Geometric construction of quasiconformal mappings in the Heisenberg group

TL;DR

This paper develops a geometric method to construct quasiconformal mappings in the Heisenberg group that minimize mean distortion, using a related problem in the Poincaré half-plane, and demonstrates its application through explicit examples.

Contribution

It introduces a novel geometric construction approach for quasiconformal maps in the Heisenberg group, including explicit examples and uniqueness conditions.

Findings

Constructed a quasiconformal map between two cylinders with proven uniqueness.

Provided geometric conditions ensuring the minimality of the constructed mappings.

Reconstructed a known map between spherical annuli as a non-trivial example.

Abstract

In this paper, we are interested in the construction of quasiconformal mappings between domains of the Heisenberg group H that minimise a mean distortion functional. We propose to construct such mappings by considering a corresponding problem between domains of Poincar\'e half-plane $\mathbb H$. 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 for the construction to be the only way to find such minimizers. Eventually, as a non trivial example of the generalisation, we manage to reconstruct the map from \cite{BFP} between two spherical annuli.