Is it correct that a disk endowed with a metric of Lorentzian signature (smooth up to the boundary) is always conformally equivalent to some simply connected domain on Minkowski plane (with the metric induced from Minkowski)?
Here is the main idea. Suppose that the metric extends smoothly to the boundary. Such a metric has two null directions at each point. Follow one until you reach the boundary, and then switch to the other one. Watch each boundary point move, according the first null direction, and then watch it move according to the second one. These are two continuous maps on the boundary. You can pretty easily come up with examples for which the resulting dynamical system is quite messy. I think that Vladimir Arnol'd once said he studied these dynamical systems in his doctoral thesis. Global invariants of these dynamical systems prove that there is an infinite dimensional moduli space of Lorentzian metrics on the closed disk. The ones on the open disk can get even worse.