ON THE GAUSSIAN CURVATURE OF MAXIMAL SURFACES AND THE CALABI–BERNSTEIN THEOREM

Abstract

In this paper, a new approach to the Calabi–Bernstein theorem on maximal surfaces in the Lorentz– Minkowski space L 3 is introduced. The approach is based on an upper bound for the total curvature of geodesic discs in a maximal surface in L 3 , 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.