Calabi-Bernstein type problems in Lorentzian Geometry

Abstract

The study of maximal hypersurfaces in Lorentzian manifolds is an interesting mathematical problem, which connects di_erential geometry, nonlinear partial di_erential equations and certain problems in mathematical relativity. One of the more celebrated result in the context of global geometry of maximal hypersurfaces is the Calabi-Bernstein theorem in the Lorentz-Minkowski spacetime. The non-parametric version of this theorem states that the only entire solutions to the maximal hypersurface equation in the Lorentz- Minkowski spacetime are spacelike a_ne hyperplanes. The present work reviews some of the classical and recent proofs of the theorem for the two dimensional case, as well as several extensions for Lorentzian warped products and other relevant spacetimes. On the other hand the problem of uniqueness of complete maximal hypersurfaces is analysed under the perspective of some new results.