Parabolicity of maximal surfaces in Lorentzian product spaces

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 M2 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 \({M^2 \times \mathbb {R}_1}\) .