Conformally homeomorphic Lorentz surfaces need not be conformally diffeomorphic

Abstract

A Lorentz surface L \mathcal {L} is an ordered pair (S, [h]) where S is an oriented C ∞ {C^\infty } 2-manifold and [h] the set of all C ∞ {C^\infty } metrics conformally equivalent to a fixed C ∞ {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 {C^1} conformally diffeomorphic. It also describes subsets of the Minkowski 2-plane which are C j {C^j} but not C j + 1 {C^{j + 1}} conformally diffeomorphic for any fixed j = 1 , 2 , … 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 {C^1} conformally diffeomorphic to any subset of the Minkowski 2-plane.