Parabolicity of maximal surfaces in Lorentzian product spaces

Details Parabolicity of maximal surfaces in Lorentzian product spaces Parabolicity of maximal surfaces in Lorentzian product spaces

2008-04-11

arxiv.org/abs/0804.1798v2

Mathematics

Differential Geometry

Final version (October 2009). To appear in Mathematische Zeitschrift. Dedicated to Professor Marcos Dajczer on the occasion of

Scientific paper

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 $<,>=<,>_M-dt^2$. 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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