Calabi–Bernstein results for maximal surfaces in Lorentzian product spaces

In this paper we establish new Calabi–Bernstein results for maximal surfaces immersed into a Lorentzian product space of the form M2×R1M2×R1, where M2M2 is a connected Riemannian surface and M2×R1M2×R1 is endowed with the Lorentzian metric 〈,〉=〈,〉M−dt2. In particular, we prove that when MM is a Riemannian surface with non-negative Gaussian curvature KMKM, any complete maximal surface in M2×R1M2×R1 must be totally geodesic. Besides, if MM is non-flat we conclude that it must be a slice M×{t0}M×{t0}, t0∈Rt0∈R (here by complete   it is meant, as usual, that the induced Riemannian metric on the maximal surface from the ambient Lorentzian metric is complete). We prove that the same happens if the maximal surface is complete with respect to the metric induced from the Riemannian product M2×RM2×R. This allows us to give also a non-parametric version of the Calabi–Bernstein theorem for entire maximal graphs in M2×R1M2×R1, under the same assumptions on KMKM. Moreover, we also construct counterexamples which show that our Calabi–Bernstein results are no longer true without the hypothesis KM≥0KM≥0. These examples are constructed via a duality result between minimal and maximal graphs.