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

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.