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 $M^2$ is a connected Riemannian surface with non-negative Gaussian curvature and $M^2\times {\mathbb{R}}_1$ is endowed with the Lorentzian product metric $<,>=<,{>}_M-d{t}^2$. 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 $M^2\times {\mathbb{R}}_1$.

arXiv commentFinal version (October 2009). To appear in Mathematische Zeitschrift. Dedicated to Professor Marcos Dajczer on the occasion of his 60th birthday

References in corpus (1)

• Calabi-Bernstein results for maximal surfaces in Lorentzian product spaces

Cited by in corpus (1)

• The number of conformally equivalent maximal graphs