Calabi–Bernstein-Type Problems in Lorentzian Geometry

Abstract

The study of maximal hypersurfaces in Lorentzian manifolds is an interesting mathematical problem, which connects differential geometry, nonlinear partial differential equations, and certain problems in mathematical relativity. One of the more celebrated results in the context of global geometry of maximal hypersurfaces is the Calabi–Bernstein theorem in the Lorentz–Minkowski spacetime. The nonparametric version of this theorem states that the only entire solutions to the maximal hypersurface equation in the Lorentz–Minkowski spacetime are spacelike affine 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 analyzed under the perspective of some new results.