# Removability and non-injectivity of conformal welding

## TL;DR

This paper constructs a specific Jordan curve and a conformal homeomorphism that maps it onto itself, illustrating non-removability and non-injectivity phenomena in conformal welding, with implications for complex analysis.

## Contribution

It provides explicit examples of non-removable Jordan curves and conformal homeomorphisms, advancing understanding of conformal welding's non-injectivity and flexibility.

## Key findings

- Existence of a non-removable Jordan curve with zero area
- Construction of a conformal homeomorphism mapping the curve onto itself
- Demonstration of non-injectivity in conformal welding

## Abstract

We construct a (non-removable) Jordan curve $\Gamma$ and a non-M\"{o}bius homeomorphism of the Riemann sphere which is conformal on the complement of $\Gamma$ and maps the curve $\Gamma$ onto itself. The curve is flexible in the sense of Bishop and may be taken to have zero area. The existence of such curves and conformal homeomorphisms is closely related to the non-injectivity of conformal welding.