Bulletin of the Belgian Mathematical Society - Simon Stevin

Spacelike Hypersurfaces with Constant Mean Curvature in the Steady State Space

A. Gervasio Colares and Henrique F. de Lima

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Abstract

In this paper we obtain height estimates concerning to a compact spacelike hypersurface $\Sigma^n$ immersed with constant mean curvature $H$ in the Steady State space $\mathcal H^{n+1}$, when its boundary is contained into some hyperplane of this spacetime. As a first application of these results, when $\Sigma^n$ has spherical boundary, we establish relations between its height and the radius of its boundary. Moreover, under a certain restriction on the Gauss map of $\Sigma^n$, we obtain a sharp estimate for $H$. Finally, we also apply our estimates to describe the end of a complete spacelike hypersurface and to get theorems of characterization concerning to spacelike hyperplanes in $\mathcal H^{n+1}$.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 17, Number 2 (2010), 287-302.

Dates
First available in Project Euclid: 26 May 2010

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1274896207

Digital Object Identifier
doi:10.36045/bbms/1274896207

Mathematical Reviews number (MathSciNet)
MR2663474

Zentralblatt MATH identifier
1198.53059

Subjects
Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 53B30: Lorentz metrics, indefinite metrics 53C50: Lorentz manifolds, manifolds with indefinite metrics

Keywords
Steady State space Spacelike hypersurface Mean curvature Height estimates Hyperbolic image

Citation

Colares, A. Gervasio; de Lima, Henrique F. Spacelike Hypersurfaces with Constant Mean Curvature in the Steady State Space. Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no. 2, 287--302. doi:10.36045/bbms/1274896207. https://projecteuclid.org/euclid.bbms/1274896207


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