Abstract
In this paper we establish some parabolicity criteria for maximal surfaces immersed into a Lorentzian product space of the form \({M^2 \times \mathbb {R}_1}\) , where M 2 is a connected Riemannian surface with non-negative Gaussian curvature and \({M^2 \times \mathbb {R}_1}\) is endowed with the Lorentzian product metric \({{\langle , \rangle}={\langle , \rangle}_M-dt^2}\) . In particular, and as an application of our main result, we deduce that every maximal graph over a starlike domain \({\Omega \subseteq M}\) is parabolic. This allows us to give an alternative proof of the non-parametric version of the Calabi–Bernstein result for entire maximal graphs in \({M^2 \times \mathbb {R}_1}\) .
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Dedicated to Professor Marcos Dajczer on the occasion of his 60th birthday.
This work was partially supported by MEC projects MTM2007-64504 and Fundación Séneca project 04540/GERM/06, Spain. This research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Regional Agency for Science and Technology (Regional Plan for Science and Technology 2007–2010).
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Albujer, A.L., Alías, L.J. Parabolicity of maximal surfaces in Lorentzian product spaces. Math. Z. 267, 453–464 (2011). https://doi.org/10.1007/s00209-009-0630-8
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DOI: https://doi.org/10.1007/s00209-009-0630-8