Quasiconformal Homeomorphisms Explicitly Determining the Basic Curve Quasi-Invariants

Abstract

The classical Belinskii theorem implies that any sufficiently regular function μ(z) on the extended complex plane C^ with a small C1+α norm generates via the two-dimensional Cauchy integral a quasiconformal automorphism w of C^ with the Beltrami coefficient μ˜=μ+O(∥μ∥2). We consider μ supported in arbitrary bounded quasiconformal disks and show that under appropriate assumptions of μ, this automorphism explicitly provides the basic curvelinear quasi-invariants associated with conformal and quasiconformal maps, advancing an old problem of quasiconformal analysis.

Keywords

• univalent function
• quasiconformal maps and reflections
• quasicircle
• universal Teichmüller space
• the Belinskii theorem
• abelian quadratic differential