Abstract
We deal with spacelike hypersurfaces immersed with some constant rth mean curvature in a steady state type spacetime, that is, a generalized Robertson–Walker spacetime of the type \({-\mathbb{R} \times_{e^t} M^n}\). In this setting, supposing that the fiber M n of the ambient space has nonnegative constant sectional curvature, we establish characterization results concerning domains of the spacelike slices \({\{t\} \times M^n}\). Afterwards, we apply such characterization results to study the uniqueness of complete spacelike hypersurfaces with one end in such a ambient space.
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Albujer A.L., Alías L.J.: Spacelike hypersurfaces with constant mean curvature in the steady state space. Proc. Am. Math. Soc. 137, 711–721 (2009)
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)
Alías L.J., Colares A.G.: Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes. Math. Proc. Cambridge Philos. Soc. 143, 703–729 (2007)
Alías L.J., Romero A., Sánchez M.: Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson–Walker spacetimes. Gen. Relat. Gravit. 27, 71–84 (1995)
Alías L.J., Romero A., Sánchez M.: Spacelike hypersurfaces of constant mean curvature and Calabi–Bernstein type problems. Tôhoku Math. J. 49, 337–345 (1997)
Aquino C.P., de Lima H.F.: Uniqueness of complete hypersurfaces with bounded higher order mean curvatures in semi-Riemannian warped products. Glasgow Math. J. 54, 201–212 (2012)
Beem J.K., Ehrlich P.E., Easley K.L.: Global Lorentzian Geometry, 2nd edn. CRC Press, New York (1996)
Bondi H., Gold T.: On the generation of magnetism by fluid motion. Monthly Not. R. Astr. Soc. 108, 252–270 (1948)
Camargo F., Caminha A., de Lima H.F.: Bernstein-type theorems in semi-Riemannian warped products. Proc. Am. Math. Soc. 139, 1841–1850 (2011)
Caminha A.: A rigidity theorem for complete CMC hypersurfaces in Lorentz manifolds. Differ. Geom. Appl. 24, 652–659 (2006)
Caminha A., de Lima H.F.: Complete spacelike hypersurfaces in conformally stationary Lorentz manifolds. Gen. Relativ. Gravit. 41, 173–189 (2009)
Colares A.G., de Lima H.F.: Spacelike hypersurfaces with constant mean curvature in the steady state space. Bull. Belg. Math. Soc. Simon Stevin 17, 287–302 (2010)
Colares A.G., de Lima H.F.: On the rigidity of spacelike hypersurfaces immersed in the steady state space \({\mathcal H^{n+1}}\). Publ. Math. Debrecen 81, 103–119 (2012)
Galloway G.J.: Cosmological spacetimes with \({\Lambda > 0}\). Contemp. Math. 359, 87–101 (2004)
Garding L.: An inequality for hyperbolic polynomials. J. Math. Mech. 8, 957–965 (1959)
Hardy G., Littlewood J.E., Pólya G.: Inequalities. Cambridge Mathematical Library, Cambridge (1989)
Hawking S.W., Ellis G.F.R.: The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge (1973)
Hoyle F.: A new model for the expanding universe. Monthly Not. R. Astr. Soc. 108, 372–382 (1948)
Latorre J.M., Romero A.: Uniqueness of noncompact spacelike hypersurfaces of constant mean curvature in generalized Robertson–Walker spacetimes. Geom. Dedicata 93, 1–10 (2002)
Montiel S.: Uniqueness of spacelike hypersurfaces of constant mean curvature in foliated spacetimes. Math. Ann. 314, 529–553 (1999)
Montiel S.: Complete non-compact spacelike hypersurfaces of constant mean curvature in de Sitter spaces. J. Math. Soc. Japan 55, 915–938 (2003)
Montiel, S., Ros, A.: Compact hypersurfaces: the Alexandrov theorem for higher order mean curvatures. In: Lawson, B., Tenenblat, K. (eds.) Differential geometry, pp. 279–296. Longman, London (1991)
Omori H.: Isometric immersions of Riemannian manifolds. J. Math. Soc. Japan 19, 205–214 (1967)
O’Neill B.: Semi-Riemannian Geometry, with Applications to Relativity. Academic Press, New York (1983)
Weinberg S.: Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. Wiley, New York (1972)
Yau S.T.: Harmonic Functions on Complete Riemannian Manifolds. Comm. Pure Appl. Math. 28, 201–228 (1975)
Yau S.T.: Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Univ. Math. J. 25, 659–670 (1976)
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Aquino, C.P., de Lima, H.F., dos Santos, F.R. et al. Spacelike hypersurfaces with constant rth mean curvature in steady state type spacetimes. J. Geom. 106, 85–96 (2015). https://doi.org/10.1007/s00022-014-0234-2
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DOI: https://doi.org/10.1007/s00022-014-0234-2