A Koebe distortion theorem for quasiconformal mappings in the Heisenberg group

Adamowicz, Tomasz; Fässler, Katrin; Warhurst, Ben

doi:10.1007/s10231-019-00871-8

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Journal article

A Koebe distortion theorem for quasiconformal mappings in the Heisenberg group

Adamowicz, Tomasz The Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland
Fässler, Katrin Department of Mathematics, University of Fribourg, Switzerland - Department of Mathematics and Statistics, University of Jyväskylä, Finland
Warhurst, Ben Institute of Mathematics, University of Warsaw, Poland

2020

Published in:

Annali di Matematica Pura ed Applicata (1923 -). - 2020, vol. 199, no. 1, p. 147–186

English We prove a Koebe distortion theorem for the average derivative of a quasiconformal mapping between domains in the sub-Riemannian Heisenberg group H1. Several auxiliary properties of quasiconformal mappings between subdomains of H1 are proven, including BMO estimates for the logarithm of the Jacobian. Applications of the Koebe theorem include diameter bounds for images of curves, comparison of integrals of the average derivative and the operator norm of the horizontal differential, as well as the study of quasiconformal densities and metrics in domains in H1. The theorems are discussed for the sub-Riemannian and the Korányi distances. This extends results due to Astala–Gehring, Astala–Koskela, Koskela and Bonk–Koskela– Rohde.

Faculty

Faculté des sciences et de médecine

Department

Département de Mathématiques

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