×

Generalized maximum principles and the unicity of complete spacelike hypersurfaces immersed in a Lorentzian product space. (English) Zbl 1306.53056

The authors prove several classification results for complete space-like hypersurfaces \(\Sigma^n\) of a Lorentzian product space \(\overline M^{n+1}:= \mathbb R _1 \times M^n,\) where \(\mathbb R _1\) is the real line equipped with a negative definite inner product and \(M^n\) is a connected oriented Riemannian manifold of dimension \(n.\) This product manifold is naturally equipped with the Lorentzian metric \(\langle \; , \; \rangle = - \pi^*_{\mathbb R}(dt^2) + \pi_M^* (\langle \; , \; \rangle_M),\) where \(\pi_{\mathbb R}\) and \(\pi_M\) denote the canonical projections from \(\overline M^{n+1}\) onto each factor. Let \(H\) denote the mean curvature of \(\Sigma^n\) in \(\overline M^{n+1},\;\) \(h = (\pi_{\mathbb R})|_{\Sigma}\) be the height function and for a fixed \(t_0 \in \mathbb R\) call \(\{ t_0 \} \times M^n\) a slice of \(\overline M^{n+1}.\) The authors consider isometric immersions \(\psi : \Sigma^n \to \overline M^{n+1} = \mathbb R_1 \times M^n\) of a complete space-like hypersurface whose sectional curvature \(K_M\) of its fiber \(M^n\) satisfies \(- \kappa \leq K_M \leq 0\) for some constant \(\kappa > 0.\)
In their first theorem the authors assume additionally that \(\Sigma^n\) lies between two slices of \(\overline M^{n+1}\) and that \(|\nabla h|\) is bounded on \(\Sigma^n.\) Then if \(H\) is bounded and if it does not change sign on \(\Sigma^n\) it follows that \(H\) is not globally bounded away from zero. In particular, if \(H\) is constant then \(\Sigma^n\) is maximal, i.e., \(H = 0.\) The technical tool for the proof is the well-known maximum principle of Omori and Yau. In their second theorem the authors assume \(- \kappa \leq K_M \leq 0\) and that \(H\) is now constant. Using a generalized maximum principle of Yau, they prove that if the height function satisfies one of the two conditions, one of which is \(|\nabla h|^2 \leq [n/k (n-1)] H^2,\) then \(\Sigma^n\) is a slice. It is also proved in this paper that a complete space-like hypersurface \(\Sigma^n\) with mean curvature \(H\) which does not change sign and for which \(|\nabla h|\) is Lebesgue integrable is necessarily maximal and with some additional conditions on the Ricci curvature that it must be a slice. These results and techniques are then applied to study the rigidity of entire vertical graphs of smooth functions \(u \in C^{\infty}(M)\) in such Lorentzian product ambient spaces.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aiyama, R.: On the Gauss map of complete space-like hypersurfaces of constant mean curvature in Minkowski space. Tsukuba J. Math. 16, 353–361 (1992) · Zbl 0782.53043
[2] Albujer, A.L.: New examples of entire maximal graphs in $$\(\backslash\)mathbb{H}\^2\(\backslash\)times \(\backslash\)mathbb{R}_1$$ . Diff. Geom. Appl. 26, 456–462 (2008) · Zbl 1147.53047 · doi:10.1016/j.difgeo.2007.11.035
[3] Albujer, A.L., Alías, L.J.: A local estimate for maximal surfaces in Lorentzian product spaces. Mat. Contemp. 34, 1–10 (2008) · Zbl 1195.53077
[4] Albujer, A.L., Alías, L.J.: Calabi-Bernstein results for maximal surfaces in Lorentzian product spaces. J. Geom. Phys. 59, 620–631 (2009) · Zbl 1173.53025
[5] Albujer, A.L., Alías, L.J.: Parabolicity of maximal surfaces in Lorentzian product spaces. Math. Z. 267, 453–464 (2011) · Zbl 1218.53062 · doi:10.1007/s00209-009-0630-8
[6] Albujer, A.L., Camargo, F., de Lima, H.F.: Complete spacelike hypersurfaces with constant mean curvature in $$-\(\backslash\)mathbb{R}\(\backslash\)times \(\backslash\)mathbb{H}\^n$$ . J. Math. Anal. Appl. 368, 650–657 (2010) · Zbl 1193.53124 · doi:10.1016/j.jmaa.2010.02.039
[7] Aledo, J.A., Alías, L.J.: On the curvatures of bounded complete spacelike hypersurfaces in the Lorentz-Minkowski space. Manuscripta Math. 101, 401–413 (2000) · Zbl 0991.53037 · doi:10.1007/s002290050223
[8] Alías, L.J., Colares, A.G.: Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in Generalized Robertson-Walker spacetimes. Math. Proc. Camb. Philos. Soc. 143, 703–729 (2007) · Zbl 1131.53035 · doi:10.1017/S0305004107000576
[9] Alías, L.J., Romero, A., Sánchez, M.: Uniqueness of complete spacelike hypersurfaces of constant mean curvature in Generalized Robertson-Walker spacetimes. Gen. Relativ. Gravit. 27, 71–84 (1995) · Zbl 0908.53034 · doi:10.1007/BF02105675
[10] Calabi, E.: Examples of Bernstein problems for some nonlinear equations. Proc. Sympos. Pure Math. 15, 223–230 (1970) · Zbl 0211.12801 · doi:10.1090/pspum/015/0264210
[11] Caminha, A., de Lima, H.F.: Complete vertical graphs with constant mean curvature in semi-Riemannian warped products. Bull. Belg. Math. Soc. Simon Stevin 16, 91–105 (2009) · Zbl 1160.53362
[12] Cheng, S.Y., Yau, S.T.: Maximal spacelike hypersurfaces in the Lorentz-Minkowski space. Ann. Math. 104, 407–419 (1976) · Zbl 0352.53021 · doi:10.2307/1970963
[13] Li, G., Salavessa, I.: Graphic Bernstein results in curved pseudo-Riemannian manifolds. J. Geom. Phys. 59, 1306–1313 (2009) · Zbl 1173.53031 · doi:10.1016/j.geomphys.2009.06.011
[14] de Lima, H.F.: On the Gauss map of complete spacelike hypersurfaces with bounded mean curvature in the Minkowski space. Bull. Belg. Math. Soc. Simon Stevin 18, 537–541 (2011) · Zbl 1235.53024
[15] Montiel, S.: Uniqueness of spacelike hypersurfaces of constant mean curvature in foliated spacetimes. Math. Ann. 314, 529–553 (1999) · Zbl 0965.53043 · doi:10.1007/s002080050306
[16] Omori, H.: Isometric immersions of Riemannian manifolds. J. Math. Soc. Japan 19, 205–214 (1967) · Zbl 0154.21501 · doi:10.2969/jmsj/01920205
[17] O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, London (1983)
[18] Salavessa, I.: Spacelike graphs with parallel mean curvature. Bull. Belg. Math. Soc. Simon Stevin 15, 65–76 (2008) · Zbl 1146.53036
[19] Xin, Y.L.: On the Gauss image of a spacelike hypersurface with constant mean curvature in Minkowski space. Comment. Math. Helv. 66, 590–598 (1991) · Zbl 0752.53038 · doi:10.1007/BF02566667
[20] Yau, S.T.: Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28, 201–228 (1975) · Zbl 0291.31002 · doi:10.1002/cpa.3160280203
[21] Yau, S.T.: Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Univ. Math. J. 25, 659–670 (1976) · Zbl 0335.53041 · doi:10.1512/iumj.1976.25.25051
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.