×

zbMATH — the first resource for mathematics

Graphic Bernstein results in curved pseudo-Riemannian manifolds. (English) Zbl 1173.53031
The classical Bernstein theorem has been generalized to graphic hypersurfaces of \(\mathbb R^{m+1}\) for \(m\leq 7\), and for higher dimensions and codimensions under various growth conditions. In the paper [J. Geom. Phys. 59, No. 5, 620–631 (2009; Zbl 1173.53025)], A. Albujer and L. Alias have proved a new Calabi-Bernstein-type result for surfaces immersed into a Lorentzian product 3-manifold of the form \(\Sigma_1\times\mathbb R\), where \(\Sigma_1\) is a Riemannian surface of nonnegative Gauss curvature. In this paper the authors generalize this result to space-like graphic submanifolds with parallel mean curvature in a non-flat pseudo-Riemannian product space of any dimension \(m+n\) and under less restrictive curvature condition.

MSC:
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Calabi, E., Examples of Bernstein problems for some nonlinear equations, Proc. sympos. pure math., 15, 223-230, (1970) · Zbl 0211.12801
[2] Cheng, S.; Yau, S.T., Maximal spacelike hypersurfaces in lorentz – minkowski space, Ann. of math., 104, 407-419, (1976) · Zbl 0352.53021
[3] Alías, L.; Romero, A.; Sanchez, 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
[4] Treibergs, A., Entire spacelike hypersurfaces of constant Mean curvature in Minkowski space, Invent. math., 66, 39-56, (1982) · Zbl 0483.53055
[5] Xin, Y., On Gauss image of a spacelike hypersurface with constant Mean curvature in Minkowski space, Comm. math. helv., 66, 590-598, (1991) · Zbl 0752.53038
[6] Xin, Y.; Ye, R., Bernstein-type theorems for spacelike surfaces with parallel Mean curvature, J. reine angew math., 489, 189-198, (1997) · Zbl 0879.53046
[7] Jost, J.; Xin, Y., Some aspects of the global geometry of entire spacelike submanifolds, Results math., 40, 233-245, (2001) · Zbl 0998.53036
[8] Albujer, A.; Alías, L., Calabi – bernstein results for maximal surfaces in Lorentz product spaces, J. geom. phys., 59, 620-631, (2009) · Zbl 1173.53025
[9] Chern, S.S., Simple proofs of two theorems on minimal surfaces, Enseignement math. II. Sér, 15, 53-61, (1969) · Zbl 0175.18603
[10] Salavessa, I.M.C., Spacelike graphs with parallel Mean curvature, Bull. belg. math. soc., 15, 65-76, (2008) · Zbl 1146.53036
[11] Wang, M., On graphic Bernstein type results in higher codimension, Trans. amer. math. soc., 355, 265-271, (2003) · Zbl 1021.53005
[12] Weyl, H., Das asymptotische verteilungsgesetz der eigenwerte linearer partieller differentialgleichungen (mit einer anwendung auf die theorie der hohlraumstrahlung), Math. ann., 71, 4, 441-479, (1912) · JFM 43.0436.01
[13] Salavessa, I.M.C., Graphs with parallel Mean curvature, Proc. amer. math. soc., 107, 2, 449-458, (1989) · Zbl 0681.53031
[14] Cheng, S.; Yau, S.T., Differential equations on Riemannian manifolds and their geometric applications, Comm. pure appl. math., 28, 333-354, (1975) · Zbl 0312.53031
[15] Yau, S.-T., Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, J. indiana univ. J., 25, 7, 659-670, (1976) · Zbl 0335.53041
[16] Alías, L.; Dajczer, M.; Ripoll, J., A Bernstein-type theorem for Riemannian manifolds with a Killing field, Ann. glob. anal. geom., 31, 363-373, (2007) · Zbl 1125.53005
[17] G. Li, I.M.C. Salavessa, Bernstein-Heinz-Chern results in calibrated manifolds, Rev. Mat. Iberoamericana (in press) · Zbl 1197.53077
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.