Parabolicity of maximal surfaces in Lorentzian product spaces

TL;DR

This paper establishes parabolicity criteria for maximal surfaces in Lorentzian product spaces, showing that maximal graphs over starlike domains are parabolic and providing an alternative proof of the Calabi-Bernstein theorem in this setting.

Contribution

It introduces new parabolicity criteria for maximal surfaces in Lorentzian product spaces and applies them to prove maximal graphs over starlike domains are parabolic, offering a new proof of the Calabi-Bernstein theorem.

Findings

Maximal graphs over starlike domains are parabolic.

Provided an alternative proof of the Calabi-Bernstein theorem.

Established parabolicity criteria for maximal surfaces in Lorentzian products.

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 $<,>=<,>_M-dt^2$. 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 $M^2\times\mathbb{R}_1$.