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.     

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.     

A Schwarz-Pick inequality for harmonic quasiconformal mappings and its applications

The main result of this paper is the sharp generalized Schwarz-Pick inequality for euclidean harmonic quasiconformal mappings with convex ranges, which generalizes a result given by Mateljevi?. As its applications, we obtain the property of quasi-isometry with respect to the Poincaré distance and an analogue of the Koebe theorem for this class of mappings.     

A note on harmonic quasiconformal mappings

In this note we show that a harmonic quasiconformal mapping f=u+iv with respect to the Poincaré metric of the upper half plane onto itself such that v(x,y)=v(y) or u(x,y)=u(x) is a conformal mapping.     

On criteria for extremality of Teichmüller mappings

Let be a Teichmüller self-mapping of the unit disk corresponding to a holomorphic quadratic differential . If satisfies the growth condition (as ), for any given 0$">, then is extremal, and for any given , there exists a subsequence of such that

is a Hamilton sequence. In addition, it is shown that there exists with bounded Bers norm such that the corresponding Teichmüller mapping is not extremal, which gives a negative answer to a conjecture by Huang in 1995.

     

Truncations of extremal quasiconformal mappings and their applications

In this paper we give a necessary and sufficient condition to decide whether the Teichmüller equivalency class [α] of a truncation α induced by a uniquely extremal Beltrami differential is a Strebel point in T. We also obtain a necessary and sufficient condition of the unique extremality of α. Using the properties of truncations we provide a method to construct Hamilton sequences. We also get a sufficient condition for the extremality of f(z,t) to be equivalent to that of F(w,t). The corresponding results in the infinitesimal case are obtained, too.     

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 v(z) in its equivalent class and a compact subset E Δ with positive measure such that the essential upper bound of v(z) on E is less than the norm of [μ].     

Unique extremality of quasiconformal mappings

In this paper, we study the conditions under which unique extremality of quasiconformal mappings occurs and provide a broader point of view of this phenomenon. Additional information is obtained by means of specialized constructions. In particular, we generalize the construction theorem in [BLMM], thus providing a more basic understanding of it. We also generalize the notion of unique extremality and give an analytic characterization of the generalized concept. Research partially supported by MNTRS, Serbia, Grant No. 144 020     

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. Project supported by the National Natural Science Foundation of China (Grant No. 19871002).     

On the uniqueness problem of harmonic quasiconformal mappings

In this paper, we give an affirmative answer to Sheretov's problem on the uniqueness of harmonic mappings and improve the unique minimal mapping theorem of Reich and Strebel. Meanwhile, we also solve a problem posed by Reich and obtain the uniqueness theorem on related weight functions.

     

On the quasi-isometries of harmonic quasiconformal mappings

We prove versions of the Ahlfors-Schwarz lemma for quasiconformal euclidean harmonic functions and harmonic mappings with respect to the Poincaré metric.     

Extremal quasiconformal mappings compatible with Fuchsian groups

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

Distortion theorems of plane quasiconformal mappings

The authors prove a conjecture on elliptic integrals and obtain sharp bounds for φK(r) and λ(K). By using Teichmüller's module theorem, the authors obtain a distortion theorem of K-quasiconformal mappings on the plane.     

On smoothness of symmetric mappings II

If the dilatation of a quasiconformal selfmap of the upper half-plane vanishes near the real line as a power of the height, the induced quasisymmetric mapping is Lipschitz with the same exponent. In this note, it is shown that the converse does not hold for any positive exponent. In addition, a sufficient condition is found to have locally a quasiconformal extension with the desired growth in the dilatation.

     

模和拟共形映射

研究了模和拟共形映射，得到了一个同胚为拟共形映射的几个条件．     

On Mori's theorem in quasiconformal theory

In this paper, it is proved that the Mori constant, the Hölder coefficient of the familyQC _K (B) ofK-quasiconformal self-mappings of the unit diskB with the origin fixed, is at most 46^1?1 K. It is also shown that the Hölder coefficient of the restriction of a mapf∈QC _K (B) to the disk |z|≤ sin 25.7° is at most 16^1?1 K.     

Local boundary dilatation of quasiconformal maps in the disk

In this paper we partly give an affirmative answer to a problem proposed by F. Gardiner and N. Lakic by studying the gluing of quasiconformal maps.

     

The invariance of inner and outer linearly locally connected sets under quasiconformal mappings

In this paper, we prove that the inner and outer linearly locally connected sets of Rn are invariant respectively under quasiconformal mappings which fixed the point at infinity. On the other hand, we prove the converses of above results to be true also.     

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.     

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.