Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Spacelike hypersurfaces with constant mean curvature in the steady state space
HTML articles powered by AMS MathViewer

by Alma L. Albujer and Luis J. Alías PDF
Proc. Amer. Math. Soc. 137 (2009), 711-721 Request permission

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.
References
  • L. V. Ahlfors, Sur le type d’une surface de Riemann, C.R. Acad. Sci. Paris 201 (1935), 30–32.
  • Kazuo Akutagawa, On spacelike hypersurfaces with constant mean curvature in the de Sitter space, Math. Z. 196 (1987), no. 1, 13–19. MR 907404, DOI 10.1007/BF01179263
  • Luis J. Alías, Alfonso Romero, and Miguel Sánchez, Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes, Gen. Relativity Gravitation 27 (1995), no. 1, 71–84. MR 1310212, DOI 10.1007/BF02105675
  • H. Bondi and T. Gold, On the generation of magnetism by fluid motion, Monthly Not. Roy. Astr. Soc. 110 (1950), 607–611. MR 44363, DOI 10.1093/mnras/110.6.607
  • A. Caminha and H. F. de Lima, Complete vertical graphs with constant mean curvature in semi-Riemannian warped products, http://arxiv.org/abs/math/0609602
  • Gregory J. Galloway, Cosmological spacetimes with $\Lambda >0$, Advances in differential geometry and general relativity, Contemp. Math., vol. 359, Amer. Math. Soc., Providence, RI, 2004, pp. 87–101. MR 2096155, DOI 10.1090/conm/359/06557
  • S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge Monographs on Mathematical Physics, No. 1, Cambridge University Press, London-New York, 1973. MR 0424186
  • F. Hoyle, A new model for the expanding universe, Monthly Not. Roy. Astr. Soc. 108 (1948), 372–382.
  • Alfred Huber, On subharmonic functions and differential geometry in the large, Comment. Math. Helv. 32 (1957), 13–72. MR 94452, DOI 10.1007/BF02564570
  • Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0238225
  • Sebastián Montiel, Complete non-compact spacelike hypersurfaces of constant mean curvature in de Sitter spaces, J. Math. Soc. Japan 55 (2003), no. 4, 915–938. MR 2003752, DOI 10.2969/jmsj/1191418756
  • Barrett O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. MR 719023
  • Jayakumar Ramanathan, Complete spacelike hypersurfaces of constant mean curvature in de Sitter space, Indiana Univ. Math. J. 36 (1987), no. 2, 349–359. MR 891779, DOI 10.1512/iumj.1987.36.36020
  • S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley & Sons, New York, 1972.
  • Shing Tung Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201–228. MR 431040, DOI 10.1002/cpa.3160280203
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53C42, 53C50
  • Retrieve articles in all journals with MSC (2000): 53C42, 53C50
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
  • 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
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 711-721
  • MSC (2000): Primary 53C42, 53C50
  • DOI: https://doi.org/10.1090/S0002-9939-08-09546-4
  • MathSciNet review: 2448594