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Spacelike hypersurfaces of constant higher order mean curvature in generalized Robertson–Walker spacetimes

Published online by Cambridge University Press:  20 October 2011

LUIS J. ALÍAS ,
DEBORA IMPERA  and
MARCO RIGOLI
LUIS J. ALÍAS
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Espinardo, Murcia, Spain. e-mail: ljalias@um.es
DEBORA IMPERA
Affiliation:
Dipartimento di Matematica, Università degli studi di Milano, Via Saldini 50, I-20133 Milano, Italy. e-mail: debora.impera@unimi.it and marco.rigoli@unimi.it
MARCO RIGOLI
Affiliation:
Dipartimento di Matematica, Università degli studi di Milano, Via Saldini 50, I-20133 Milano, Italy. e-mail: debora.impera@unimi.it and marco.rigoli@unimi.it
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Abstract

In this paper we analyse the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes. We consider first the case of compact spacelike hypersurfaces, completing some previous results given in [2]. We next extend these results to the complete noncompact case. In that case, our approach is based on the use of a generalized version of the Omori–Yau maximum principle for trace type differential operators, recently given by the authors in [3].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 152 , Issue 2 , March 2012 , pp. 365 - 383
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[1]Alías, L. J., Brasil, A. Jr. and Colares, A.G.Integral formulae for spacelike hypersurfaces in conformally stationary spacetimes and applications. Proc. Edinb. Math. Soc. (2) 46 (2003), 465–488.CrossRefGoogle Scholar
[2]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), 705–729.CrossRefGoogle Scholar
[3]Alías, L. J., Impera, D. and Rigoli, M. Hypersurfaces of constant higher order mean curvature in warped product spaces. Preprint 2010. Available at http://arxiv.org/abs/1109.6474.Google Scholar
[4]Alías, L. J. and Montiel, S. Uniqueness of spacelike hypersurfaces with constant mean curvature in generalized Robertson–Walker spacetimes. Differential geometry, Valencia, 2001, 59–69. (World Sci. Publ., 2002).Google Scholar
[5]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
[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.CrossRefGoogle Scholar
[7]Caballero, M., Romero, A. and Rubio, R. M. Constant mean curvature spacelike hypersurfaces in Lorentzian manifolds with a timelike gradient conformal vector field. To appear in Class. Quantum Gravity (2011).CrossRefGoogle Scholar
[8]Elbert, M. F.Constant positive 2-mean curvature hypersurfaces. Illinois J. Math. 46 (2002), 247–267.CrossRefGoogle Scholar
[9]Garding, L.An inequality for hyperbolic polynomials. J. Math. Mech. 8 (1959), 957–965.Google Scholar
[10]Gilbarg, D and Trudinger, N.SElliptic partial differential equations of second order. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224 (Springer-Verlag, 1983).Google Scholar
[11]Montiel, S.Uniqueness of spacelike hypersurfaces of constant mean curvature in foliated spacetimes. Math. Ann. 314 (1999), 529–553.CrossRefGoogle Scholar
[12]O'Neill, B.Semi-Riemannian geometry. Pure and Applied Mathematics, 103 (Academic Press Inc., 1983).Google Scholar
[13]Pigola, S., Rigoli, M. and Setti, A. G.A Liouville-type result for quasi-linear elliptic equations on complete Riemannian manifolds. J. Funct. Anal. 219 (2005), 400–432.CrossRefGoogle Scholar
[14]Pigola, S., Rigoli, M. and Setti, A. G.Maximum principles on Riemannian manifolds and applications. Mem. Amer. Math. Soc. 174 (2005), no. 822, x+99.Google Scholar
[15]Reilly, R. C.Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differential Geometry 8 (1973), 465–477.CrossRefGoogle Scholar
[16]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