TAILIEUCHUNG - Đề tài " Conformal welding and Koebe’s theorem "

It is well known that not every orientation-preserving homeomorphism of the circle to itself is a conformal welding, but in this paper we prove several results which state that every homeomorphism is “almost” a welding in a precise way. The proofs are based on Koebe’s circle domain theorem. We also give a new proof of the well known fact that quasisymmetric maps are conformal weldings. 1. Introduction Let D ⊂ R2 be the open unit disk, let D∗ = S 2 \D and let T = ∂D = ∂D∗ be the unit circle. . | Annals of Mathematics Conformal welding and Koebe s theorem By Christopher J. Bishop Annals of Mathematics 166 2007 613 656 Conformal welding and Koebe s theorem By Christopher J. Bishop Abstract It is well known that not every orientation-preserving homeomorphism of the circle to itself is a conformal welding but in this paper we prove several results which state that every homeomorphism is almost a welding in a precise way. The proofs are based on Koebe s circle domain theorem. We also give a new proof of the well known fact that quasisymmetric maps are conformal weldings. 1. Introduction Let D c R2 be the open unit disk let D S2 D and let T dD dD be the unit circle. Given a closed Jordan curve r let f D Q and g D Q be conformal maps onto the bounded and unbounded complementary components of r respectively. Then h g-1 o f T T is a homeomorphism. Moreover any homeomorphism arising in this way is called a conformal welding. The map r h from closed curves to circle homeomorphisms is well known to be neither onto nor 1-to-1 see Remarks 1 and 2 but in this paper we will show it is almost onto every h is close to a conformal welding and far from 1-to-1 there are h s which correspond to a dense set of r s . We say that h is a generalized conformal welding on the set E c T if there are conformal maps f D Q g D Q onto disjoint domains such that f has radial limits on E g has radial limits on h E and these limits satisfy f g o h on E. Generalized conformal welding was invented by David Hamilton in 19 see his papers 20 and 21 for applications to Kleinian groups and Julia sets . For E c T let E denote its Lebesgue measure normalized so that T 1 and cap E its logarithmic capacity see 2 . Theorem 1. Given any orientation-preserving homeomorphism h T T and any e 0 there are a set E c T with E h E e and a conformal welding homeomorphism H T T such that h x H x for all x E T E. The author is partially supported by NSF Grant DMS 0705455. 614 CHRISTOPHER J. BISHOP In particular .

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