Rigidity of circle domains whose boundary has \(\sigma\)-finite linear measure (Q1319405)

Let \(\Omega\) be a circle domain in the Riemann sphere \(\mathbb{C}\) whose boundary has \(\sigma\)-finite linear measure. The authors prove that \(\Omega\) is rigid in the sense that any conformal homeomorphism of \(\Omega\) onto any other circle domain is equal to the restriction of Möbius transform. This beautiful result is strongly related to the Koebe uniformization conjecture and should be valuable for anybody interested in rigidity type theorems.