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Conformal welding and Koebe’s theorem. (English) Zbl 1144.30007

The paper is aimed to prove several results stating that every homeomorphism of the unit circle \(\mathbb T\) is close to a conformal welding. Let \(\mathbb D\) be the open unit disk, and \(\mathbb D^*\) the complement of the closure \(\overline{\mathbb D}\) of \(\mathbb D\). Given a closed Jordan curve \(\Gamma\), let \(f: \mathbb D\to\Omega\) and \(g: \mathbb D^*\to\Omega^*\) be conformal maps onto the bounded and unbounded complementary components of \(\Gamma\) respectively. Then \(h=g^{-1}\circ f: \mathbb T\to\mathbb T\) is a homeomorphism called a conformal welding. The map \(\Gamma\to h\) from closed curves to circle homeomorphisms is neither onto nor one-to-one. Additionally, \(h\) is called a generalized conformal welding on \(E\subset\mathbb T\) if there are conformal maps \(f: \mathbb D\to\Omega\), \(g: \mathbb D^*\to\Omega^*\) onto disjoint domains such that \(f\) has radial limits on \(E\), \(g\) has radial limits on \(h(E)\) and these limits satisfy \(f=g\circ h\) on \(E\). Let \(| E| \) denote the Lebesgue measure (normalized so that \(| \mathbb T| =1\)) of \(E\subset\mathbb T\) and \(\text{cap}(E)\) its logarithmic capacity.
The paper is organized as a long way to prove the main Theorem 1: Given any orientation-preserving homeomorphism \(h: \mathbb T\to\mathbb T\) and any \(\varepsilon>0\), there are a set \(E\subset\mathbb T\) with \(| E| +| h(E)| <\varepsilon\) and a conformal welding homeomorphism \(H: \mathbb T\to\mathbb T\) such that \(h(x)=H(x)\) for all \(x\in\mathbb T\setminus E\). In particular, every such \(h\) is a generalized conformal welding on a set \(E\) with Lebesgue measure as close to 1 as we wish.
The long proof way consists of several theorems many of which are of independent interest. For example, Corollary 6: Suppose \(h: \mathbb T\to\mathbb T\) is a orientation-preserving homeomorphism such that \(E\) has zero logarithmic capacity if and only if \(h(E)\) does. Then \(h\) is a generalized conformal welding on \(\mathbb T\setminus F\), where \(F\) has zero logarithmic capacity.
Corollary 7: Suppose \(h: \mathbb T\to\mathbb T\) is a orientation-preserving homeomorphism such that \(| h(F)| =| h^{-1}(F)| =0\) provided \(\text{cap}(F)=0\). Then \(h\) is a generalized conformal welding on a set of \(E\) such that both \(E\) and \(h(E)\) have full Lebesgue measure. These results were conjectured by D. Hamilton.
The proofs are based on Koebe’s circle domain theorem. Besides, the author gives a new proof of the well known fact that quasisymmetric maps are conformal weldings. It is worth to notice that one of the main proof steps (Theorem 3) proposes a condition for a homeomorphism \(h\) to be a conformal welding in terms of \(h\) being sufficiently “wild” while previously known criteria say \(h\) is a welding if it is sufficiently “nice”, e.g., \(h\) is quasisymmetric or some weakening of quasisymmetric.

MSC:

30C20 Conformal mappings of special domains
30C62 Quasiconformal mappings in the complex plane
30C85 Capacity and harmonic measure in the complex plane
37E10 Dynamical systems involving maps of the circle
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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