×

Compact spacelike hypersurfaces with constant mean curvature in the anti de Sitter space. (English) Zbl 1178.83006

Summary: We obtain a height estimate concerning to a compact spacelike hypersurface \(\Sigma ^{n}\) immersed with constant mean curvature \(H\) in the anti-de Sitter space \(\mathbb H_{1}^{n+1}\), when its boundary \(\partial\Sigma \) is contained into an umbilical spacelike hypersurface of this spacetime which is isometric to the hyperbolic space \(\mathbb H^{n}\). Our estimate depends only on the value of \(H\) and on the geometry of \(\partial \Sigma\). As applications of our estimate, we obtain a characterization of hyperbolic domains of \(\mathbb H_{1}^{n+1}\) and nonexistence results in connection with such types of hypersurfaces.

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
53Z05 Applications of differential geometry to physics
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] J. Marsden and F. Tipler, “Maximal hypersurfaces and foliations of constant mean curvature in general relativity,” Bulletin of the American Physical Society, vol. 23, p. 84, 1978.
[2] S. M. Stumbles, “Hypersurfaces of constant mean extrinsic curvature,” Annals of Physics, vol. 133, no. 1, pp. 28-56, 1981. · Zbl 0472.53063 · doi:10.1016/0003-4916(81)90240-2
[3] E. Calabi, “Examples of Bernstein problems for some nonlinear equations,” Proceedings of Symposia in Pure Mathematics, vol. 15, pp. 223-230, 1970. · Zbl 0211.12801
[4] S. Y. Cheng and S. T. Yau, “Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces,” Annals of Mathematics, vol. 104, no. 3, pp. 407-419, 1976. · Zbl 0352.53021 · doi:10.2307/1970963
[5] L. J. Alías and J. M. Malacarne, “Spacelike hypersurfaces with constant higher order mean curvature in Minkowski space-time,” Journal of Geometry and Physics, vol. 41, no. 4, pp. 359-375, 2002. · Zbl 1023.53053 · doi:10.1016/S0393-0440(01)00081-X
[6] L. J. Alías and J. A. Pastor, “Constant mean curvature spacelike hypersurfaces with spherical boundary in the Lorentz-Minkowski space,” Journal of Geometry and Physics, vol. 28, no. 1-2, pp. 85-93, 1998. · Zbl 0945.53036 · doi:10.1016/S0393-0440(98)00014-X
[7] L. J. Alías and J. A. Pastor, “Spacelike hypersurfaces with constant scalar curvature in the Lorentz-Minkowski space,” Annals of Global Analysis and Geometry, vol. 18, no. 1, pp. 75-83, 2000. · Zbl 0949.53040 · doi:10.1023/A:1006660924994
[8] L. J. Alías, R. López, and J. A. Pastor, “Compact spacelike surfaces with constant mean curvature in the Lorentz-Minkowski 3-space,” The Tohoku Mathematical Journal, vol. 50, no. 4, pp. 491-501, 1998. · Zbl 0934.53040 · doi:10.2748/tmj/1178224893
[9] R. López, “Area monotonicity for spacelike surfaces with constant mean curvature,” Journal of Geometry and Physics, vol. 52, no. 3, pp. 353-363, 2004. · Zbl 1074.53051 · doi:10.1016/j.geomphys.2004.03.008
[10] D. Hoffman, J. H. S. de Lira, and H. Rosenberg, “Constant mean curvature surfaces in M2\times \Bbb R,” Transactions of the American Mathematical Society, vol. 358, no. 2, pp. 491-507, 2006. · Zbl 1079.53088 · doi:10.1090/S0002-9947-05-04084-5
[11] H. F. de Lima, “A sharp height estimate for compact spacelike hypersurfaces with constant r-mean curvature in the Lorentz-Minkowski space and application,” Differential Geometry and Its Applications, vol. 26, no. 4, pp. 445-455, 2008. · Zbl 1144.53080 · doi:10.1016/j.difgeo.2007.11.033
[12] H. F. de Lima, “Spacelike hypersurfaces with constant higher order mean curvature in de Sitter space,” Journal of Geometry and Physics, vol. 57, no. 3, pp. 967-975, 2007. · Zbl 1111.53049 · doi:10.1016/j.geomphys.2006.07.005
[13] A. G. Colares and H. F. de Lima, “Space-like hypersurfaces with positive constant r-mean curvature in Lorentzian product spaces,” General Relativity and Gravitation, vol. 40, no. 10, pp. 2131-2147, 2008. · Zbl 1152.83323 · doi:10.1007/s10714-008-0621-9
[14] A. Caminha and H. F. de Lima, “Complete spacelike hypersurfaces in conformally stationary Lorentz manifolds,” General Relativity and Gravitation, vol. 41, no. 1, pp. 173-189, 2009. · Zbl 1162.83304 · doi:10.1007/s10714-008-0663-z
[15] A. Caminha and H. F. de Lima, “Complete vertical graphs with constant mean curvature in semi-Riemannian warped products,” Bulletin of the Belgian Mathematical Society, vol. 16, no. 1, pp. 91-105, 2009. · Zbl 1160.53362
[16] A. L. Albujer and L. J. Alías, “Spacelike hypersurfaces with constant mean curvature in the steady state space,” Proceedings of the American Mathematical Society, vol. 137, no. 2, pp. 711-721, 2009. · Zbl 1162.53043 · doi:10.1090/S0002-9939-08-09546-4
[17] R. K. Sachs and H. H. Wu, General Relativity for Mathematicians, Graduate Texts in Mathematics, Springer, New York, NY, USA, 1977. · Zbl 0373.53001
[18] J. M. Latorre and A. Romero, “Uniqueness of noncompact spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes,” Geometriae Dedicata, vol. 93, pp. 1-10, 2002. · Zbl 1029.53072 · doi:10.1023/A:1020341512060
[19] B. O’Neill, Semi-Riemannian geometry, vol. 103, Academic Press, London, UK, 1983. · Zbl 0531.53051
[20] S. Carroll, Spacetime and Geometry: An Introduction to General Relativity, Addison Wesley, San Francisco, Calif, USA, 2004. · Zbl 1131.83001
[21] R. M. Wald, General Relativity, University of Chicago Press, Chicago, Ill, USA, 1984. · Zbl 0549.53001
[22] S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley & Sons, New York, NY, USA, 1972.
[23] L. J. Alías, A. Romero, and M. Sánchez, “Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes,” General Relativity and Gravitation, vol. 27, no. 1, pp. 71-84, 1995. · Zbl 0908.53034 · doi:10.1007/BF02105675
[24] A. L. Besse, Einstein Manifolds, vol. 10 of Results in Mathematics and Related Areas (3), Springer, Berlin, Germany, 1987. · Zbl 0613.53001
[25] S. Montiel, “Uniqueness of spacelike hypersurfaces of constant mean curvature in foliated spacetimes,” Mathematische Annalen, vol. 314, no. 3, pp. 529-553, 1999. · Zbl 0965.53043 · doi:10.1007/s002080050306
[26] L. J. Alías, A. Brasil Jr., and A. G. Colares, “Integral formulae for spacelike hypersurfaces in conformally stationary spacetimes and applications,” Proceedings of the Edinburgh Mathematical Society, vol. 46, no. 2, pp. 465-488, 2003. · Zbl 1053.53038 · doi:10.1017/S0013091502000500
[27] L. J. Alías and A. G. Colares, “Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 143, no. 3, pp. 703-729, 2007. · Zbl 1131.53035 · doi:10.1017/S0305004107000576
[28] A. Barros, A. Brasil Jr., and A. Caminha, “Stability of spacelike hypersurfaces in foliated spacetimes,” Differential Geometry and Its Applications, vol. 26, no. 4, pp. 357-365, 2008. · Zbl 1145.53046 · doi:10.1016/j.difgeo.2007.11.028
[29] S. W. Hawking and G. F. R. Ellis, The large Scale Structure of Space-Time, Cambridge University Press, Cambridge, UK, 1973. · Zbl 0265.53054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.