This paper describes subsets of the Minkowski 2-plane which are onformally 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 \(j=1,2,\dots\). 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.
In the paper, examples like the regions \(R_f=\{(x,y)\mid 0<y<f(x),0<x<1\}\), with a strictly increasing \(C^j\) bijection \(f: [0,1]\to[0,1]\), are considered in order to establish some of the results: a conformal diffeomorphism (homeomorphism) is defined to preserve isotropic lines – as the \(C^j\) maps \(F:R_f\to R_{id}\), \((x,y)\mapsto (f(x),y)\) do.