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Complete CMC hypersurfaces in the hyperbolic space with prescribed Gauss mapping


Authors: Abdênago Barros, Cícero Aquino and Henrique de Lima
Journal: Proc. Amer. Math. Soc. 142 (2014), 3597-3604
MSC (2010): Primary 53C42; Secondary 53C50
DOI: https://doi.org/10.1090/S0002-9939-2014-12118-6
Published electronically: June 10, 2014
MathSciNet review: 3238435
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Abstract: Our aim in this paper is to show that a complete hypersurface $ x:M^{n}\to \mathbb{H}^{n+1}$ immersed with constant mean curvature into the hyperbolic space $ \mathbb{H}^{n+1}$ is totally umbilical provided that its Gauss mapping $ \nu $ has some suitable behavior. In this setting, our first result requires that the image $ \nu (M)$ lies in a totally umbilical spacelike hypersurface of the de Sitter space $ \mathbb{S}_1^{n+1}$, while in our second one we suppose that $ M^n$ has scalar curvature bounded from below and that $ \nu (M)$ is contained in the closure of a domain enclosed by a totally umbilical spacelike hypersurface of $ \mathbb{S}_1^{n+1}$ determined by some vector $ a$ of the Minkowski space $ \mathbb{L}^{n+2}$, with the tangential component of $ a$ with respect to $ M^n$ having Lebesgue integrable norm.


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Abdênago Barros
Affiliation: Departamento de Matemática, Universidade Federal do Ceará, 60455-760 Fortaleza, Ceará, Brazil
Email: abbarros@mat.ufc.br

Cícero Aquino
Affiliation: Departamento de Matemática, Universidade Federal do Piauí, 64049-550 Teresina, Piauí, Brazil
Email: cicero.aquino@ufpi.edu.br

Henrique de Lima
Affiliation: Departamento de Matemática e Estatística, Universidade Federal de Campina Grande, 58109-970 Campina Grande, Paraíba, Brazil
Email: henrique@dme.ufcg.edu.br

DOI: https://doi.org/10.1090/S0002-9939-2014-12118-6
Keywords: Hyperbolic space, complete hypersurfaces, spacelike hypersurfaces, totally umbilical hypersurfaces, mean curvature, Gauss mapping
Received by editor(s): August 6, 2012
Received by editor(s) in revised form: October 10, 2012
Published electronically: June 10, 2014
Communicated by: Lei Ni
Article copyright: © Copyright 2014 American Mathematical Society