Maximal surface equation on a Riemannian 2-manifold with finite total curvature

The differential equation of maximal surfaces on a complete Riemannian 2-manifold with finite total curvature is studied. Uniqueness theorems that widely extend the classical Calabi-Bernstein's theorem in non-parametric version, as well as previous results on complete maximal graphs into Lorentzian warped products, are given. All entire solutions of maximal equation in certain natural Lorentzian warped product, as well as non-existence results, are provided.