The author constructs a non-removable Jordan curve \(\Gamma\) and a non-Möbius homeomorphism of the Riemann sphere \(\hat{\mathbb C}\) which is conformal on the complement of \(\Gamma\) and \(F(\Gamma)=\Gamma\). Denote \(\mathbb D:=\{z\in\mathbb C:|z|<1\}\), \(\mathbb D^*:=\{z\in\mathbb C:|z|>1\}\) and \(\mathbb T=\partial\mathbb D\). Given a closed Jordan curve \(\Gamma\), let \(f\) and \(g\) be conformal maps from \(\mathbb D\) and \(\mathbb D^*\) onto \(\Omega\) and \(\Omega^*\) which are bounded and unbounded complementary components of \(\Gamma\), respectively. Extend \(f\) and \(g\) continuously on the closure of their respective domains, so that \(h_{\Gamma}=g^{-1}\circ f:\mathbb T\to\mathbb T\) defines an orientation-preserving homeomorphism of \(\mathbb T\), called the conformal welding homeomorphism of \(\Gamma\).
A compact set \(E\subset\mathbb C\) is said to be conformally removable if every \(F\in\text{CH}(E)\) is Möbius, where \(\text{CH}(E)\) is the collection of homeomorphisms \(F:\hat{\mathbb C}\to\hat{\mathbb C}\) which are conformal on \(\hat{\mathbb C}\setminus E\). If a Jordan curve \(\Gamma\) is conformally removable, then \(\Gamma\) uniquely corresponds to its conformal welding homeomorphism, modulo Möbius equivalence. The author aims to answer the question whether the converse holds.
Question 1.2: If \(\Gamma\) is a non-removable Jordan curve, does there necessarily exist another curve having the same welding homeomorphism, but which is not a Möbius image of \(\Gamma\)?
The author proves the affirmative.
Theorem 1.3: There exists a Jordan curve \(\Gamma\) and a non-Möbius homeomorphism \(F:\hat{\mathbb C}\to\hat{\mathbb C}\) conformal on \(\hat{\mathbb C}\setminus\Gamma\) such that \(F(\Gamma)=\Gamma\). Moreover, the curve \(\Gamma\) may be taken to have zero area.
The construction is based on Bishop’s notion of so-called flexible curves \(\Gamma\) defined by the two conditions: given any Jordan curve \(\tilde\Gamma\) and \(\epsilon>0\), there exists \(F\in\text{CH}(\Gamma)\) such that \(d(F(\Gamma),\tilde\Gamma)<\epsilon\), where \(d\) is the Hausdorff distance, and, given points \(z_1,z_2\) in each complementary component of \(\Gamma\) and points \(w_1,w_2\) in each complementary component of \(\tilde\Gamma\), we can choose \(F\) so that \(F(z_1)=w_1\) and \(F(z_2)=w_2\).
Finally, the author discusses the non-injectivity of conformal welding for curves of positive area and poses open problems related to Question 1.2.

Reviewer: Dmitri V. Prokhorov (Saratov)

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.