Abstract
Our purpose in this paper is to study the geometry of complete linear Weingarten spacelike hypersurfaces immersed with two distinct principal curvatures in a locally symmetric Lorentz space, which is supposed to obey standard curvature constrains. In this setting, we apply some appropriated generalized maximum principles to a suitable Cheng-Yau modified operator in order to guarantee that such a spacelike hypersurface must be isometric to an isoparametric hypersurface of the ambient space.
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Abe, N., Koike, N., Yamaguchi, S.: Congruence theorems for proper semi-Riemannian hypersurfaces in a real space form. Yokohama Math. J. 35, 123–136 (1987)
Akutagawa, K.: On spacelike hypersurfaces with constant mean curvature in the de Sitter space. Math. Z. 196, 13–19 (1987)
Alías, L.J., Brasil Jr., A., Colares, A.G.: Integral formulae for spacelike hypersurfaces in conformally stationary spacetimes and applications. Proc. Edinburgh Math. Soc. 46, 465–488 (2003)
Calabi, E.: Examples of Bernstein problems for some nonlinear equations. Proc. Symp. Pure Math. 15, 223–230 (1970)
Caminha, A.: The geometry of closed conformal vector fields on Riemannian spaces. Bull. Braz. Math. Soc. 42, 277–300 (2011)
Cao, L., Wei, G.: A new characterization of hyperbolic cylinder in anti-de Sitter space \(\mathbb{H}_1^{n+1}(-1)\). J. Math. Anal. Appl. 329, 408–414 (2007)
Cartan, É.: Familles de surfaces isoparamétriques dans les espaces à courbure constante. Ann. Mat. Pura Appl. 17, 177–191 (1938)
Chao, X.: Complete spacelike hypersurfaces in the de Sitter space. Osaka J. Math. 50, 715–723 (2013)
Chaves, R.M.B., Sousa Jr., L.A.M., Valério, B.C.: New characterizations for hyperbolic cylinders in anti-de Sitter spaces. J. Math. Anal. Appl. 393, 166–176 (2012)
Cheng, Q.M.: Complete space-like hypersurfaces of a de Sitter space with \(r=kH\). Mem. Fac. Sci. Kyushu Univ. 44, 67–77 (1990)
Cheng, S.Y., Yau, S.-T.: Maximal spacelike hypersurfaces in the Lorentz-Minkowski space. Ann. Math. 104, 407–419 (1976)
Cheng, S.Y., Yau, S.-T.: Hypersurfaces with constant scalar curvature. Math. Ann. 225, 195–204 (1977)
Choi, S.M., Lyu, S.M., Suh, T.J.: Complete space-like hypersurfaces in a Lorentz manifolds. Math. J. Toyama Univ. 22, 53–76 (1999)
de Lima, H.F., de Lima, J.R.: Characterizations of linear Weingarten spacelike hypersurfaces in Einstein spacetimes. Glasgow Math. J. 55, 567–579 (2013)
de Lima, H.F., de Lima, J.R.: Complete linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Lorentz space. Results Math. 63, 865–876 (2013)
de Lima, H.F., Velásquez, M.A.L.: On the geometry of linear Weingarten spacelike hypersurfaces in the de Sitter space. Bull. Braz. Math. Soc. 44, 49–65 (2013)
Goddard, A.J.: Some remarks on the existence of spacelike hypersurfaces of constant mean curvature. Math. Proc. Camb. Philos. Soc. 82, 489–495 (1977)
Gomes, J.N.V., de Lima, H.F., Santos, F.R., Velásquez, M.A.L.: On the complete linear Weingarten spacelike hypersurfaces with two distinct principal curvatures in Lorentzian space forms. J. Math. Anal. Appl. 418, 248–263 (2014)
Hou, Z.H., Yang, D.: Linear Weingarten spacelike hypersurfaces in de Sitter space. Bull. Belg. Math. Soc. Simon Stevin 17, 769–780 (2010)
Karp, L.: On Stokes’ theorem for noncompact manifolds. Proc. Am. Math. Soc. 82, 487–490 (1981)
Li, H.: Global rigidity theorems of hypersurfaces. Ark. Math. 35, 327–351 (1997)
Montiel, S.: An integral inequality for compact spacelike hypersurfaces in de Sitter space and applications to the case of constant mean curvature. Indiana Univ. Math. J. 37, 909–917 (1988)
Nishikawa, S.: On maximal spacelike hypersurfaces in a Lorentzian manifold. Nagoya Math. J. 95, 117–124 (1984)
Omori, H.: Isometric immersions of Riemannian manifolds. J. Math. Soc. Jpn. 19, 205–214 (1967)
Perdomo, O.: New examples of maximal spacelike surfaces in the anti-de Sitter space. J. Math. Anal. Appl. 353, 403–409 (2009)
Suh, T.J., Choi, S.M., Yang, H.Y.: On space-like hypersurfaces with constant mean curvature in a Lorentz manifold. Houston J. Math. 28, 47–70 (2002)
Treibergs, A.E.: Entire spacelike hypersurfaces of constant mean curvature in Minkowski space. Invent. Math. 66, 39–56 (1982)
Yau, S.-T.: Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Univ. Math. J. 25, 659–670 (1976)
Acknowledgments
The first author is partially supported by CNPq, Brazil, grant 300769/2012-1. The second author is partially supported by CAPES, Brazil. The third author is partially supported by PRONEX/CNPq/FAPEAM, Brazil, Grant 716.UNI52.1769.03062009. The authors would like to thank the referee for giving valuable suggestions which improved the paper.
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de Lima, H.F., dos Santos, F.R., Gomes, J.N. et al. On the complete spacelike hypersurfaces immersed with two distinct principal curvatures in a locally symmetric Lorentz space. Collect. Math. 67, 379–397 (2016). https://doi.org/10.1007/s13348-015-0145-z
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DOI: https://doi.org/10.1007/s13348-015-0145-z
Keywords
- Locally symmetric Lorentz spaces
- Einstein spacetimes
- Complete linear Weingarten spacelike hypersurfaces
- Isoparametric hypersurfaces