Conformal mappings on the Grushin plane

TL;DR

This paper explores conformal mappings in the Grushin plane, characterizing their properties via Sobolev mappings, linking them to complex plane conformality, and discussing applications like length-distortion and extension theorems.

Contribution

It introduces new characterizations of conformal mappings on the Grushin plane and connects them with classical conformality through the Meyerson map.

Findings

Characterizations of conformal mappings via Sobolev spaces

Connection between Grushin plane and complex plane conformality

Applications to length-distortion estimates and extension theorems

Abstract

We study conformal mappings in the Grushin plane and provide a number of their characterizations in terms of the Sobolev mappings and their geometry. Furthermore, we connect conformality on the Grushin plane with conformality on the complex plane by using the Meyerson map. Among applications we discuss admissible curves and length-distortion estimates in the Grushin plane, as well as the Carath\'eodory extension theorem.