两类拟共形映射的掩盖定理

In this paper we define two classes of quasiconformal mappings,and study their covering 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.     

多元K-拟全纯函数及其正规定则

研究函数族的正规性定则是复变函数论中的一个重要且有意义的工作．引进多元K-拟全纯函数的定义，并给出了多元K-拟全纯函数族的一个正规定则．     

外部边界球可达域上拟共形映射的Lipschitz连续性

定义了外部边界球可达域,利用曲线族的模获得如下结果:设D是R~n中的有界拟凸域,f:D→B~n是K-拟共形映射,若D是外部边界球可达域,则f∈Lip_α(D),其中α=K~(1/(1-n)).     

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.     

Bohr radius for locally univalent harmonic mappings

We consider the class of all sense‐preserving harmonic mappings of the unit disk , where h and g are analytic with , and determine the Bohr radius if any one of the following conditions holds:

• 1. h is bounded in .
• 2. h satisfies the condition in with .
• 3. both h and g are bounded in .
• 4. h is bounded and .

We also consider the problem of determining the Bohr radius when the supremum of the modulus of the dilatation of f in is strictly less than 1. In addition, we determine the Bohr radius for the space of analytic Bloch functions and the space of harmonic Bloch functions. The paper concludes with two conjectures.     

Nonlinear Cauchy-Riemann operators in

This paper has arisen from an effort to provide a comprehensive and unifying development of the -theory of quasiconformal mappings in . The governing equations for these mappings form nonlinear differential systems of the first order, analogous in many respects to the Cauchy-Riemann equations in the complex plane. This approach demands that one must work out certain variational integrals involving the Jacobian determinant. Guided by such integrals, we introduce two nonlinear differential operators, denoted by and , which act on weakly differentiable deformations of a domain .

Solutions to the so-called Cauchy-Riemann equations and are simply conformal deformations preserving and reversing orientation, respectively. These operators, though genuinely nonlinear, possess the important feature of being rank-one convex. Among the many desirable properties, we give the fundamental -estimate

In quest of the best constant , we are faced with fascinating problems regarding quasiconvexity of some related variational functionals. Applications to quasiconformal mappings are indicated.

     

Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings

In this paper we shall show that the boundary of the hyperbolic building considered by M. Bourdon admits Poincaré type inequalities. Then by using Heinonen-Koskela's work, we shall prove Loewner capacity estimates for some families of curves of and the fact that every quasiconformal homeomorphism is quasisymmetric. Therefore by these results, the answer to questions 19 and 20 of Heinonen and Semmes (Thirty-three YES or NO questions about mappings, measures and metrics, Conform Geom. Dyn. 1 (1997), 1-12) is NO.

     

Convergence Groups, Hausdorff Dimension, and a Theorem of Sullivan and Tukia

We show that a discrete, quasiconformal group preserving ⁿ has the property that its exponent of convergence and the Hausdorff dimension of its limit set detect the existence of a non-empty regular set on the sphere at infinity to ⁿ . This generalizes a result due separately to Sullivan and Tukia, in which it is further assumed that the group act isometrically on ⁿ , i.e. is a Kleinian group. From this generalization we are able to extract geometric information about infinite-index subgroups within certain of these groups.     

On the extremality of quasiconformal mappings and quasiconformal deformations

Given a family of quasiconformal deformations such that has a uniform bound , the solution of the Löwner-type differential equation

is an -quasiconformal mapping. An open question is to determine, for each fixed , whether the extremality of is equivalent to that of . The note gives this a negative approach in both directions.