Rigidity results for stochastically complete maximal hypersurfaces in Generalized Robertson-Walker spacetimes
María Á. Medina, José A. S. Pelegrín
TL;DR
The article studies rigidity of stochastically complete maximal hypersurfaces in Generalized Robertson-Walker spacetimes under the Null Convergence Condition. It derives a key relation between the height function $\tau$ and the warping function $f$, showing that stochastic completeness enforces sign constraints on $\frac{f'}{f}(\tau)$ and leads to containment in totally geodesic spacelike slices under NCC with bounded hyperbolic angle. It extends Calabi–Bernstein-type uniqueness to noncompact settings, including Lorentz–Minkowski and de Sitter spacetimes, via curvature-based Bochner formulas and maximum principles. Finally, it proves a Calabi–Bernstein-type theorem for entire maximal graphs in GRW spacetimes with suitable warp-function conditions, ensuring only trivial solutions under completeness and curvature hypotheses.
Abstract
In this article we obtain new rigidity results for stochastically complete maximal hypersurfaces in Generalized Robertson-Walker spacetimes that satisfy the Null Energy Condition. Under appropiate geometric assumptions we prove new parametric uniqueness and nonexistence results as well as obtain a Calabi-Bernstein type result for the maximal hypersurface equation in these ambient spacetimes.