Conformally Homeomorphic Lorentz Surfaces need not be Conformally Diffeomorphic

Abstract

A Lorentz surface $\mathcal {L}$ is an ordered pair (S, [h]) where S is an oriented ${C^\infty }$ 2-manifold and [h] the set of all ${C^\infty }$ metrics conformally equivalent to a fixed ${C^\infty }$ Lorentzian metric h on S. (Thus Lorentz surfaces are the indefinite metric analogs of Riemann surfaces.) This paper describes subsets of the Minkowski 2-plane which are conformally homeomorphic, but not even ${C^1}$ conformally diffeomorphic. It also describes subsets of the Minkowski 2-plane which are ${C^j}$ but not ${C^{j + 1}}$ conformally diffeomorphic for any fixed $j = 1,2, \ldots$. Finally, the paper describes a Lorentz surface conformally homeomorphic to a subset of the Minkowski 2-plane, but not ${C^1}$ conformally diffeomorphic to any subset of the Minkowski 2-plane.