Uniqueness for John-domain uniformization

Determine whether, for every John domain, any two conformal maps from the domain onto circle domains differ by the restriction of a Möbius transformation.

Background

The existence problem asks whether a domain can be conformally mapped onto a circle domain, whereas the uniqueness problem asks whether such a map is unique up to postcomposition with a Möbius transformation. Uniqueness is known for finitely connected and countably connected domains, and it has also been established for certain geometrically controlled circle domains.

The paper proves existence of circle-domain uniformization for John domains, but explains that the resulting circle domain need not itself retain the geometric properties commonly used in known uniqueness theorems. It therefore explicitly leaves the uniqueness question unresolved for John domains.