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

TL;DR

This paper investigates maximal and constant mean curvature spacelike hypersurfaces in pp-wave spacetimes with a parallel lightlike vector, establishing non-existence results and extending classical theorems in this geometric context.

Contribution

It provides new non-existence results for compact hypersurfaces with non-zero mean curvature and extends the Calabi-Bernstein theorem to pp-wave spacetimes.

Findings

Non-existence of compact spacelike hypersurfaces with non-zero mean curvature

Every compact maximal hypersurface is totally geodesic

Extension of the Calabi-Bernstein theorem to pp-wave spacetimes

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.