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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Spacelike hypersurfaces with constant mean curvature in the steady state space


Authors: Alma L. Albujer and Luis J. Alías
Journal: Proc. Amer. Math. Soc. 137 (2009), 711-721
MSC (2000): Primary 53C42, 53C50
Published electronically: September 4, 2008
MathSciNet review: 2448594
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Abstract: We consider complete spacelike hypersurfaces with constant mean curvature in the open region of de Sitter space known as the steady state space. We prove that if the hypersurface is bounded away from the infinity of the ambient space, then the mean curvature must be $ H=1$. Moreover, in the 2-dimensional case we obtain that the only complete spacelike surfaces with constant mean curvature which are bounded away from the infinity are the totally umbilical flat surfaces. We also derive some other consequences for hypersurfaces which are bounded away from the future infinity. Finally, using an isometrically equivalent model for the steady state space, we extend our results to a wider family of spacetimes.


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Additional Information

Alma L. Albujer
Affiliation: Departamento de Matemáticas, Universidad de Murcia, E-30100 Espinardo, Murcia, Spain
Email: albujer@um.es

Luis J. Alías
Affiliation: Departamento de Matemáticas, Universidad de Murcia, E-30100 Espinardo, Murcia, Spain
Email: ljalias@um.es

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09546-4
PII: S 0002-9939(08)09546-4
Keywords: de Sitter space, steady state space, spacelike hypersurface, mean curvature, parabolicity
Received by editor(s): May 31, 2007
Received by editor(s) in revised form: February 5, 2008
Published electronically: September 4, 2008
Additional Notes: The first author was supported by FPU Grant AP2004-4087 from Secretaría de Estado de Universidades e Investigación, MEC Spain.
This research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Regional Agency for Science and Technology (Regional Plan for Science and Technology 2007-2010).
This research was partially supported by MEC project MTM2007-64504 and Fundación Séneca project 04540/GERM/06, Spain.
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.