Uncountably many 𝐶⁰ conformally distinct Lorentz surfaces and a finiteness theorem

Abstract

This paper describes an uncountable family of Lorentz surfaces realized as rectangular regions in the Minkowski 2-planeE12E^2_1. A simpleC0C^0conformal invariant is defined which assigns a different real value to each Lorentz surface in the family. While these surfaces provide uncountably manyC0C^0conformally distinct, bounded, convex subsets ofE12E^2_1which are each symmetric about a properly embedded timelike curve and about a properly embedded spacelike curve, it is shown that there are only 21C0C^0conformally distinct, bounded, convex subsets ofE12E^2_1which are symmetric about some null line.