On entire f-maximal graphs in the Lorentzian product Gn×R1

In the Lorentzian product Gn×R1, we give a comparison theorem between the f-volume of an entire f-maximal graph and the f-volume of the hyperbolic Hr+ under the condition that the gradient of the function defining the graph is bounded away from 1. This condition comes from an example of non-planar entire f-maximal graph in Gn×R1 and is equivalent to the hyperbolic angle function of the graph being bounded. As a consequence, we obtain a Calabi-Bernstein type theorem for f-maximal graphs in Gn×R1.