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Bernstein-type theorems in semi-Riemannian warped products. (English) Zbl 1223.53045
If \(M^n\) is an \(n\)-dimensional Riemannian manifold, the Lorentzian warped product \(-\mathbb{R}\times_{e^t}M^n\) is called a steady-state type space while the Riemannian warped product \(\mathbb{R}\times_{e^t}M^n\) is known as a hyperbolic type space. For \(M^n=\mathbb{R}^n\), the classical steady state and the hyperbolic space are obtained, respectively. Assume that \(M^n\) is a complete and connected Riemannian manifold. For a given \(t_0\in\mathbb{R}\), a slice \(\Sigma_{t_0}=\{t_0\}\times M^n\) is a complete, connected (space-like) hypersurface with mean curvature \(H=1\) (after choosing a suitable orientation).
In this paper, the authors study complete hypersurfaces, in both steady-state type spaces and in hyperbolic type spaces, with bounded mean curvature and with the gradient of the height function satisfying suitable conditions. By using a technique of S. T. Yau, they obtain Bernstein-type results for this class of immersions.
For instance, the authors prove that, if \(\Sigma\) is a complete connected space-like hypersurface of a steady-state type space-time, contained in a slab bounded by two slices, with bounded mean curvature \(H\geq 1\), and such that the gradient of its height function has integrable norm, then \(\Sigma\) must be a slice. Analogously, in the Riemannian setting the authors prove that, if \(\Sigma\) is a complete, connected hypersurface of a hyperbolic type space-time contained in a slab, with bounded mean curvature \(0<H\leq1\), and satisfying the condition that the gradient of the height function has integrable norm, then \(\Sigma\) has to be a slice.
The last part of the paper is devoted to extend, under similar assumptions, the previous results to the case in which the conditions on the mean curvature are substituted by the following conditions on the \(r\)-th mean curvatures of the hypersurfaces: \(0<H_r\leq H_{r+1}\) in the Lorentzian case; and \(H_r\geq H_{r+1}>0\) in the Riemannian case.

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53Z05 Applications of differential geometry to physics
Full Text: DOI
[1] Alma L. Albujer and Luis J. Alías, Spacelike hypersurfaces with constant mean curvature in the steady state space, Proc. Amer. Math. Soc. 137 (2009), no. 2, 711 – 721. · Zbl 1162.53043
[2] Luis J. Alías and A. Gervasio Colares, Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes, Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 3, 703 – 729. · Zbl 1131.53035 · doi:10.1017/S0305004107000576 · doi.org
[3] Luis J. Alías and Marcos Dajczer, Uniqueness of constant mean curvature surfaces properly immersed in a slab, Comment. Math. Helv. 81 (2006), no. 3, 653 – 663. · Zbl 1110.53039 · doi:10.4171/CMH/68 · doi.org
[4] Luis J. Alías, Marcos Dajczer, and Jaime Ripoll, A Bernstein-type theorem for Riemannian manifolds with a Killing field, Ann. Global Anal. Geom. 31 (2007), no. 4, 363 – 373. · Zbl 1125.53005 · doi:10.1007/s10455-006-9045-5 · doi.org
[5] Luis J. Alías, Alfonso Romero, and Miguel Sánchez, Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes, Gen. Relativity Gravitation 27 (1995), no. 1, 71 – 84. · Zbl 0908.53034 · doi:10.1007/BF02105675 · doi.org
[6] João Lucas Marques Barbosa and Antônio Gervasio Colares, Stability of hypersurfaces with constant \?-mean curvature, Ann. Global Anal. Geom. 15 (1997), no. 3, 277 – 297. · Zbl 0891.53044 · doi:10.1023/A:1006514303828 · doi.org
[7] H. Bondi - T. Gold. On the generation of magnetism by fluid motion. Monthly Not. Roy. Astr. Soc. 108 (1948), 252-270. · Zbl 0031.09603
[8] A. Caminha and H. F. de Lima, Complete vertical graphs with constant mean curvature in semi-Riemannian warped products, Bull. Belg. Math. Soc. Simon Stevin 16 (2009), no. 1, 91 – 105. · Zbl 1160.53362
[9] A. Caminha, P. Sousa - F. Camargo. Complete foliations of space forms by hypersurfaces. Accepted for publication in Bull. Braz. Math. Soc. · Zbl 1226.53055
[10] S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge University Press, London-New York, 1973. Cambridge Monographs on Mathematical Physics, No. 1. · Zbl 0265.53054
[11] F. Hoyle. A new model for the expanding universe. Monthly Not. Roy. Astr. Soc. 108 (1948), 372-382. · Zbl 0031.38304
[12] Sebastián Montiel, 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), no. 4, 909 – 917. · Zbl 0677.53067 · doi:10.1512/iumj.1988.37.37045 · doi.org
[13] Sebastián Montiel, Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds, Indiana Univ. Math. J. 48 (1999), no. 2, 711 – 748. · Zbl 0973.53048 · doi:10.1512/iumj.1999.48.1562 · doi.org
[14] Sebastián Montiel, Uniqueness of spacelike hypersurfaces of constant mean curvature in foliated spacetimes, Math. Ann. 314 (1999), no. 3, 529 – 553. · Zbl 0965.53043 · doi:10.1007/s002080050306 · doi.org
[15] Sebastián Montiel, Complete non-compact spacelike hypersurfaces of constant mean curvature in de Sitter spaces, J. Math. Soc. Japan 55 (2003), no. 4, 915 – 938. · Zbl 1049.53044 · doi:10.2969/jmsj/1191418756 · doi.org
[16] Barrett O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. · Zbl 0531.53051
[17] Harold Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math. 117 (1993), no. 2, 211 – 239. · Zbl 0787.53046
[18] S. Weinberg. Gravitation and cosmology: Principles and applications of the general theory of relativity. John Wiley & Sons, New York (1972).
[19] Shing Tung Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), no. 7, 659 – 670. · Zbl 0335.53041 · doi:10.1512/iumj.1976.25.25051 · doi.org
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