NUMERICAL ANALYSIS OF QUASICONFORMAL MAPPINGS BY CIRCLE PACKINGS

Thurston proposed that conformal mappings can be approximated by circle packing isomorphisms and the approach can be implemented efficiently. Based on the circle packing methods the rate of convergence of approximating solutions for quasiconformal mappings in the plane is discussed.

1-quasiconformal mappings on a (2,2)-type quadric

This paper gets the Beltrami equations satisfied by a 1-quasiconformal mapping, which are exactly CR or anti-CR equations on （2,2）-type quadric Q0. This means a 1-quasiconformal mapping on Q0 is CR or anti-CR. This reduces the determination of 1- quasiconformal mappings to a problem on the theory of several complex analysis. The result about the group of CR automorphisms is used to determine the unit component of group of 1-quasiconformal mappings.

PARAMETRIC REPRESENTATIONS OF QUASICONFORMAL MAPPINGS

In this article,we first give two simple examples to illustrate that two types of parametric representation of the family ofΣ0 K have some gaps.Then we also find that the area derivative formula(1.6),which is used to estimate the area distortion ofΣ0 K,cannot be derived from[6],but that formula still holds forΣ0 K through our amendatory parametric representation for the one obtained by Eremenko and Hamilton.

Extremal Quasiconformal Mappings Compatible with Fuchsian Groups

For every torsion free Fuchsian group F with Poincaré’s -operator norm ║г║=1, it is proved that there exists an extremal Beltrami differential of F which is also extremal under its own boundary correspondence. It is also proved that the imbedding of the Teichmüller space T(Γ) into the universal Teichmüller space T is not a global isometry unless Γ is an elementary group.

On Locally Quasiconformal Mappings

A homeomorphism w=f（z） of a domain D is called a locally quasiconformal mapping, if for each subdomain D＇ of D with ＇D, the restriction of f（z） on D＇ is a quasiconformal mapping. We give some conditions for a measurable function μ（z） on the unit disc to be the complex dilatation of a locally quasiconformal mapping f which can be homeomorphically extended to the closed unit disc.

Quadrilaterals, extremal quasiconformal extensions and Hamilton sequences

The relationship between Strebel boundary dilatation of a quasisymmetric function h of the unit circle and the dilatation indicated by the change in the modules of the quadrilaterals with vertices on the circle intrigues many mathematicians. It had been a conjecture for some time that the dilatations Ko（h） and K1（h） of h are equal before Anderson and Hinkkanen disproved this by constructing concrete counterexamples. The independent work of Wu and of Yang completely characterizes the condition for Ko（h） = K1 （h） when h has no substantial boundary point. In this paper, we give a necessary and sufficient condition to determine the equality for h admitting a substantial boundary point.

ON THE CONVERGENCE OF CIRCLE PACKINGS TO THE QUASICONFORMAL MAP

Rodin and Sullivan （1987） proved Thurston＇s conjecture that a scheme based on the Circle Packing Theorem converges to the Riemann mapping, thereby proved a refreshing geometric view of the Riemann Mapping Theorem. Naturally, we consider to use the ellipses to pack the bounded simply connected domain and obtain similarly a sequence simplicial homeomorphism between the ellipse packing and the circle packing. In this paper, we prove that these simplicial homeomorphism approximate a quasiconformal mapping from the bounded simply connected domain onto the unit disk with the modulus of their complex dilatations tending to 1 almost everywhere in the domain when the ratio of the longer axis and shorter axis of the ellipse tending to ∞.

REMARKS ON JOHN DISKS

Let D■R2 be a Jordan domain,D*=R2\D,the exterior of D.In this article,the authors obtained the following results:(1)If D is a John disk,then D is an outer linearly locally connected domain;(2)If D* is a John disk,then D is an inner linearly locally connected domain;(3)A homeomorphism f:R 2 →R 2 is a quasiconformal mapping if and only if f(D)is a John disk for any John disk D■R 2 ;and(4)If D is a bounded quasidisk,then D is a John disk,and there exists an unbounded quasidisk which is not a John disk.

THE VALUE DISTRIBUTION OF GAUSS MAPS OF IMMERSED HARMONIC SURFACES WITH RAMIFICATION

Motivated by the result of Chen-Liu-Ru[1],we investigate the value distribution properties for the generalized Gauss maps of weakly complete harmonic surfaces immersed in R^(n) with ramification,which can be seen as a generalization of the results in the case of the minimal surfaces.In addition,we give an estimate of the Gauss curvature for the K-quasiconfomal harmonic surfaces whose generalized Gauss map is ramified over a set of hyperplanes.

A NEW PROOF OF THE DELTA INEQUALITY

The purpose of this paper is to give a relatively elementary and direct proof of the Delta Inequality, which plays a very important role in the study of the extremal problem of quasiconformal mappings.

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.

关于双曲三角形间的拟共形映射

Quasiconformal mappings between hyperbolic triangles are considered.We give an explicit estimate of the dilation of the quasiconformal mappings,which generalizes Bishop's results.

On Mori's Theorem in Quasiconformal Theory

In this paper,it is proved that the Mori constant,the Hlder coeflicient of the family QCK（B）of K-quasiconformal self-mappings of the unit disk B with the origin fixed,is at most 461-1/K.It is also shown that the Hlder coefficient of the restriction of a map f∈QCK（B）to the disk|z|≤sin 25.7°is at most 161-1/K.

An Explicit Example of a Reich Sequence for a Uniquely Extremal Quasiconformal Mapping

An explicit example of a Reich sequence for a uniquely extremal quasiconformal mapping in a borderline case between uniqueness and non-uniqueness is given.

Some Remarks on the Geodesics in Infinite Dimensional Teichmuller Spaces

Let B denote the set of those points p in an infinite dimensional Teichmiiller space such that a geodesic between the zero point and p is not unique. Some properties of the set B are given.

On the Compactness of Homeomorphisms with Integrable Dilatations

In this paper,a distortion theorem and some equicontinuity and compactness theorems are obtained for the homeomorphism f with the sphere dilatation H（x,f）∈Lloc（D）.

The ACL Property of Homeomorphisms under Weak Conditions

Suppose that D is a domain in (n 2) and y= f(x): D → R^n is a homeomorphism. We prove that if the modulus dilatation K(x, f) satisfies the condition A then f(x) is ACL.

On a Relation between the Universal Teichmu¨ller Space and the Grunsky Operator

Being a Strebel point gives a sufficient condition for that the extremal Beltrami coefficient is uniquely determined in a Teichmiiller equivalence class. We consider how Strebel points are characterized. In this paper, we will give a new characterization of Strebel points in a certain subset of the universal Teichmfiller space by a property of the Grunsky operator.

On the Geometry of Infinite Dimensional Teichmüller Spaces

We prove that in any infinite dimensional Teichmüller space,there exists a minimal geodesic lying in two distinct geodesic disks.

Twists and Gromov Hyperbolicity of Riemann Surfaces

The main aim of this paper is to study whether the Gromov hyperbolicity is preserved under some transformations on Riemann surfaces （with their Poincare metrics）. We prove that quasiconformal maps between Riemann surfaces preserve hyperbolicity; however, we also show that arbitrary twists along simple closed geodesics do not preserve it, in general.