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On the volume and the Gauss map image of spacelike hypersurfaces in de Sitter space


Authors: Juan A. Aledo and Luis J. Alías
Journal: Proc. Amer. Math. Soc. 130 (2002), 1145-1151
MSC (2000): Primary 53C42; Secondary 53B30, 53C50
DOI: https://doi.org/10.1090/S0002-9939-01-06133-0
Published electronically: September 19, 2001
MathSciNet review: 1873790
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Abstract: In this paper we prove that a complete spacelike hypersurface $M^n$ in de Sitter space such that its image under the Gauss map is contained in a hyperbolic geodesic ball of radius $\varrho$is necessarily compact and its $n$-dimensional volume satisfies $\omega_n/\mathrm{cosh}(\varrho)\leq \mathrm{vol}(M)\leq\omega_n\mathrm{ cosh}^{n}(\varrho)$, where $\omega_n$ denotes the volume of a unitary round $n$-sphere. We also characterize the case where these inequalities become equalities. As an application of our result, we also conclude that Goddard's conjecture is true under the assumption that the hyperbolic image of the hypersurface is bounded.


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  • 1. R. Aiyama, On the Gauss map of complete space-like hypersurfaces of constant mean curvature in Minkowski space, Tsukuba J. Math. 16 (1992), 353-361. MR 93m:53062
  • 2. K. Akutagawa, On spacelike hypersurfaces with constant mean curvature in the de Sitter space, Math. Z. 196 (1987), 13-19. MR 88h:53052
  • 3. J.A. Aledo and L.J. Alías, On the curvatures of bounded complete spacelike hypersurfaces in the Lorentz-Minkowski space, Manuscripta Math. 101 (2000), 401-413. MR 2001a:53091
  • 4. J.A. Aledo, L.J. Alías and A. Romero, Integral formulas for compact spacelike hypersurfaces in de Sitter space. Applications to the case of constant higher order mean curvature, J. Geom. Phys. 31 (1999), 195-208. MR 2000e:53092
  • 5. L.J. Alías, On the Ricci curvature of compact spacelike hypersurfaces in de Sitter space, Geom. Dedicata 77 (1999), 297-304. MR 2000h:53087
  • 6. Q-M. Cheng and S. Ishikawa, Spacelike hypersurfaces with constant scalar curvature, Manuscripta Math. 95 (1998), 499-505. MR 99e:53087
  • 7. A.J. Goddard, Some remarks on the existence of spacelike hypersurfaces of constant mean curvature, Math. Proc. Cambridge Phil. Soc. 82 (1977), 489-495. MR 56:16547
  • 8. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. II, Interscience, New York, 1969. MR 38:6501
  • 9. H. Li Global rigidity theorems of hypersurfaces, Ark. Mat. 35 (1997), 327-351. MR 98j:53074
  • 10. S. Montiel, An integral inequality for compact spacelike hypersurfaces in de Sitter space and applications to the case of constant mean curvature, Indiana Univ. Math. J. 37 (1988), 909-917. MR 90c:53167
  • 11. B. Palmer, The Gauss map of a spacelike constant mean curvature hypersurface of Minkowski space, Comment. Math. Helv. 65 (1990), 52-57. MR 91a:53014
  • 12. Y.L. Xin, On the Gauss image of a spacelike hypersurface with constant mean curvature in Minkowski space, Comment. Math. Helv. 66 (1991), 590-598. MR 93d:53016
  • 13. J. Ramanathan, Complete spacelike hypersurfaces of constant mean curvature in de Sitter space, Indiana Univ. Math. J. 36 (1987), 349-359. MR 88g:53067
  • 14. Y. Zheng, On spacelike hypersurfaces in the de Sitter space, Ann. Global Anal. Geom. 13 (1995), 317-321. MR 96j:53062
  • 15. Y. Zheng, Spacelike hypersurfaces with constant scalar curvature in the de Sitter spaces, Differ. Geom. Appl. 6 (1996), 51-54. MR 97e:53124

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Additional Information

Juan A. Aledo
Affiliation: Departamento de Matemáticas, Universidad de Castilla-La Mancha, Escuela Politécnica Superior de Albacete, 02071 Albacete, Spain
Email: jaledo@pol-ab.uclm.es

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

DOI: https://doi.org/10.1090/S0002-9939-01-06133-0
Keywords: de Sitter space, spacelike hypersurface, volume, Gauss map, hyperbolic image, mean curvature
Received by editor(s): April 17, 2000
Received by editor(s) in revised form: October 5, 2000
Published electronically: September 19, 2001
Additional Notes: This work has been partially supported by DGICYT Grant No PB97-0784-C03-02 and Consejería de Educación y Cultura CARM, Programa Séneca (PRIDTYC)
Communicated by: Wolfgang Ziller
Article copyright: © Copyright 2001 American Mathematical Society

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