Calabi-Bernstein results for maximal surfaces in Lorentzian product spaces

Abstract

In this paper we establish new Calabi-Bernstein results 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 and $M^2\times\mathbb{R}_1$ is endowed with the Lorentzian metric $<,>=<,>_{M}-dt^2$. In particular, when $M$ is a Riemannian surface with non-negative Gaussian curvature $K_M$, we prove that any complete maximal surface in $M^2\times\mathbb{R}_1$ must be totally geodesic. Besides, if $M$ is non-flat we conclude that it must be a slice $M\times\{t_0\}$, $t_0\in\mathbb{R}$ (here by "complete" it is meant, as usual, that the induced Riemannian metric on the maximal surface from the ambient Lorentzian metric is complete). We prove that the same happens if the maximal surface is complete with respect to the metric induced from the Riemannian product $M^2\times\mathbb{R}$. This allows us to give also a non-parametric version of the Calabi-Bernstein theorem for entire maximal graphs in $M^2\times\mathbb{R}_1$, under the same assumptions on $K_M$. Moreover, we also construct counterexamples which show that our Calabi-Bernstein results are no longer true without the hypothesis $K_M\geq 0$. These examples are constructed via a duality result between minimal and maximal graphs.