Hypersurfaces with light-like points in a Lorentzian manifold II

Abstract

In the authors' previous work, it was shown that if a zero mean curvature C⁴-differentiable hypersurface in an arbitrarily given Lorentzian manifold admits a degenerate light-like point, then the hypersurface contains a light-like geodesic segment passing through the point. The purpose of this paper is to point out that the same conclusion holds with just C³-differentiability of the hypersurfaces.