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CHARACTERIZATIONS OF LINEAR WEINGARTEN SPACELIKE HYPERSURFACES IN EINSTEIN SPACETIMES

Published online by Cambridge University Press:  25 February 2013

HENRIQUE F. DE LIMA  and
JOSEÍLSON R. DE LIMA
HENRIQUE F. DE LIMA
Affiliation:
Departamento de Matemática e Estatística, Universidade Federal de Campina Grande, 58429-970 Campina Grande, Paraíba, Brazil e-mails: henrique@dme.ufcg.edu.br, joseilson@dme.ufcg.edu.br
JOSEÍLSON R. DE LIMA
Affiliation:
Departamento de Matemática e Estatística, Universidade Federal de Campina Grande, 58429-970 Campina Grande, Paraíba, Brazil e-mails: henrique@dme.ufcg.edu.br, joseilson@dme.ufcg.edu.br
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Abstract

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Our purpose is to study the geometry of linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Einstein spacetime, whose sectional curvature is supposed to obey some standard restrictions. In this setting, by using as main analytical tool a generalized maximum principle for complete non-compact Riemannian manifolds, we establish sufficient conditions to guarantee that such a hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures, one of which is simple. Applications to the de Sitter space are given.

Keywords

Primary 53C42Secondary 53A1053B30 and 53C50
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue 3 , September 2013 , pp. 567 - 579
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Abe, N., Koike, N. and Yamaguchi, S., Congruence theorems for proper semi-Riemannian hypersurfaces in a real space form, Yokohama Math. J. 35 (1987), 123–136.Google Scholar
2.Akutagawa, K., On spacelike hypersurfaces with constant mean curvature in the de Sitter space, Math. Z. 196 (1987), 13–19.Google Scholar
3.Alencar, H. and do Carmo, M., Hypersurfaces with constant mean curvature in spheres, Proc. Am. Math. Soc. 120 (1994), 1223–1229.CrossRefGoogle Scholar
4.Alías, L. J., Brasil, A. Jr. and Colares, A. G., Integral formulae for spacelike hypersurfaces in conformally stationary spacetimes and applications, Proc. Edinburgh Math. Soc. 46 (2003), 465–488.Google Scholar
5.Brasil, A. Jr., Colares, A. G. and Palmas, O., A gap theorem for complete constant scalar curvature hypersurfaces in the de Sitter space, J. Geom. Phys. 37 (2001), 237–250.Google Scholar
6.Brasil, A. Jr., Colares, A. G. and Palmas, O., Complete spacelike hypersurfaces with constant mean curvature in the de Sitter space: A gap theorem, Illinois J. Math. 47 (2003), 847–866.CrossRefGoogle Scholar
7.Camargo, F. E. C., Chaves, R. M. B. and Sousa, L. A. M. Jr., Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in de Sitter space, Diff. Geom. Appl. 26 (2008), 592–599.CrossRefGoogle Scholar
8.Caminha, A., A rigidity theorem for complete CMC hypersurfaces in Lorentz manifolds, Diff. Geom. Appl. 24 (2006), 652–659.Google Scholar
9.Caminha, A., The geometry of closed conformal vector fields on Riemannian spaces, Bull. Braz. Math. Soc. 42 (2011), 277–300.CrossRefGoogle Scholar
10.Cartan, É., Familles de surfaces isoparamétriques dans les espaces à courbure constante, Ann. Mat. Pura Appl. 17 (1938), 177–191.CrossRefGoogle Scholar
11.Cheng, Q. M. and Ishikawa, S., Spacelike hypersurfaces with constant scalar curvature, Manuscripta Math. 95 (1998), 499–505.Google Scholar
12.Cheng, S. Y. and Yau, S. T., Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), 195–204.CrossRefGoogle Scholar
13.Choi, S. M., Lyu, S. M. and Suh, Y. J., Complete space-like hypersurfaces in a Lorentz manifold, Math. J. Toyama Univ. 22 (1999), 53–76.Google Scholar
14.Goddard, A. J., Some remarks on the existence of spacelike hypersurfaces of constant mean curvature, Math. Proc. Camb. Phil. Soc. 82 (1977), 489–495.CrossRefGoogle Scholar
15.Hu, Z.-J., Scherfner, M. and Zhai, S.-J., On spacelike hypersurfaces with constant scalar curvature in the de Sitter space, Diff. Geom. Appl. 25 (2007), 594–611.Google Scholar
16.Li, H., Global rigidity theorems of hypersurfaces, Ark. Mat. 35 (1997), 327–351.Google Scholar
17.Li, H., Suh, Y. J. and Wei, G., Linear Weingarten hypersurfaces in a unit sphere, Bull. Korean Math. Soc. 46 (2009), 321–329.Google Scholar
18.Lichnerowicz, A., L'integration des équations de la gravitation relativiste et le problème n corps, J. Math. Pures Appl. 23 (1944), 37–63.Google Scholar
19.Liu, J. and Sun, Z., On spacelike hypersurfaces with constant scalar curvature in locally symmetric Lorentz spaces, J. Math. Anal. Appl. 364 (2010), 195–203.Google Scholar
20.Montiel, S., An integral inequality for compact spacelike hypersurfaces in the de Sitter space and applications to the case of constant mean curvature, Indiana Univ. Math. J. 37 (1988), 909–917.CrossRefGoogle Scholar
21.Montiel, S., A characterization of hyperbolic cylinders in the de Sitter space, Tôhoku Math. J. 48 (1996), 23–31.Google Scholar
22.Ok Baek, J., Cheng, Q. M. and Suh, Y. Jin, Complete space-like hypersurfaces in locally symmetric Lorentz spaces, J. Geom. Phys. 49 (2004), 231–247.Google Scholar
23.Okumura, M., Hypersurfaces and a pinching problem on the second fundamental tensor, Am. J. Math. 96 (1974), 207–213.CrossRefGoogle Scholar
24.Suh, Y. J., Choi, Y. S. and Yang, H. Y., On space-like hypersurfaces with constant mean curvature in a Lorentz manifold, Houston J. Math. 28 (2002), 47–70.Google Scholar
25.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
26.Zhang, S. and Wu, B., Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in locally symmetric Lorentz spaces, J. Geom. Phys. 60 (2010), 333–340.Google Scholar