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.

On the Estimates of Hyperbolic Area Distortion of Quasiconformal Mappings

The distortion property of hyperbolic area of planar quasiconformal mappings is studied in this paper. In the case of radial quasiconformal mappings and angular deformed quasiconformal mappings their hyperbolic area distortions are estimated quite sharply. The result can be applied to judge whether the hyperbolic area of a planar subset is explodable.

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.

On Quasiconformal Mappings Between Hyperbolic Triangles

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 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 ∞.

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.

Covering Theorems on Two Classes of Quasi-conformal Mappings

In this paper we define two classes of quasiconformal mappings, and study theircovering properties by methods of module. We obtain some new results. In the meantime,we give new methods to prove Koebe 1/2 covering theorem on convex conformal mappings.

Hamilton sequences for extremal quasiconformal mappings in the unit disk

By studying the mapping by heights for quadratic differentials introduced by Strebel, some relations have been established between the maximal norm sequence for quasisymmetric functions and the Hamilton sequence for extremal quasiconformal mappings in the unit disk. Consequently it is proved that a Hamilton sequence is only determined by e quasisymmetric function.

On the extremal sets of extremal quasiconformal mappings

The properties of the extremal sets of extremal quasiconformal mappings are discussed. It is proved that if an extremal Beltrami coefficient μ(z) is not uniquely extremal, then there exists an extremal Beltrami coefficient ?(z) in its equivalent class and a compact subset E ? △ with positive measure such that the essential upper bound of ?(z) on E is less than the norm of [μ].

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 the Bloch Constant for K-Quasiconformal Mappings in Several Complex Variables

We study the Bloch constant for K-quasiconformal holomorphic mappings of the unit ball B of Cn into Cn. The final result we prove in this paper is: If f is a K-quasiconformal holomorphic mapping of B into Cn such that det（f’（0））=1, then f（B） contains a schlicht ball of radius at least （CnK）1-n integral from n=0 to 1(（1+t）n-1/（1-t）2 exp{-（n+1）t/（1-t）}dt, where Cn】1 is a constant depending on n only, and Cn→101/2 as n→∞.

On Harmonic K-Quasiconformal Mappings Associated with Asymmetric Vertical Strips

In this paper,we discuss the sense-preserving univalent harmonic mappings from the unit disk D onto asymmetrical vertical strips Ωα={ω：α-π/2sinαR（ω）α/2sinα},π/2≤απSuch results as analytic representation formula,coefficient estimates,distortion theorem and area theorem are derived.

Extremal polygonal quasiconformal mappings and biLipschitz mappings

We show that the extremal polygonal quasiconformal mappings are biLipschitz with respect to the hyperbolic metric in the unit disk.

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.

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.

FEASIBILITY OF THE REICH PROCEDURE IN THE DECOMPOSITION OF PLANE QUASICONFORMAL MAPPINGS

In the decomposition problems, studied by Retch, of quasiconformal self mappings of the unit disc which keep the boundary points fixed, the construction of the first one requires the application of the Hahn-Banach theorem (so it is abstract) and it is only a variational decomposition (a small weight one), and that of the second one avoids the Hahn-Banach theorem and gets rid of the restriction to the variational decomposition. But the success of the second decomposition procedure (the Retch procedure) is guaranteed only when minimal maximal dilatation K(f) is sufficiently small. Therefore, it can not guarantee even a variational decomposition. Huang Xinzhong then proved that the inverse Reich procedure was successful for ally X(f). But the inverse Retch procedure is not so natural as the Retch procedure and the corresponding decomposition can not replace the first one. It is still an open problem whether the Reich procedure is successful for any X(f). The present paper gives an affirmative answer to this problem.

A Local Hlder Estimate of(K_1, K_2)-Quasiconformal Mappings between Hypersurfaces

In this paper, we prove a local HSlder estimate of （K1, K2）-quasiconformal mappings be- tween n-dimensional hypersurfaces of Rn＋l under an assumption of bounded mean curvature of the original hypersurface M. With some new ingredients of the isoperimetric inequality and the co-area formula on manifolds, we extend Simon＇s work of quasiconformal mappings on surfaces of R3 to the setting of n-dimensional hypersurfaces of Rn＋1.

Douady-Earle延拓中用到的拟共形映照参数表示

假设f^(μ(z))(z)是单位圆D到自身保持-1,i,1不动,具有复特征μ(z)的Douady-Earle延拓。借助于上半平面到自身保持0,1,∞三点不动的拟共形映射的参数表示,利用单位圆到上半平面的共形映射,给出Douady-Earle延拓f^(μ(z))(z)的参数表示。

Singular values, quasiconformal maps and the Schottky upper bound

Some properties and asymptotically sharp bounds are obtained for singdar values of Ramanujan’s generalized modular equation. from which infinite-product representations of the Hersch-Pfluger ?dimtortion function ? K (r) and the Agard η-distortion function η K (t) follow. By these results, the explicit quasiconformal Schwan lemma is improved, several properties are obtained for the Schottky upper bound, and a conjecture on the linear distortion function λ (K) is proved to be true.