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Generalized maximum principles and the rigidity of complete spacelike hypersurfaces

Published online by Cambridge University Press:  16 August 2012

FERNANDA CAMARGO ,
ANTONIO CAMINHA ,
HENRIQUE DE LIMA  and
ULISSES PARENTE
FERNANDA CAMARGO
Affiliation:
Departamento de Matemática, Universidade Federal do Ceará, Fortaleza, Ceará, 60455-760, Brazil. e-mail: [email protected]
ANTONIO CAMINHA
Affiliation:
Departamento de Matemática, Universidade Federal do Ceará, Fortaleza, Ceará, 60455-760, Brazil. e-mail: [email protected]
HENRIQUE DE LIMA
Affiliation:
Departamento de Matemática e Estatística, Universidade Federal de Campina Grande, 58109-970 Campina Grande, Paraíba, Brazil. e-mail: [email protected]
ULISSES PARENTE
Affiliation:
Faculdade de Educação, Ciências e Letras do Sertão Central Universidade Estadual do Ceará, Quixadá, Ceará, 63900-000, Brazil. e-mail: [email protected]
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Abstract

In this paper, we apply several forms of generalized maximum principles to the study of the uniqueness of complete, non-compact spacelike hypersurfaces immersed in a class of Lorentzian warped products obeying a suitable convergence condition.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 153 , Issue 3 , November 2012 , pp. 541 - 556
Copyright
Copyright © Cambridge Philosophical Society 2012

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References

REFERENCES

[1]Aiyama, R.On the Gauss map of complete space-like hypersurfaces of constant mean curvature in Minkowski space. Tsukuba J. Math. 16 (1992), 353–361.CrossRefGoogle Scholar
[2]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
[3]Albujer, A. L., Camargo, F. E. C. and de Lima, H. F.Complete spacelike hypersurfaces in a Robertson–Walker spacetime. Math. Proc. Camb. Phil. Soc. 151 (2011), 271–282.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.CrossRefGoogle Scholar
[5]Alías, L. J. and Colares, A. G.Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes. Math. Proc. Camb. Phil. Soc. 143 (2007), 703–729.CrossRefGoogle Scholar
[6]Alías, L. J., Impera, D. and Rigoli, M.spacelike hypersurfaces of constant higher order mean curvature in generalized Robertson–Walker spacetimes. Math. Proc. Camb. Phil. Soc. 152 (2012), 365–383.CrossRefGoogle Scholar
[7]Alías, L. J., Impera, D. and Rigoli, M. Hypersurfaces of constant higher order mean curvature in warped products. To appear in Trans. Amer. Math. Soc.Google Scholar
[8]Alías, L. J., Romero, A. and Sánchez, M.Uniqueness of complete spacelike hypersurfaces of constant mean curvature in Generalized Robertson–Walker spacetimes. Gen. Relativity Gravitation 27 (1995), 71–84.CrossRefGoogle Scholar
[9]Caballero, M., Romero, A. and Rubio, R. M.Constant mean curvature spacelike surfaces in three–dimensional generalized Robertson–Walker spacetimes. Lett. Math. Phys. 93 (2010), 85–105.CrossRefGoogle Scholar
[10]Caballero, M., Romero, A. and Rubio, R. M.Uniqueness of maximal surfaces in generalized Robertson–Walker spacetimes and Calabi–Bernstein type problems. J. Geom. Phys. 60 (2010), 394–402.CrossRefGoogle Scholar
[11]Caballero, M., Romero, A. and Rubio, R. M.Complete cmc spacelike surfaces with bounded hyperbolic angle in generalized Robertson–Walker spacetimes. Int. J. Geom. Meth. Mod. Phys. 7 (2010), 961–978.CrossRefGoogle Scholar
[12]Calabi, E.Examples of Bernstein problems for some nonlinear equations. Proc. Sympos. Pure Math. 15 (1970), 223–230.CrossRefGoogle Scholar
[13]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.CrossRefGoogle Scholar
[14]Camargo, F. E. C., Chaves, R. M. B. and de Sousa, L. A. M. JrRigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in de Sitter space. Diff. Geom. Appl. 26 (2008), 592?–599.CrossRefGoogle Scholar
[15]Camargo, F. E. C. and de Lima, H. F.New characterizations of totally geodesic hypersurfaces in the anti-de Sitter space ℍ1n + 1. J. of Geom. and Physics 60 (2010), 1326–1332.CrossRefGoogle Scholar
[16]Caminha, A.A rigidity theorem for complete CMC hypersurfaces in Lorentz manifolds. Diff. Geom. Appl. 24 (2006), 652–659.CrossRefGoogle Scholar
[17]Caminha, A. and de Lima, H. F.Complete spacelike hypersurfaces in conformally stationary Lorentz manifolds. Gen. Relativity Gravitation 41 (2009), 173–189.CrossRefGoogle Scholar
[18]Caminha, A. and de Lima, H. F.Complete vertical graph with constant mean curvature in semi-Riemannian warped products. Bull. of the Belgian Math. Soc. 16 (2009), 91–105.Google Scholar
[19]Caminha, A., Sousa, P. and Camargo, F. E. C.Complete foliations of space forms by hypersurfaces. Bull. Braz. Math. Soc. 41 (3) (2010), 339–353.CrossRefGoogle Scholar
[20]Cheng, S. Y. and Yau, S. T.Maximal spacelike hypersurfaces in the Lorentz–Minkowski space. Ann. of Math. 104 (1976), 407–419.CrossRefGoogle Scholar
[21]Cheng, S. Y. and Yau, S. T.Hypersurfaces with constant scalar curvature. Math. Ann. 225 (1977), 195–204.CrossRefGoogle Scholar
[22]Garding, L.An inequality for hyperbolic polynomials. J. Math. Mech. 8 (1959), 957–965.Google Scholar
[23]Hardy, G., Littlewood, J. E. and Pólya, G.Inequalities. Cambridge Mathematical Library (Cambridge, 1989).Google Scholar
[24]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 (1988), 909–917.CrossRefGoogle Scholar
[25]Montiel, S.Uniqueness of spacelike hypersurfaces of constant mean curvature in foliated spacetimes. Math. Ann. 314 (1999), 529–553.CrossRefGoogle Scholar
[26]Montiel, S.Complete non-compact spacelike hypersurfaces of constant mean curvature in de Sitter space. J. Math. Soc. Japan 55 (2003), 915–938.CrossRefGoogle Scholar
[27]Montiel, S. and Ros, A.Compact hypersurfaces: the Alexandrov theorem for higher order mean curvatures. in Differential Geometry (ed. Lawson, B. and Tenenblat, K.), pp. 279–296 (Longman, 1991).Google Scholar
[28]Omori, H.Isometric immersions of Riemannian manifolds. J. Math. Soc. Japan 19 (1967), 205–214.CrossRefGoogle Scholar
[29]O'Neill, B. Semi-Riemannian Geometry with Applications to Relativity. (Academic Press, London, 1983).Google Scholar
[30]Romero, A. and Rubio, R. M.On the mean curvature of spacelike surfaces in certain three-dimensional Robertson–Walker spacetimes and Calabi–Bernstein's type problems. Ann. Global Anal. Geom. 37 (2010), 21–31.CrossRefGoogle Scholar
[31]Rosenberg, H.Hypersurfaces of constant curvature in space forms. Bull. Sci. Math. 117 (1993), 217–239.Google Scholar
[32]Xin, Y. L.On the Gauss image of a spacelike hypersurface with constant mean curvature in Minkowski space. Comment. Math. Helv. 66 (1991), 590–598.CrossRefGoogle Scholar
[33]Yau, S. T.Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28 (1975), 201–228.CrossRefGoogle Scholar
[34]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