A koebe distortion theorem for quasiconformal mappings in the Heisenberg group. (English) Zbl 1435.30160
Summary: We prove a Koebe distortion theorem for the average derivative of a quasiconformal mapping between domains in the sub-Riemannian Heisenberg group \({\mathbb{H}}^1 \). Several auxiliary properties of quasiconformal mappings between subdomains of \({\mathbb{H}}^1\) 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 \({\mathbb{H}}^1 \). 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.
MSC:
| 30L10 | Quasiconformal mappings in metric spaces |
| 30C65 | Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations |
| 30F45 | Conformal metrics (hyperbolic, Poincaré, distance functions) |
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