Complete spacelike hypersurfaces immersed in pp-wave spacetimes

Abstract

We study some aspects of the geometry of complete spacelike hypersurfaces immersed into a pp-wave spacetime, namely, into a connected Lorentzian manifold admitting a parallel lightlike vector field. Initially, by applying suitable versions of the classical Hopf and Stokes theorems and a criterion of parabolicity for complete Riemannian manifolds, we obtain sufficient conditions which guarantee that a complete spacelike hypersurface is either maximal, 1-maximal or totally geodesic. As a consequence of these results, we also establish some results of nonexistence concerning such spacelike hypersurfaces. Finally, considering constant mean curvature closed spacelike hypersurfaces immersed in a pp-wave spacetime, we study a notion of stability via the first nonzero eigenvalue of the Laplacian.