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ON THE UMBILICITY OF HYPERSURFACES IN THE HYPERBOLIC SPACE

Published online by Cambridge University Press:  16 November 2016

C. P. AQUINO
Affiliation:
Departamento de Matemática, Universidade Federal do Piauí, 64049-550 Teresina, Piauí, Brazil email cicero.aquino@ufpi.edu.br
M. BATISTA
Affiliation:
Instituto de Matemática, Universidade Federal de Alagoas, 57072-970 Maceió, Alagoas, Brazil email mhbs@mat.ufal.br
H. F. DE LIMA
Affiliation:
Departamento de Matemática, Universidade Federal de Campina Grande, 58429-970 Campina Grande, Paraíba, Brazil email henrique@mat.ufcg.edu.br

Abstract

In this paper, we establish new characterization results concerning totally umbilical hypersurfaces of the hyperbolic space $\mathbb{H}^{n+1}$ , under suitable constraints on the behavior of the Lorentzian Gauss map of complete hypersurfaces having some constant higher order mean curvature. Furthermore, working with different warped product models for $\mathbb{H}^{n+1}$ and supposing that certain natural inequalities involving two consecutive higher order mean curvature functions are satisfied, we study the rigidity and the nonexistence of complete hypersurfaces immersed in $\mathbb{H}^{n+1}$ .

Type
Research Article
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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Footnotes

The first author is partially supported by CNPq, Brazil, grant number 302738/2014-2. The second author is partially supported by CNPq, Brazil, grant number 456755/2014-4. The third author is partially supported by CNPq, Brazil, grant number 303977/2015-9.

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ON THE UMBILICITY OF HYPERSURFACES IN THE HYPERBOLIC SPACE
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