Quasisymmetric Koebe uniformization

Resumen:

A planar domain  is said to be a circle (resp. slit) domain if every boundary component of  is either a circle (resp. a slit) or a point. In 1918 Paul Koebe proved that every finitely connected planar domain is conformal to a circle domain. The still open Koebe's conjecture from 1908 states that every domain  is conformal to a circle domain. The countable case of Koebe's conjecture was settled in 1993 by He and Schramm.

We study the quasisymmetric analogue of Koebe's conjecture. Informally, a map  between metric spaces is quasisymmetric if it doesn't distort shapes too much. Unlike the conformal case, however, not every domain (even countable one) is quasisymmetric to a circle domain. For a large class of metric spaces we obtain a characterization of those which admit such a Quasisymmetric Koebe Uniformization by circle domains. This is based on joint works with Wenbo Li and Jonathan Rehmert

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