Uncountably many $C^0$ conformally distinct Lorentz surfaces and a finiteness theorem

This paper describes an uncountable family of Lorentz surfaces realized as rectangular regions in the Minkowski 2-plane $E^2_1$. A simple $C^0$ conformal invariant is defined which assigns a different real value to each Lorentz surface in the family. While these surfaces provide uncountably many $C^0$ conformally distinct, bounded, convex subsets of $E^2_1$ which are each symmetric about a properly embedded timelike curve and about a properly embedded spacelike curve, it is shown that there are only 21 $C^0$ conformally distinct, bounded, convex subsets of $E^2_1$ which are symmetric about some null line.