Hypersurfaces with light-like points

Abstract

Consider a constant mean curvature immersion $F:U(\subset \boldsymbol{R}^n)\to M$ into an arbitrary Lorentzian $(n+1)$-manifold $M$. A point $o\in U$ is called a light-like point if the first fundamental form $ds^2$ of $F$ degenerates at $o$. We denote by $B_F$ the determinant function of the symmetric matrix associated to $ds^2$ with respect to a local coordinate system at $o$. A light-like point $o$ is said to be degenerate if the exterior derivative of $B_F$ vanishes at $o$. We show that if $o$ is a degenerate light-like point, then the image of $F$ contains a light-like geodesic segment of $M$ passing through $f(o)$ (cf.\ Theorem E). This explains why several known examples of constant mean curvature hypersurface in the Lorentz-Minkowski $(n+1)$-space form $\boldsymbol{R}^{n+1}_1$ contain light-like lines on their sets of light-like points, under a suitable regularity condition of $F$. Several related results are also given.