Entire $f$-maximal graphs in the lorentzian product ${\mathbb{G}}^n\times {\mathbb{R}}_1$

Abstract

In the lorentzian product ${\mathbb{G}}^n\times {\mathbb{R}}_1,$ we give a comparison between the $f$-volume of an entire $f$-maximal graph and the $f$-volume of the hyperbolic $H_r^{+}$ under the assumption that the gradient of the function defining the graph is bounded away from 1. As a consequence, we obtain a Bernstein type theorem for $f$-maximal graphs in ${\mathbb{G}}^n\times {\mathbb{R}}_1.$ Without the condition on the gradient of the function, an example of non-planar entire $f$-maximal graph in the Lorentzian product ${\mathbb{G}}^n\times {\mathbb{R}}_1$ is given. This example shows that the assumption on the gradient of the function defining the graph in the volume comparison as well as in the Bernstein type theorem is essential.

References in corpus (2)

• Calabi-Bernstein results for maximal surfaces in Lorentzian product spaces
• Bernstein type theorem for entire weighted minimal graphs in ${\mathbb{G}}^n\times \mathbb{R}$