New examples of entire maximal graphs in H2×R1H2×R1

In this paper we obtain new explicit examples of complete and non-complete entire maximal graphs in H2×R1H2×R1. The existence of these entire maximal graphs shows that entire maximal graphs in this Lorentzian product space are not necessarily complete, on the contrary that in the Lorentz–Minkowski space. Moreover, in [A.L. Albujer, L.J. Alías, Calabi–Bernstein results for maximal surfaces in Lorentzian product spaces, Preprint, 2006], the author jointly with Alías gave a Calabi–Bernstein theorem for maximal surfaces immersed into the Lorentzian product space M2×R1M2×R1, where M2M2 is a connected Riemannian surface of non-negative Gaussian curvature, and these examples show that the assumption on KMKM is necessary.