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Quasiconformal maps of cylindrical domains. (English) Zbl 0674.30017

Contrary to the plane case there is no simple topological or metric characterization of domains D in \({\mathbb{R}}^ 3\) which are quasiconformally equivalent to the unit ball \(B^ 3\). In this paper necessary and sufficient conditions for cylindrical domains \(D=G\times {\mathbb{R}}\) are given. Here G is a simply connected plane domain.
For example, D is quasiconformally equivalent to \(B^ 3\) iff G is finitely connected on the boundary and satisfies an internal chord arc condition or G is BLD-homeomorphic to a disk or to a half plane. The internal chord arc condition means that if u and v are points on the prime end boundary \(\partial^*G\), then \(\sigma (u,v)\leq c\delta_ G(u,v)\) where \(\sigma\) denotes the length of the shorter component of \(\partial^*G\setminus \{u,v\}\) and \(\delta_ G\) is the internal distance in the prime end compactification of G. A BLD-homeomorphism is simply a locally L-bilipschitz homeomorphism for some \(L\geq 1\). The proofs use extensively various John-type properties of G [O. Martio and J. Sarvas, Ann. Acad. Sci. Fenn., Ser. A I 4, 383-401 (1979; Zbl 0406.30013)].
Reviewer: O.Martio

MSC:

30C62 Quasiconformal mappings in the complex plane
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