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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 \(\partial U\).

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