Characterisation of plane regions that support quasiconformal mappings to their domes

We prove that the nearest point retraction of a region P, not the whole plane, to its dome is long-range bilipschitz if and only if P is uniformly perfect. From this we prove that P uniformly perfect is necessary and sufficient for the existence of a K-quasiconformal map from P to its dome which extends to be the identity on the boundary and is finite distance to the nearest point retraction. Thus our work extends the classic theorem of Sullivan for simply-connected regions to regions of arbitrary connectivity. In particular, our study results in a simple, transparent proof of the original theorem.