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On the Gaussian Curvature of Maximal Surfaces and the Calabi–Bernstein Theorem

In this paper, a new approach to the Calabi–Bernstein theorem on maximal surfaces in the Lorentz–Minkowski space L3 is introduced. The approach is based on an upper bound for the total curvature of geodesic discs in a maximal surface in L3, involving the local geometry of the surface and its hyperbolic image. As an application of this, a new proof of the Calabi–Bernstein theorem is provided.