Space-like maximal surfaces containing entire null lines in Lorentz-Minkowski 3-space

Abstract

Consider a surface $S$ immersed in the Lorentz-Minkowski 3-space $\mathbf{R}^{3}_{1}$. A complete light-like line in $\mathbf{R}^{3}_{1}$ is called an \textit{entire null line} on the surface $S$ in $\mathbf{R}^{3}_{1}$ if it lies on $S$ and consists of only null points with respect to the induced metric. In this paper, we show the existence of embedded space-like maximal graphs containing entire null lines. If such a graph is defined on a convex domain in $\mathbf{R}^{2}$, then it must be contained in a light-like plane (cf. Remark~3.3). Our example is critical in the sense that it is defined on a certain non-convex domain.