Conformally natural reflections in Jordan curves with applications to Teichmüller spaces

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Abstract

In his fundamental paper [1] Ahlfors initiated the study of quasiconformal reflections. Using the results of Beurling and Ahlfors [6] he showed that every quasicircle that passes through ∞ permits a quasiconformal reflection that satisfies a global Lipschitz condition (with exponent one) in the plane. Using that result he proved by a direct construction that the Bers embedding of the universal Teichmüller space has an open image. Lipschitz continuous quasiconformal reflections also play a crucial role in Bers’s subsequent proof (see [4] and [5]) that for any Teichmüller space the Bers embedding not only has an open image but also has local cross sections. That result is one of the cornerstones of Teichmüller theory.