Abstract
An extremal quasiconformal homeomorphisms in a class of homeomorphisms between two CR 3-manifolds is an one which has the least conformal distortion among this class. This paper studies extremal quasiconformal homeomorphisms between CR 3-manifolds which admit transversal CR circle actions. Equivariant $K$-quasiconformal homeomorphisms are characterized by an area-preserving property and the $K$-quasiconformality of their quotient maps on the spaces of $S^1$-orbits. A large family of invariant CR structures on $S^3$ is constructed so that the extremal quasiconformal homeomorphisms among the equivariant mappings between them and the standard structure are completely determined. These homeomorphisms also serve as examples showing that the extremal quasiconformal homeomorphisms between two invariant CR manifolds are not necessarily equivariant.
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