An Approach to Studying Quasiconformal Mappings on Generalized Grushin Planes

TL;DR

This paper establishes quasisymmetric equivalence between the complex plane and generalized Grushin planes, developing an analytic framework for quasisymmetry and characterizing conformal mappings in these spaces.

Contribution

It introduces an analytic definition of quasisymmetry on generalized Grushin planes and links it to conformal mappings, expanding understanding of geometric function theory in these spaces.

Findings

Complex plane and generalized Grushin planes are quasisymmetrically equivalent.

An analytic definition of quasisymmetry on $G_r$ spaces is developed.

Characterizations of conformal mappings on generalized Grushin planes are provided.

Abstract

We demonstrate that the complex plane and a class of generalized Grushin planes $G_r$, where $r$ is a function satisfying specific requirements, are quasisymmetrically equivalent. Then using conjugation we are able to develop an analytic definition of quasisymmetry for homeomorphisms on $G_r$ spaces. In the last section we show our analytic definition of quasisymmetry is consistent with earlier notions of conformal mappings on the Grushin plane. This leads to several characterizations of conformal mappings on the generalized Grushin planes.