Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-19T19:12:12.576Z Has data issue: false hasContentIssue false
  • English
  • Français

Complete spacelike hypersurfaces in a Robertson–Walker spacetime

Published online by Cambridge University Press:  13 July 2011

ALMA L. ALBUJER ,
FERNANDA E. C. CAMARGO  and
HENRIQUE F. DE LIMA
ALMA L. ALBUJER
Affiliation:
Departamento de Matemáticas, Campus Universitario de Rabanales, Universidad de Córdoba, 14071 Córdoba, Spain. e-mail: alma.albujer@uco.es
FERNANDA E. C. CAMARGO
Affiliation:
Departamento de Matemática, Universidade Federal do Ceará, 60455-760 Fortaleza, Ceará, Brazil. e-mail: feccamargo@yahoo.com.br
HENRIQUE F. DE LIMA
Affiliation:
Departamento de Matemática e Estatística, Universidade Federal de Campina Grande, 58109-970 Campina Grande, Paraíba, Brazil. e-mail: henrique@dme.ufcg.edu.br
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, as a suitable application of the well-known generalized maximum principle of Omori–Yau, we obtain uniqueness results concerning to complete spacelike hypersurfaces with constant mean curvature immersed in a Robertson–Walker (RW) spacetime. As an application of such uniqueness results for the case of vertical graphs in a RW spacetime, we also get non-parametric rigidity results.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 151 , Issue 2 , September 2011 , pp. 271 - 282
Copyright
Copyright © Cambridge Philosophical Society 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Aiyama, R.On the Gauss map of complete space-like hypersurfaces of constant mean curvature in Minkowski space. Tsukuba J. Math. 16 (1992), 353–361.CrossRefGoogle Scholar
[2]Albujer, A. L.New examples of entire maximal graphs in ℍH2 × ℝ1. Diff. Geom. Appl. 26 (2008), 456–462.CrossRefGoogle Scholar
[3]Albujer, A. L. and Alías, L. J.Calabi–Bernstein results for maximal surfaces in Lorentzian product spaces. J. Geom. Phys. 59 (2009), 620–631.CrossRefGoogle Scholar
[4]Albujer, A. L. and Alías, L. J.Spacelike hypersurfaces with constant mean curvature in the steady state space. Proc. Amer. Math. Soc. 137 (2009), 711–721.CrossRefGoogle Scholar
[5]Albujer, A. L., Camargo, F. E. C. and de Lima, H. F.Complete spacelike hypersurfaces with constant mean curvature in ℍ × ℝn. J. Math. Anal. Appl. 368 (2010), 650–657.CrossRefGoogle Scholar
[6]Alías, L.J., Brasil, A. Jr. and Colares, A. G.Integral formulae for spacelike hypersurfaces in conformally stationary spacetimes and applications. Proc. Edinb. Math. Soc. 46 (2003), 465–488.CrossRefGoogle Scholar
[7]Alías, L. J. and Colares, A. G.Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes. Math. Proc. Cam. Phil. Soc. 143 (2007), 703–729.CrossRefGoogle Scholar
[8]Alías, L. J., Romero, A. and Sánchez, M.Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson–Walker spacetimes. Gen. Relativity Gravitation 27 (1995), 71–84.CrossRefGoogle Scholar
[9]Alías, L. J., Romero, A. and Sánchez, M.Spacelike hypersurfaces of constant mean curvature and Calabi–Bernstein type problems. Tôhoku Math. J. 49 (1997), 337–345.CrossRefGoogle Scholar
[10]Caballero, M., Romero, A. and Rubio, R. M.Uniqueness of maximal surfaces in generalized Robertson–Walker spacetimes and Calabi–Bernstein type problems. J. Geom. Phys. 60 (2010), 394–402.CrossRefGoogle Scholar
[11]Caballero, M., Romero, A. and Rubio, R. M.Complete cmc spacelike surfaces with bounded hyperbolic angle in generalized Robertson–Walker spacetimes. Int. J. Geom. Methods Mod. Phys. 7 (2010), 961–978.CrossRefGoogle Scholar
[12]Caballero, M., Romero, A. and Rubio, R. M.Constant mean curvature spacelike surfaces in three-dimensional generalized Robertson–Walker spacetimes. Lett. Math. Phys. 93 (2010), 85–105.CrossRefGoogle Scholar
[13]Calabi, E.Examples of Bernstein problems for some nonlinear equations. Proc. Sympos. Pure Math. 15 (1970), 223–230.CrossRefGoogle Scholar
[14]Cheng, S. Y. and Yau, S. T.Maximal spacelike hypersurfaces in the Lorentz–Minkowski space. Ann. of Math. 104 (1976), 407–419.CrossRefGoogle Scholar
[15]Hawking, S. W. and Ellis, G. F. R.The Large Scale Structure of Spacetime, (Cambridge University Press, 1973).CrossRefGoogle Scholar
[16]Li, G. and Salavessa, I.Graphic Bernstein results in curved pseudo-Riemannian manifolds. J. Geom. Phys. 59 (2009), 1306–1313.CrossRefGoogle Scholar
[17]Montiel, S.Uniqueness of spacelike hypersurfaces of constant mean curvature in foliated spacetimes. Math. Ann. 314 (1999), 529–553.CrossRefGoogle Scholar
[18]Omori, H.Isometric immersions of Riemannian manifolds. J. Math. Soc. Japan 19 (1967), 205–214.CrossRefGoogle Scholar
[19]O'Neill, B.Semi-Riemannian Geometry with applications to Relativity (Academic Press, 1983).Google Scholar
[20]Rainer, M. and Schmidt, H-J.Inhomogeneous cosmological models with homogeneous inner hypersurface geometry. Gen. Relativity Gravitation 27 (1995), 1265–1293.CrossRefGoogle Scholar
[21]Romero, A. and Rubio, R. M.On the mean curvature of spaclike surfaces in certain three-dimensional Robertson–Walker spacetimes and Calabi–Bernstein's type problems. Ann. Glob. Anal. Geom. 37 (2010), 21–31.CrossRefGoogle Scholar
[22]Xin, Y. L.On the Gauss image of a spacelike hypersurface with constant mean curvature in Minkowski space. Comment. Math. Helv. 66 (1991), 590–598.CrossRefGoogle Scholar
[23]Yau, S. T.Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28 (1975), 201–228.CrossRefGoogle Scholar