Regularity And Extremality Of Quasiconformal Homeomorphisms On CR 3-Manifolds

Abstract

This paper first studies the regularity of conformal homeomorphisms on smooth locally embeddable strongly pseudoconvex CR manifolds. Then moduli of curve families are used to estimate the maximal dilatations of quasiconformal homeomorphisms. On certain CR 3-manifolds, namely, CR circle bundles over flat tori, extremal quasiconformal homeomorphisms in some homotopy classes are constructed. These extremal mappings have similar behaviors to Teichm\"uller mappings on Riemann surfaces.