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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Geometric construction of quasiconformal mappings in the Heisenberg group
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by Robin Timsit PDF
Conform. Geom. Dyn. 22 (2018), 99-140 Request permission

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
  • 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