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UNIQUENESS OF COMPLETE HYPERSURFACES WITH BOUNDED HIGHER ORDER MEAN CURVATURES IN SEMI-RIEMANNIAN WARPED PRODUCTS

Published online by Cambridge University Press:  09 December 2011

C. P. AQUINO  and
H. F. DE LIMA
C. P. AQUINO
Affiliation:
Departamento de Matemática, Universidade Federal do Piauí, Teresina, Piauí, 64049-550, Brazil Departamento de Matemática, Universidade Federal do Ceará, Fortaleza, Ceará, 60455-760, Brazil e-mail: [email protected]
H. F. DE LIMA
Affiliation:
Departamento de Matemática e Estatística, Universidade Federal de Campina Grande, Campina Grande, Paraíba, 58109-970, Brazil e-mail: [email protected]
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Abstract

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In this paper, we deal with complete hypersurfaces immersed with bounded higher order mean curvatures in steady state-type spacetimes and in hyperbolic-type spaces. By applying a generalised maximum principle for the Yau's square operator [11], we obtain uniqueness results in each of these ambient spaces.

Keywords

Primary 53C42Secondary 53B3053C5053Z0583C99
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 1 , January 2012 , pp. 201 - 212
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Albujer, A. L. and Alías, L. J., Spacelike hypersurfaces with constant mean curvature in the steady state space, Proc. Amer. Math. Soc. 137 (2009), 711–721.CrossRefGoogle Scholar
2.Alías, L. J. and Colares, A. G., Uniqueness of space-like hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes, Math. Proc. Camb. Philos. Soc. 143 (2007), 703–729.Google Scholar
3.Alías, L. J. and Dajczer, M., Uniqueness of constant mean curvature surfaces properly immersed in a slab, Comment. Math. Helv. 81 (2006), 653–663.CrossRefGoogle Scholar
4.Alías, L. J., Dajczer, M. and Ripoll, J., A Bernstein-type theorem for Riemannian manifolds with a Killing field, Ann. Global Anal. Geom. 31 (2007), 363–373.CrossRefGoogle Scholar
5.Alías, L. J., Romero, A. and Sánchez, M., Uniqueness of complete space-like hypersurfaces with constant mean curvature in Generalized Robertson–Walker spacetimes, Gen. Relat. Grav. 27 (1995), 71–84.Google Scholar
6.Barbosa, J. L. M. and Colares, A. G., Stability of hypersurfaces with constant r-mean curvature, Ann. Global Anal. Geom. 15 (1997), 277–297.Google Scholar
7.Bondi, H. and Gold, T., On the generation of magnetism by fluid motion, Month. Not. Roy. Astr. Soc. 108 (1948), 252–270.Google Scholar
8.Camargo, F. E. C., Caminha, A. and de Lima, H. F., Bernstein-type theorems in semi-Riemannian warped products, Proc. Amer. Math. Soc. 139 (2011), 1841–1850.Google Scholar
9.Caminha, A. and de Lima, H. F., Complete vertical graphs with constant mean curvature in semi-Riemannian warped products, Bull. Belgian Math. Soc. 16 (2009), 91–105.Google Scholar
10.Caminha, A. and de Lima, H. F., Complete space-like hypersurfaces in conformally stationary Lorentz manifolds, Gen. Relativ. Gravit. 41 (2009), 173–189.CrossRefGoogle Scholar
11.Cheng, S. Y. and Yau, S. T., Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), 195–204.CrossRefGoogle Scholar
12.Colares, A. G. and de Lima, H. F., On the rigidity of spacelike hypersurfaces immersed in the steady state space , to appear in Publ. Math. Debrecen (2011).Google Scholar
13.de Lima, H. F., Rigidity theorems in the hyperbolic space, preprint (2010).Google Scholar
14.Hawking, S. W. and Ellis, G. F. R., The large scale structure of spacetime (Cambridge University Press, Cambridge, UK, 1973).Google Scholar
15.Hoyle, F., A new model for the expanding universe, Month. Not. Roy. Astr. Soc. 108 (1948), 372–382.Google Scholar
16.López, R. and Montiel, S., Existence of constant mean curvature graphs in hyperbolic space, Calc. Var. 8 (1999), 177–190.Google Scholar
17.Montiel, S., An integral inequality for compact space-like hypersurfaces in de Sitter space and applications to the case of constant mean curvature, Indiana Univ. Math. J. 37 (1988), 909–917.Google Scholar
18.Montiel, S., Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds, Indiana Univ. Math. J. 48 (1999), 711–748.CrossRefGoogle Scholar
19.Montiel, S., Uniqueness of space-like hypersurfaces of constant mean curvature in foliated spacetimes, Math. Ann. 314 (1999), 529–553.Google Scholar
20.Montiel, S., Complete non-compact space-like hypersurfaces of constant mean curvature in de Sitter Space, J. Math. Soc. Japan. 55 (2003), 915–938.Google Scholar
21.Omori, H., Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan 19 (1967), 205–214.Google Scholar
22.O'Neill, B., Semi-Riemannian geometry, with applications to relativity (Academic Press, New York 1983).Google Scholar
23.Rosenberg, H., Hypersurfaces of constant curvature in space forms, Bull. Sc. Math. 117 (1993), 217–239.Google Scholar
24.Weinberg, S., Gravitation and cosmology: Principles and applications of the general theory of relativity (John Wiley, New York, 1972).Google Scholar
25.Yau, S. T., Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201–228.CrossRefGoogle Scholar
26.Yau, S. T., Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana Univ. Math. J. 25 (1976), 659–670.CrossRefGoogle Scholar