Smoothability of the Conformal Boundary of a Lorentz Surface Implies 'Global Smoothability'

In 1985, Kulkarni defined the conformal boundary of a simply connected and time-oriented Lorentzian surface . He also introduced a notion of 'smoothability' of this boundary, depending only on local properties of . In this paper we show that smoothability of is in fact a global property of . In doing so, we classify Lorentzian surfaces with smoothable boundaries up to conformal homeomorphism. To be specific, suppose that the Minkowski plane E ² ₁ is the x,y-plane with metric dxdy. Our main theorem states that if is smoothable then is conformally homeomorphic to the interior U of a Jordan curve in E ² ₁ that is locally the graph of a continuous function over either the x-axis or the y-axis at each point of U.