×

Calabi-Bernstein results for maximal surfaces in Lorentzian product spaces. (English) Zbl 1173.53025

Let \((N=M^2\times \mathbb R\), \(g=g_M-dt^2)\) be a Lorentzian 3-manifold which is a direct product of a Riemannian surface \((M, g_M)\) and the straight line. The authors prove the following version of the Calabi-Bernstein theorem:
If \((M,g_M)\) has non negative curvature \(K\), then any maximal (i.e. with zero mean curvature) space-like surface \(\Sigma\) is totally geodesic. In particular, if \(K>0\) at some point, then \(\Sigma=M\times \{t_0\}\) is a slice.
A non-parametric version of the Calabi-Bernstein theorem for entire maximal graph in \(M\times\mathbb R\) is also proved and some counter-examples, showing that the condition \(K\geq 0\) is essential, are given.

MSC:

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

References:

[1] Frankel, T., Applications of Duschek’s formula to cosmology and minimal surfaces, Bull. Amer. Math. Soc., 81, 579-582 (1975) · Zbl 0302.53003
[2] Brill, D.; Flaherty, F., Isolated maximal surfaces in spacetime, Comm. Math. Phys., 50, 157-165 (1976) · Zbl 0337.53051
[3] Marsden, J. E.; Tipler, F. J., Maximal hypersurfaces and foliations of constant mean curvature in general relativity, Phys. Rep., 66, 109-139 (1980)
[4] Calabi, E., Examples of Bernstein problems for some nonlinear equations, (Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, CA, 1968) (1970), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 223-230 · Zbl 0211.12801
[5] Cheng, S. Y.; Yau, S. T., Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math. (2), 104, 407-419 (1976) · Zbl 0352.53021
[6] Kobayashi, O., Maximal surfaces in the 3-dimensional Minkowski space \(L^3\), Tokyo J. Math., 6, 297-309 (1983) · Zbl 0535.53052
[7] L.V. McNertey, One-parameter families of surfaces with constant curvature in Lorentz 3-space, Ph.D. Thesis, Brown University, USA, 1980; L.V. McNertey, One-parameter families of surfaces with constant curvature in Lorentz 3-space, Ph.D. Thesis, Brown University, USA, 1980
[8] Estudillo, F. J.M.; Romero, A., On maximal surfaces in the \(n\)-dimensional Lorentz-Minkowski space, Geom. Dedicata, 38, 167-174 (1991) · Zbl 0732.53048
[9] Estudillo, F. J.M.; Romero, A., Generalized maximal surfaces in Lorentz-Minkowski space \(L^3\), Math. Proc. Cambridge Philos. Soc., 111, 515-524 (1992) · Zbl 0824.53061
[10] Estudillo, F. J.M.; Romero, A., On the Gauss curvature of maximal surfaces in the 3-dimensional Lorentz-Minkowski space, Comment. Math. Helv., 69, 1-4 (1994) · Zbl 0810.53050
[11] Romero, A., Simple proof of Calabi-Bernstein’s theorem on maximal surfaces, Proc. Amer. Math. Soc., 124, 1315-1317 (1996) · Zbl 0853.53042
[12] Alías, L. J.; Palmer, B., On the Gaussian curvature of maximal surfaces and the Calabi-Bernstein theorem, Bull. London Math. Soc., 33, 454-458 (2001) · Zbl 1041.53038
[13] Albujer, A. L., New examples of entire maximal graphs in \(H^2 \times R_1\), Differential Geom. Appl., 26, 456-462 (2008) · Zbl 1147.53047
[14] Fernández, I.; Mira, P., Complete maximal surfaces in static Robertson-Walker 3-spaces, Gen. Relativity Gravitation, 39, 2073-2077 (2007) · Zbl 1136.53003
[15] Alías, L. J.; Palmer, B., A duality result between the minimal surface equation and the maximal surface equation, An. Acad. Brasil. Ciênc., 73, 161-164 (2001) · Zbl 0999.53007
[16] Alías, L. J.; Romero, A.; Sánchez, M., Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes, Gen. Relativity Gravitation, 27, 71-84 (1995) · Zbl 0908.53034
[17] Kobayashi, S.; Nomizu, K., Foundations of Differential Geometry, Vol. II (1969), Interscience: Interscience New York · Zbl 0175.48504
[18] Ahlfors, L. V., Sur le type d’une surface de Riemann, C. R. Acad. Sci. Paris, 201, 30-32 (1935) · JFM 61.0365.01
[19] Huber, A., On subharmonic functions and differential geometry in the large, Comment. Math. Helv., 32, 13-72 (1957) · Zbl 0080.15001
[20] Latorre, J. M.; Romero, A., New examples of Calabi-Bernstein problems for some nonlinear equations, Differential Geom. Appl., 15, 153-163 (2001) · Zbl 1041.53039
[21] Chern, S. S., Simple proofs of two theorems in minimal surfaces, Enseign. Math. II. Sér., 15, 53-61 (1969) · Zbl 0175.18603
[22] Montaldo, S.; Onnis, I. I., A note on surfaces in \(H^2 \times R\), Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 10, 939-950 (2007) · Zbl 1183.53055
[23] I.I. Onnis, Superficies em certos espacos homogeneos tridimensionais, Ph.D. Thesis, Universidade Estadual de Campinas, Brazil, 2005; I.I. Onnis, Superficies em certos espacos homogeneos tridimensionais, Ph.D. Thesis, Universidade Estadual de Campinas, Brazil, 2005
[24] Duc, D. M.; Hieu, N. V., Graphs with prescribed mean curvature on Poincaré disk, Bull. London Math. Soc., 27, 353-358 (1995) · Zbl 0840.53007
[25] Nelli, B.; Rosenberg, H., Minimal surfaces in \(H^2 \times R\), Bull. Braz. Math. Soc. (N.S.), 33, 263-292 (2002) · Zbl 1038.53011
[26] A.L. Albujer, Geometría global de superficies espaciales en espacios producto lorentzianos, Ph.D. Thesis, Universidad de Murcia, Spain, 2008; A.L. Albujer, Geometría global de superficies espaciales en espacios producto lorentzianos, Ph.D. Thesis, Universidad de Murcia, Spain, 2008
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.