TWO GEOMETRIC CHARACTERISTICS OF QUASICIRCLES

In this paper,the following two results are obtained:(1) If Γ is a Jordan curve of 2,∞∈Γ,then Γ is a quasicircle if and only if there exists a constant k,1≤k<+∞,such that for any four points z 1,z 2,w 1,w 2∈Γ,there exists a k-quasiconformal mapping h of 2 with h(∞)=∞,h(Γ)=Γ and h(z j)=w j(j=1,2).(2)If Γ is a Jordan curve of 2, then Γ is a quasicircle if and only if Γ is a bounded circular distortion curve.

The Faber Operator Acting on BMOA,BMO-Teichmüller Space and Chord-arc Curves

After reviewing Grunsky operator and Faber operator acting on Dirichlet space,we discuss the boundedness of Faber operator on BMOA,a new subject which turns out to be closely related to the BMO theory of the universal Teichmüller space.In particular,we show that the Faber operator acts as a bounded operator on BMOA if the symbol conformal map stays nearly to the base point in the BMO-Teichmüller space.Meanwhile,we obtain several results on quasiconformal mappings,BMOTeichm¨uller space and chord-arc curves as well.As by-products,this provides a complex analysis approach to the boundedness of the Cauchy integral acting on BMO functions on a chord-arc curve near to the unit circle in the BMO-Teichmüller space.

Bounded type Siegel disks of finite type maps with few singular values Dedicated to the Memory of Professor Lei Tan

Let U be an open subset of the Riemann sphere C. We give sufficient conditions for which a finite type map f : U →C with at most three singular values has a Siegel disk compactly contained in U and whose boundary is a quasicircle containing a unique critical point. The main tool is quasiconformal surgery à la Douady-Ghys-Herman-(S′)wia tek. We also give sufficient conditions for which, instead, ? has not compact closure in U. The main tool is the Schwarzian derivative and area inequalities à la Graczyk-(S′)wia tek.

On the Dynamics of a Family of Entire Functions

We prove that for any bounded type irrational number 0 〈 0 〈 1, the boundary of theSiegel disk of fα（z） = e^2πiθsin（z） ＋ αsin^3（z）, a E C, which centered at the origin, is a quasicircle passing through 2, 4 or 6 critical points of fα counted with multiplicity.

Generalized Beurling-Ahlfors'Theorem

A generalized Beurling-Ahlfors’ Theorem for the self homeomorphism f of the upper half plane with the sphere dilatation H(z,f) L (H) is established and the property of weighted quasi-isometry for the generalized Beurling-Ahlfors’ extension is studied.

Sewing Homeomorphism and Conformal Invariants

This paper is devoted to the study of some fundamental properties of the sewing home- omorphism induced by a Jordan domain. In particular, using conformal invariants such as harmonic measure, extremal distance, and reduced extremal distance, we give several necessary and sumcient conditions for the sewing homeomorphism to be bi-Lipschitz or bi-Holder. Furthermore, equivalent conditions for a Jordan curve to be a quasicircle are also obtained.

Notes on Univalent Functions with Quasiconforml Extensions

We make improvements to some of the constants concerned with univalent functions with quasiconformal extensions that have appeared in the literature.