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

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- by Juan A. Aledo and Luis J. Alías
- Proc. Amer. Math. Soc.
**130**(2002), 1145-1151 - DOI: https://doi.org/10.1090/S0002-9939-01-06133-0
- Published electronically: September 19, 2001
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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.## References

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## Bibliographic 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
- 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
- © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1873790