A Koebe distortion theorem for quasiconformal mappings in the Heisenberg group
| Autor: | Ben Warhurst, Tomasz Adamowicz, Katrin Fässler |
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| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
Quasiconformal mapping Logarithm Mathematics::Complex Variables Applied Mathematics 010102 general mathematics Derivative 01 natural sciences Distortion (mathematics) symbols.namesake 0103 physical sciences Jacobian matrix and determinant symbols Heisenberg group 010307 mathematical physics 0101 mathematics Operator norm Differential (mathematics) Mathematics |
| Zdroj: | Annali di Matematica Pura ed Applicata (1923 -). 199:147-186 |
| ISSN: | 1618-1891 0373-3114 |
| DOI: | 10.1007/s10231-019-00871-8 |
| Popis: | 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 Koranyi distances. This extends results due to Astala–Gehring, Astala–Koskela, Koskela and Bonk–Koskela–Rohde. |
| Databáze: | OpenAIRE |
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