Quasiconformal mappings on the Grushin plane

TL;DR

This paper characterizes quasisymmetric, metrically quasiconformal, and geometrically quasiconformal mappings on the Grushin plane, establishing their equivalence and exploring related Euclidean quasiconformal properties.

Contribution

It proves the equivalence of different quasiconformal notions on the Grushin plane and analyzes the properties of quasisymmetric parametrizations.

Findings

Quasisymmetric maps on the Grushin plane are equivalent to metrically and geometrically quasiconformal maps.

A quasisymmetric parametrization of the Grushin plane by the Euclidean plane must be geometrically quasiconformal.

Euclidean quasiconformal maps exhibit absolute continuity on almost every compact curve, unlike in the Grushin case.

Abstract

We prove that a self-homeomorphism of the Grushin plane is quasisymmetric if and only if it is metrically quasiconformal and if and only if it is geometrically quasiconformal. As the main step in our argument, we show that a quasisymmetric parametrization of the Grushin plane by the Euclidean plane must also be geometrically quasiconformal. We also discuss some aspects of the Euclidean theory of quasiconformal maps, such as absolute continuity on almost every compact curve, not satisfied in the Grushin case.