Entire $f$-maximal graphs in the lorentzian product $\Bbb G^n\times\Bbb R_1$

TL;DR

This paper investigates entire $f$-maximal graphs in Lorentzian products, establishing volume comparison results and a Bernstein type theorem under gradient bounds, and provides examples showing the necessity of these bounds.

Contribution

It introduces a volume comparison and Bernstein theorem for $f$-maximal graphs in Lorentzian products, highlighting the importance of gradient bounds for these results.

Findings

Volume comparison between $f$-maximal graphs and hyperbolic models.

A Bernstein type theorem for $f$-maximal graphs with bounded gradient.

Existence of non-planar entire $f$-maximal graphs without gradient bounds.

Abstract

In the lorentzian product $\Bbb G^n\times\Bbb R_1,$ 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 $\Bbb G^n\times\Bbb R_1.$ Without the condition on the gradient of the function, an example of non-planar entire $f$-maximal graph in the Lorentzian product $\Bbb G^n\times\Bbb R_1$ 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.