Romero, Alfonso; Rubio, Rafael M. New proof of the Calabi-Bernstein theorem. (English) Zbl 1198.53061 Geom. Dedicata 147, 173-176 (2010). Summary: By means of a local integral formula on a maximal surface, which involves the hyperbolic angle between the unit normal vector field and a fixed time-like direction, we give a new proof of the parametric Calabi-Bernstein theorem. A suitable modification of the integral formula gives an independent proof to the non-parametric case of this result. Cited in 10 Documents MSC: 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics Keywords:space-like surface; zero mean curvature; Calabi-Bernstein theorem PDFBibTeX XMLCite \textit{A. Romero} and \textit{R. M. Rubio}, Geom. Dedicata 147, 173--176 (2010; Zbl 1198.53061) Full Text: DOI References: [1] Alías L. J., Palmer B.: Zero mean curvature surfaces with non-negative curvature in flat Lorentzian 4-spaces. Proc. R. Soc. Lond. A 455, 631–636 (1999) · Zbl 0970.53034 [2] Alías L. J., Palmer B.: On the Gaussian curvature of maximal surfaces and the Calabi-Bernstein theorem. Bull. Lond. Math. Soc. 33, 454–458 (2001) · Zbl 1041.53038 [3] Calabi E.: Examples of Bernstein problems for some non-linear equations. Proc. Sympos. Pure Math. 15, 223–230 (1970) · Zbl 0211.12801 [4] Cheng S.-Y., Yau S.-T.: Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces. Ann. Math. 104, 407–419 (1976) · Zbl 0352.53021 [5] Chern S. S.: Simple proofs of two theorems on minimal surfaces. Enseign. Math. 15, 53–61 (1969) · Zbl 0175.18603 [6] Estudillo F. J. M., Romero A.: Generalized maximal surfaces in Lorentz-Minkowski space \({\mathbb{L}^3}\) . Math. Proc. Camb. Phil. Soc. 111, 515–524 (1992) · Zbl 0824.53061 [7] 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 [8] Kazdan J. L. : Parabolicity and the Liouville property on complete Riemannian manifolds aspects of math E10. In: Tromba, A.J. (eds) Friedr., pp. 153–166. Vieweg and Sohn, Bonn (1987) [9] Kobayashi O.: Maximal surfaces in the 3-dimensional Minkowski space \({\mathbb{L}^3}\) . Tokyo J. Math. 6, 297–309 (1983) · Zbl 0535.53052 [10] Romero A.: Simple proof of Calabi-Bernstein’s theorem. Proc. Am. Math. Soc. 124, 1315–1317 (1996) · Zbl 0853.53042 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.