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Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes

Published online by Cambridge University Press:  01 November 2007

LUIS J. ALÍAS  and
A. GERVASIO COLARES
LUIS J. ALÍAS*
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, E-30100 Espinardo, Murcia, Spain. e-mail: [email protected]
A. GERVASIO COLARES*
Affiliation:
Departamento de Matemática, Universidade Federal do Ceará, 60455-760 Fortaleza-Ce, Brazil. e-mail: [email protected]
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Abstract

In this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker (GRW) spacetimes. In particular, we consider the following question: under what conditions must a compact spacelike hypersurface with constant higher order mean curvature in a spatially closed GRW spacetime be a spacelike slice? We prove that this happens, essentially, under the so called null convergence condition. Our approach is based on the use of the Newton transformations (and their associated differential operators) and the Minkowski formulae for spacelike hypersurfaces.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 143 , Issue 3 , November 2007 , pp. 703 - 729
Copyright
Copyright © Cambridge Philosophical Society 2007

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References

REFERENCES

[1]Aledo, J. A., Alías, L. J. and Romero, A.. Integral formulas for compact spacelike hypersurfaces in de Sitter space. Applications to the case of constant higher order mean curvature. J. Geom. Phys. 31 (1999), 195–208.CrossRefGoogle Scholar
[2]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
[3]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 Scientific Publishing, 2002).CrossRefGoogle Scholar
[4]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
[5]Alías, L. J., Romero, A. and Sánchez, M.. Spacelike hypersurfaces of constant mean curvature and Calabi-Bernstein type problems. Tôhoku Math. J. 49 (1997), 337–345.CrossRefGoogle Scholar
[6]Alías, L. J., Romero, A. and Sánchez, M.. Spacelike hypersurfaces of constant mean curvature in certain spacetimes. Nonlinear Anal. 30 (1997), 655–661.CrossRefGoogle Scholar
[7]Besse, A. L.. Einstein Manifolds (Springer-Verlag, 1987).CrossRefGoogle Scholar
[8]Barbosa, J. L. and Colares, A. G.. Stability of hypersurfaces with constant r-mean curvature. Ann. Global Anal. Geom. 15 (1997), 277–297.CrossRefGoogle Scholar
[9]Bartnik, R.. Existence of maximal surfaces in asymptotically flat spacetimes. Comm. Math. Phys. 94 (1984), 155–175.CrossRefGoogle Scholar
[10]Bartnik, R., Chrusciel, P. T. and Murchadha, N. Ó. On maximal surfaces in asymptotically flat space-times. Comm. Math. Phys. 130 (1990), 95–109.CrossRefGoogle Scholar
[11]Bartnik, R. and Simon, L.. Spacelike hypersurfaces with prescribed boundary values and mean curvature. Comm. Math. Phys. 87 (1982/83), 131–152.CrossRefGoogle Scholar
[12]Brill, D. and Flaherty, F.. Isolated maximal surfaces in spacetime. Comm. Math. Phys. 50 (1976), 157–165.CrossRefGoogle Scholar
[13]Calabi, E.. Examples of Bernstein problems for some nonlinear equations. 1970 \itGlobal Analysis, Proc. Sympos. Pure Math. Vol. XV (1968) 223–230.Google Scholar
[14]Cheng, X. and Rosenberg, H.. Embedded positive constant r-mean cuvature hypersurfaces in M m× R. An. Acad. Brasil. Ciênc. 77 (2005), 183–199.CrossRefGoogle Scholar
[15]Cheng, S. Y. and Yau, S. T.. Maximal space-like hypersurfaces in the Lorentz–Minkowski spaces. Ann. of Math. (2) 104 (1976), 407–419.CrossRefGoogle Scholar
[16]Choquet–Bruhat, Y.. Quelques propriétés des sousvariétés maximales d'une variété lorentzienne. C. R. Acad. Sci. Paris Sér. A-B 281 (1975), A577–A580.Google Scholar
[17]Choquet–Bruhat, Y.. Maximal submanifolds and submanifolds with constant mean extrinsic curvature of a Lorentzian manifold. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), 361–376.Google Scholar
[18]Choquet–Bruhat, Y., Fischer, A. E. and Marsden, J. E.. Maximal hypersurfaces and positivity of mass. In Isolated Gravitating Systems in General Relativity. Proceedings of the International School of Physics “Enrico Fermi” (Ehlers, J., ed.) (North–Holland Publishing Co., 1979), pp. 396–456.Google Scholar
[19]Choquet–Bruhat, Y. and York, J. W. Jr. The Cauchy Problem. in General Relativity and Gravitation, Vol. 1 (Plenum, 1980), pp. 99–172.Google Scholar
[20]Elbert, F.. Constant positive 2-mean curvature hypersurfaces. Illinois J. Math. 46 (2002), 247–267.CrossRefGoogle Scholar
[21]Garding, L.. An inequality for hyperbolic polynomials. J. Math. Mech. 8 (1959), 957–965.Google Scholar
[22]Gerhardt, C.. H-surfaces in Lorentzian manifolds. Comm. Math. Phys. 89 (1983), 523–553.CrossRefGoogle Scholar
[23]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
[24]Hardy, G., Littlewood, J. E. and Póyla, G.. Inequalities, 2nd. ed. (Cambridge Mathematical Library, 1989).Google Scholar
[25]Hsiung, C. C.. Some integral formulas for closed hypersurfaces. Math. Scand. 2 (1954), 286–294.CrossRefGoogle Scholar
[26]Marsden, J. E. and Tipler, F. J.. Maximal hypersurfaces and foliations of constant mean curvature in general relativity. Phys. Rep. 66 (1980), 109–139.CrossRefGoogle Scholar
[27]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
[28]Montiel, S.. Uniqueness of spacelike hypersurfaces of constant mean curvature in foliated spacetimes. Math. Ann. 314 (1999), 529–553.CrossRefGoogle Scholar
[29]Montiel, S. and Ros, A.. Compact hypersurfaces: The Alexandrov theorem for higher order mean curvatures. In Differential Geometry (ed. Lawson, B. and Tenenblat, K.) (Longman, 1991), pp. 279–296.Google Scholar
[30]O'Neill, B.. Semi-Riemannian Geometry with Applications to Relativity (Academic Press, 1983).Google Scholar
[31]Reilly, R. C.. Variational properties of functions of the mean curvature for hypersurfaces in space forms. J. Differential Geom. 8 (1973), 465–477.CrossRefGoogle Scholar
[32]Rosenberg, H.. Hypersurfaces of constant curvature in space forms. Bull. Sci. Math. 117 (1993), 211–239.Google Scholar
[33]Stumbles, S.. Hypersurfaces of constant mean extrinsic curvature. Ann. Physics 133 (1981), 28–56.CrossRefGoogle Scholar