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In this paper we establish new Calabi-Bernstein results for maximal surfacesimmersed into a Lorentzian product space of the form $M^2\times\mathbb{R}_1$,where $M^2$ is a connected Riemannian surface and $M^2\times\mathbb{R}_1$ isendowed with the Lorentzian metric $<,>=<,>_{M}-dt^2$. In particular, when $M$is a Riemannian surface with non-negative Gaussian curvature $K_M$, we provethat any complete maximal surface in $M^2\times\mathbb{R}_1$ must be totallygeodesic. Besides, if $M$ is non-flat we conclude that it must be a slice$M\times\{t_0\}$, $t_0\in\mathbb{R}$ (here by "complete" it is meant, as usual,that the induced Riemannian metric on the maximal surface from the ambientLorentzian metric is complete). We prove that the same happens if the maximalsurface is complete with respect to the metric induced from the Riemannianproduct $M^2\times\mathbb{R}$. This allows us to give also a non-parametricversion of the Calabi-Bernstein theorem for entire maximal graphs in$M^2\times\mathbb{R}_1$, under the same assumptions on $K_M$. Moreover, we alsoconstruct counterexamples which show that our Calabi-Bernstein results are nolonger true without the hypothesis $K_M\geq 0$. These examples are constructedvia a duality result between minimal and maximal graphs.

Calabi-Bernstein results for maximal surfaces in Lorentzian product spaces