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Smoothability of the conformal boundary of a Lorentz surface implies ‘global smoothability’. (English) Zbl 1003.53053
Author’s abstract: In [Proc. R. Soc. Lond., Ser. A 401, 117-130 (1985; Zbl 0574.53040)] R. Kulkarni defined the conformal boundary $$\partial{\mathcal L}$$ of a simply connected and time-oriented Lorentzian surface $${\mathcal L}$$. He also introduced a notion of ‘smoothability’ of this boundary, depending only on local properties of $$\partial{\mathcal L}$$. In this paper we show that smoothability of $$\partial{\mathcal L}$$ is in fact a global property of $${\mathcal L}$$. 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 $$\partial{\mathcal L}$$ is smoothable then $${\mathcal L}$$ 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 $$\partial U$$.

##### MSC:
 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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