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Bulletin of the London Mathematical Society Advance Access originally published online on December 11, 2007
Bulletin of the London Mathematical Society 2007 39(6):962-972; doi:10.1112/blms/bdm101
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© 2007 London Mathematical Society

Characterisation of plane regions that support quasiconformal mappings to their domes

A. Marden

Department of Mathematics
University of Minnesota
Minneapolis, MN 55455
USA

V. Markovic

Department of Mathematics
Stony Brook University
Stony Brook, NY 11794
USA
and
Mathematics Institute
University of Warwick
Coventry CV4 7A1
United Kingdom
V.Markovic@warwick.ac.uk

Received 12 February 2007.

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.

2000 Mathematics Subject Classification 30F (primary), 30C62, 30F40, 30F60, 32G05 (secondary).


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