拟共形映射的偏差定理

本文得到了一个空间拟共形映射的偏差定理,这一结果是F.W.Gehring的平面偏差定理的空间拓广。     

一类近于凸映照子族精确的偏差定理

首先建立了C~n中单位多圆柱上一类近于凸映照子族精确的偏差定理,同时在复Banach空间单位球上也建立了该类映照精确的偏差定理的下界估计.其次在复Banach空间单位球上建立了准星形映照精确的偏差定理.所得结果将单复变中近于凸函数和星形函数的偏差定理推广至高维情形,并且对龚升提出的一个公开问题给出肯定的回答.     

拟共形映照的双曲面积偏差估计

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.     

Modular equations and distortion functions

Modular equations occur in number theory, but it is less known that such equations also occur in the study of deformation properties of quasiconformal mappings. The authors study two important plane quasiconformal distortion functions, obtaining monotonicity and convexity properties, and finding sharp bounds for them. Applications are provided that relate to the quasiconformal Schwarz Lemma and to Schottky’s Theorem. These results also yield new bounds for singular values of complete elliptic integrals.      

有限偏差函数与调和函数

分别借助解析函数与调和函数两类函数的Dirichlet积分,利用相关文献给定边界值的拟共形映射极值伸缩商的估计方法,通过有限偏差函数和拟共形映射的关系估计了具有给定边界值的有限偏差函数的极值伸缩商.得到了解析函数的Dirichlet积分在有限偏差函数下具有拟不变性,同时给出有限偏差函数极值伸缩商的下界估计.     

Growth Theorem and Distortion Theorem for a Sub class of Parab olic Starlike Mappings

In this article, we firstly introduce a subclass of parabolic starlike mappings writen as P S (β;ρ). Secondly, the growth theorem for P S (β;ρ) on the unit ball in complex Banach space is obtained. Finaly, as the application of the growth theorem of P S (β;ρ), the distortion theorem along a unit direction is also established.     

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.     

Uniqueness of extremal quasiconformal mappings of Teichmüller type

We prove a theorem concerning the uniqueness of extremal quasiconformal mappings, generalizing thereby a well-known theorem of Teichmüller.     

Sufficient conditions for the univalence of quasiconformal mappings

Using two methods, quasiconformal continuation involving a theorem of Hadamard and direct estimation of f(z₂)?f(z₁), we obtain sufficient conditions for the univalence of continuously differentiable mappings f(z) of plane domains which, in the case of conformal mappings, reduce to both well-known and new results.     

Analogs of the Ikoma–Schwartz lemma and Liouville theorem for mappings with unbounded characteristic

We obtain results on the local behavior of open discrete mappings f:D ? \mathbbRⁿ f:D \to {\mathbb{R}^n} , n ≥ 2, that satisfy certain conditions related to the distortion of capacities of condensers. It is shown that, in an infinitesimal neighborhood of zero, the indicated mapping cannot grow faster than an integral of a special type that corresponds to the distortion of the capacity under this mapping, which is an analog of the well-known Ikoma growth estimate proved for quasiconformal mappings of the unit ball into itself and of the classic Schwartz lemma for analytic functions. For mappings of the indicated type, we also obtain an analog of the well-known Liouville theorem for analytic functions.     

拟共形映射面积偏差条件下的Schwarz型定理

关于拟共形映射在一些面积偏差条件下,得到了Schwarz型定理．     

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.     

Homeomorphisms of the Heisenberg group preserving BMO

We provide a new geometric proof of Reimann’s theorem characterizing quasiconformal mappings as the ones preserving functions of bounded mean oscillation. While our proof is new already in the Euclidean spaces, it is applicable in Heisenberg groups as well as in more general stratified nilpotent Carnot groups.     

Canonical reducing curves of surface homeomorphism

In this paper we will introduce the concept of canonical reducing set of a surface homeomorphism, and prove that it is unique up to an isotopy. As an application, we will give a simple proof of Thurston's theorem on classifying mappings on non-orientable surface, using the techniques of quasiconformal mappings and some known results in the orientable case, especially the Thurston theorem on orientable surfaces.     

A new proof of the Ahlfors five islands theorem

We deduce the Ahlfors five islands theorem from a corresponding result of Nevanlinna concerning perfectly branched values, a rescaling lemma for non-normal families and an existence theorem for quasiconformal mappings. We also give a proof of Nevanlinna’s result based on the rescaling lemma and a version of Schwarz’s lemma.     

Hausdorff dimension distribution of quasiconformal mappings on the Heisenberg group

We construct quasiconformal mappings on the Heisenberg group which change the Hausdorff dimension of Cantor-type sets in an arbitrary fashion. On the other hand, we give examples of subsets of the Heisenberg group whose Hausdorff dimension cannot be lowered by any quasiconformal mapping. For a general set of a certain Hausdorff dimension we obtain estimates of the Hausdorff dimension of the image set in terms of the magnitude of the quasiconformal distortion.     

复Banach空间中单位球上双全纯凸映射的偏差定理

本文讨论一般复Banach空间上单位球B的Caratheodory度量和Kobayashi 度量的性质,并据此将Cn(n≥1)中单位球Bn上双全纯凸映射的矩阵形式偏差定理 推广到一般复Banach空间的单位球B上.     

高维空间中的有界凸域

证明n维空间中的有界凸域D能被拟共形映射到n维单位球B~n(0,1),即D是拟球,从而说明拟共形映射中的黎曼定理在n维空间中的有界凸域类中是成立的．     

Differentiability of mappings in the geometry of Carnot manifolds

We study the differentiability of mappings in the geometry of Carnot-Carathéodory spaces under the condition of minimal smoothness of vector fields. We introduce a new concept of hc-differentiability and prove the hc-differentiability of Lipschitz mappings of Carnot-Carathéodory spaces (a generalization of Rademacher’s theorem) and a generalization of Stepanov’s theorem. As a consequence, we obtain the hc-differentiability almost everywhere of the quasiconformal mappings of Carnot-Carathéodory spaces. We establish the hc-differentiability of rectifiable curves by way of proof. Moreover, the paper contains a new proof of the functorial property of the correspondence “a local basis ? the nilpotent tangent cone.”     

Angle distortion and conformality

We offer an alternative to the usual proofs for the equivalence between conformality and analyticity with nonzero derivative. Our proof uses the z̄-derivative to quantify angle distortion, and thereby foreshadows some important ideas in the study of quasiconformal mappings.