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

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.