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Conformal Geometry and Dynamics

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Geometric construction of quasiconformal mappings in the Heisenberg group


Author: Robin Timsit
Journal: Conform. Geom. Dyn. 22 (2018), 99-140
MSC (2010): Primary 30L10, 30C75
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.


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Additional 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

DOI: https://doi.org/10.1090/ecgd/323
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
Article copyright: © Copyright 2018 American Mathematical Society

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