Geometric construction of quasiconformal mappings in the Heisenberg group
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- by Robin Timsit
- Conform. Geom. Dyn. 22 (2018), 99-140
- DOI: https://doi.org/10.1090/ecgd/323
- Published electronically: August 22, 2018
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Abstract:
In this paper, we are interested in the construction of quasiconformal mappings between domains of the Heisenberg group $\mathbf {H}$ that minimize a mean distortion functional. We propose to construct such mappings by considering a corresponding problem between domains of Poincaré half-plane $\mathbb H$ and then, lifting every of its solutions to $\mathbf H$. The first map we construct is a quasiconformal map between two cylinders. We explain the method used to find it and prove its uniqueness up to rotations. Then, we give geometric conditions which ensure that a minimizer (in $\mathbf {H}$) comes as a lift of a minimizer between domains of $\mathbb H$. Finally, as a non-trivial example of the generalization, we manage to reconstruct the map from [Ann. Acad. Sci. Fenn. Math. 38 (2013), pp. 149–180] between two spherical annuli and prove its uniqueness as a minimizer.References
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Bibliographic Information
- Robin Timsit
- Affiliation: Sorbonne Université, Université Paris Diderot, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG, F-75005, Paris, France
- Email: robin.timsit@imj-prg.fr
- Received by editor(s): December 6, 2016
- Received by editor(s) in revised form: May 29, 2017, and March 23, 2018
- Published electronically: August 22, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Conform. Geom. Dyn. 22 (2018), 99-140
- MSC (2010): Primary 30L10, 30C75
- DOI: https://doi.org/10.1090/ecgd/323
- MathSciNet review: 3845546