). When \({\overline{M}^{n+1}}\) is a Lorentzian product space, we obtain a new Calabi–Bernstein type result concerning the CMC spacelike hypersurface equation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Akutagawa K.: On spacelike hypersurfaces with constant mean curvature in the de Sitter space. Math. Z. 196, 13–19 (1987)
A.L. Albujer, New examples of entire maximal graphs in \({\mathbb{H}^{2} \times \mathbb{R}_{1}}\), Diff. Geom. App. 26 (2008), 456–462.
A.L. Albujer, F. Camargo and H.F. de Lima, Complete spacelike hypersurfaces in a Robertson–Walker spacetime, Math. Proc. Cambridge Phil. Soc. 151 (2011), 271–282.
Aledo J.A., Romero A., Rubio R.M.: Constant mean curvature spacelike hypersurfaces in Lorentzian warped products and Calabi–Bernstein type problems. Nonl. Anal. 106, 57–69 (2014)
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)
Alías L.J., Colares A.G.: Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in Generalized Robertson–Walker spacetimes. Math. Proc. Cambridge Philos. Soc. 143, 703–729 (2007)
Alías L.J., Romero A., Sánchez M.: Uniqueness of complete spacelike hypersurfaces with constant mean curvature in Generalized Robertson–Walker spacetimes. Gen. Relat. Grav. 27, 71–84 (1995)
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)
D. Bak and S.J. Rey, Cosmic Holography, Classical Quant. Grav. 17 (2000), L83–L89.
Bochner S.: Vector fields and Ricci curvature. Bull. Am. Math. Soc. 52, 776–797 (1946)
Caballero M., Romero A., Rubio R.M.: Constant mean curvature spacelike surfaces in three-dimensional generalized Robertson–Walker spacetimes. Lett. Math. Phys. 93, 85–105 (2010)
Caballero M., Romero A., Rubio R.M.: Uniqueness of maximal surfaces in generalized Robertson–Walker spacetimes and Calabi–Bernstein type problems. J. Geom. Phys. 60, 394–402 (2010)
Bousso R.: The holographic principle. Rev. Mod. Phys. 74, 825–874 (2002)
Calabi E.: Examples of Bernstein problems for some nonlinear equations. Proc. Sympos. Pure Math. 15, 223–230 (1970)
Cheng S.Y., Yau S.T.: Maximal Spacelike Hypersurfaces in the Lorentz–Minkowski Space. Ann. of Math. 104, 407–419 (1976)
Chiu H.Y.: A cosmological model of universe. Ann. Phys. 43, 1–41 (1967)
Latorre J.M., Romero A.: Uniqueness of Noncompact Spacelike Hypersurfaces of Constant Mean Curvature in Generalized Robertson–Walker Spacetimes. Geom. Ded. 93, 1–10 (2002)
J. Marsdan and F. Tipler, Maximal hypersurfaces and foliations of constant mean curvature in general relativity, Bull. Am. Phys. Soc. 23, (1978) 84.
Omori H.: Isometric immersions of Riemannian manifolds. J. Math. Soc. Japan 19, 205–214 (1967)
B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London, 1983.
A. Romero and R.M. Rubio, On the mean curvature of spacelike surfaces in certain three-dimensional Robertson–Walker spacetimes and Calabi–Bernstein’s type problems, Ann. Global Anal. Geom. 37 (2010), 21–31.
A. Romero, R. Rubio and J.J. Salamanca, Uniqueness of complete maximal hypersurfaces in spatially parabolic generalized Robertson–Walker spacetimes, Classical Quantum Gravity 30 (2013), 115007 pp. 13.
Romero A., Rubio R., Salamanca J.J.: A new approach for uniqueness of complete maximal hypersurfaces in spatially parabolic GRW spacetimes. J. Math. Anal. Appl. 419, 355–372 (2014)
Stumbles S.: Hypersurfaces of constant mean extrinsic curvature. Ann. Phys. 133, 28–56 (1980)
Yau S.T.: Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Univ. Math. J. 25, 659–670 (1976)
Author information
Authors and Affiliations
Corresponding author
Additional information
C. Aquino is partially supported by CNPq/Brazil, grant 302738/2014-2.
H. Baltazar is partially supported by CNPq/Brazil.
H.F. de Lima is partially supported by CNPq/Brazil, grant 303977/2015-9.
Rights and permissions
About this article
Cite this article
Aquino, C., Baltazar, H. & de Lima, H. A New Calabi–Bernstein Type Result in Spatially Closed Generalized Robertson–Walker Spacetimes. Milan J. Math. 85, 235–245 (2017). https://doi.org/10.1007/s00032-017-0271-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00032-017-0271-z