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

Consider a surface $S$ immersed in the Lorentz-Minkowski 3-space $\boldsymbol R^3_1$. A complete light-like line in $\boldsymbol R^3_1$ is called an entire null line on the surface $S$ in $\boldsymbol R^3_1$ 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 $\boldsymbol R^2$, then it must be a light-like plane. Our example is critical in the sense that it is defined on a certain non-convex domain.