On maximal hypersurfaces in Lorentz manifolds admitting a parallel lightlike vector field

Abstract

We study constant mean curvature spacelike hypersurfaces and in particular maximal hypersurfaces immersed in pp-wave spacetimes satisfying the timelike convergence condition. We prove the non-existence of compact spacelike hypersurfaces whose constant mean curvature is non-zero and also that every compact maximal hypersurface is totally geodesic. Moreover, we give an extension of the classical Calabi–Bernstein theorem to this class of pp-wave spacetimes.