Abstract
We apply the well know Omori–Yau generalized maximum principle (Omori in J Math Soc Jpn 19:205–214, 1967; Yau in Commun Pure Appl Math 28:201–228, 1975), as well as a suitable extension of it that was established in a joint work with Caminha (Caminha and de Lima in Gen Relat Grav 41:173–189, 2009), in order to investigate Bernstein-type properties of complete spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime, which is supposed to obey a standard convergence condition.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Albujer A.L., Camargo F.E.C., de Lima H.F.: Complete spacelike hypersurfaces in a Robertson–Walker spacetime. Math. Proc. Cambridge Phil. Soc. 151, 271–282 (2011)
Alías L.J., Brasil A. Jr, Colares A.G.: Integral formulae for spacelike hypersurfaces in conformally stationary spacetimes and applications. Proc. Edinburgh Math. Soc. 46, 465–488 (2003)
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., Impera D., Rigoli M.: Spacelike hypersurfaces of constant higher order mean curvature in generalized Robertson–Walker spacetimes. Math. Proc. Cambridge Philos. Soc. 152, 365–383 (2012)
Alías L.J., Romero A., Sánchez M.: Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson–Walker spacetimes. Gen. Relat. Grav. 27, 71–84 (1995)
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.: Complete cmc spacelike surfaces with bounded hyperbolic angle in generalized Robertson–Walker spacetimes. Int. J. Geom. Meth. Mod. Phys. 7, 961–978 (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)
Calabi E.: Examples of Bernstein problems for some nonlinear equations. Proc. Sympos. Pure Math. 15, 223–230 (1970)
Caminha A., de Lima H.F.: Complete spacelike hypersurfaces in conformally stationary Lorentz manifolds. Gen. Relat. Grav. 41, 173–189 (2009)
Cheng S.Y., Yau S.T.: Maximal spacelike hypersurfaces in the Lorentz–Minkowski Space. Ann. Math. 104, 407–419 (1976)
Omori H.: Isometric immersions of Riemannian manifolds. J. Math. Soc. Jpn. 19, 205–214 (1967)
O’Neill B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, London (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. Global Anal. Geom. 37, 21–31 (2010)
Yau S.T.: Harmonic functions on complete Riemannian manifolds. Commun. Pure Appl. Math. 28, 201–228 (1975)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
de Lima, H.F. On Bernstein-type properties of complete spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime. J. Geom. 103, 219–229 (2012). https://doi.org/10.1007/s00022-012-0122-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00022-012-0122-6