Let us consider the Minkowski spacetime $\mathbb R^{1+2}$ equipped with the metric $$ \eta(t,x) = -(dt)^2+ (dx^1)^2+(dx^2)^2.$$ Let $\Omega$ be a bounded simply connected domain in $\mathbb R^2$ with a smooth boundary representing a surface at time slice $t=0$ and consider its future Cauchy development to be denoted by $D^+(\Omega)$. Finally, let $S= \partial D^+(\Omega)$. It is known that $S$ is a continuous surface.
Is the following statement true? Either $\Omega$ is a disk, or there is a continuous curve $\gamma$ on $S$ with the properties that (i) given any point $p\in \gamma$ there are at least two distinct null geodesic segments $\beta_p$ and $\zeta_p$ lying on $S$ that connect $p$ to $\partial \Omega$ and (ii) writing $f(p)$ and $g(p)$ for the points of intersections of $\beta_p$ and $\zeta_p$ with $\partial \Omega$ respectively, the functions $f:\gamma \to \partial \Omega$ and $g:\gamma \to \partial \Omega$ are surjective.
