拟共形映射与调和函数

本文利用调和函数的Ｄｉｒｉｃｈｌｅｔ积分来估计具有给定边界值的拟共形映射的极值伸缩商,回答Ｄ．Ｐａｒｔｙｋａ关于拟对称函数谱值和特征值的一个问题。     

拟共形映射的极值问题

研究了拟共形映射的极值问题.通过对一类具有边界对应的拟共形扩张函数的伸缩商的上界估计,得到了一些新的方法和新的结果.     

具有不可缩伸缩商的拟共形映射

在研究拟共形映射的唯一极值性时,Reich引入了具有不可缩伸缩商的拟共形映射的概念.本文将继续讨论这类映射,并给出无穷小情形下的一些类似结果.     

具有不可缩伸缩商的拟共形映射

在研究拟共形映射的唯一极值性时,Reich引入了具有不可缩伸缩商的拟共形映射的概念.本文将继续讨论这类映射,并给出无穷小情形下的一些类似结果     

调和映射的可积拟共形延拓

本文研究了平面调和映射的可积拟共形延拓问题.利用经典的共形映射的拟共形延拓方法和调和映射的性质,获得了一些条件使得调和映射可以拟共形延拓至整个平面且其复伸缩商关于双曲度量是p次可积的,推广了解析单叶函数的相关结果.     

线性测度与K-拟共形调和映射

本文主要讨论调和映射与线性测度之间的关系.首先,证明单位圆周上任意可测集在单叶且保向的调和映射作用下的边界线性测度的最佳偏差定理.其次,建立K-拟共形调和映射的Schwarz型引理,并利用K-拟共形调和映射来刻画M-Lavrentiev域.最后,讨论具有有限径向长度的K-拟共形调和映射的系数估计以及径向长度与曲线周长之间的比值.     

拟圆周的Hausdorff维数

考察了中的闭集的Hausdorff维数在渐近共形映射下的不变性,以及拟圆周的Hausdorff维数和拟共形映射的边界伸缩商的关系,证明了中的闭集的Hausdorff维数在渐近共形映射下是不变的,给出了拟圆周的Hausdorff维数与边界伸缩商的一个不等式的简单证明,在某种意义上推广了相关的两个结果.     

四边形的模与本质边界点

本文研究了单位圆周上一类具有本质边界点的拟对称同胚,证明了它的极值拟共形延拓的最大伸缩商等于曲边四边形模之比的上确界.     

四边形的模与本质边界点

本文研究了单位圆周上一类具有本质边界点的拟对称同胚,证明了它的极值拟共形延拓的最大伸缩商等于曲边四边形模之比的上确界．     

Grunsky算子的本性模

利用一个推广的Grunsky不等式,借助于单叶函数的拟共形延拓的边界伸缩商,我们给出Grunsky算子的本性模的一些估计.作为推论,我们推出Grunsky算子的紧性准则.     

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 EXTREMALITY AND UNIQUE EXTREMALITY OF TEICHMULLER MAPPINGS

ＯＮＥＸＴＲＥＭＡＬＩＴＹＡＮＤＵＮＩＱＵＥＥＸＴＲＥＭＡＬＩＴＹＯＦＴＥＩＣＨＭＵＬＬＥＲＭＡＰＰＩＮＧＳ￥ＬＡＩＷＡＮＣＡＩ；ＷＵＺＨＥＭＩＮＡｂｓｔｒａｃｔ：ＣｏｎｓｉｄｅｒｔｈｅＴｅｉｃｈｍｕｌｌｅｒｍａｐｐｉｎｇｆａｓｓｏｃｉａｔｅｄｗｉｔ...     

ALMOST QUASICONFORMAL MAPPINGS WITH GIVEN BOUNDARY VALUES AND A COMPLEX DILATATION BOUND

In the extremal problems of quasiconformal mappings with given boundary valuesand a complex dilatation bound which are discussed by Reich,the extremal mapping isrequired to have no conformal point set of positive measure on the defining set T of thecomplex dilatation bound b(w).Under the additional assumptions that T\T has measurezero and b(w)is continuous a.e.Chen Jixiu proved that the extremal mapping may berelaxed to have a conformal positive measure set and a finite number of singularity pointson T.In this paper,the author proves that when the additional assumptions are given up,the same relaxations still hold and the extremal mapping is also allowed to have a countablenumber of singularity points on T.     

Boundary Dilatation and Degenerating Hamilton Sequences

In this paper, we study the boundary dilatation of quasiconformal mappings in the unit disc. By using Strebel mapping by heights theory we show that a degenerating Hamilton sequence is determined by a quasisymmetric function.     

VARIABILITY SETS AND HAMILTON SEQUENCES

This paper studies extremal quasiconformal mappings. Some properties of the variability set are obtained and the Hamilton sequences which are induced by point shift differentials are also discussed.     

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.     

Distortion and topology

For a self mapping f: D→D of the unit disk in C which has finite distortion, we give a separation condition on the components of the set where the distortion is very large - say greater than a given constant - which implies that f still extends homeomorphically and quasisymmetrically to the boundary S = ?D. Thus f shares its boundary values with a quasiconformal mapping whose distortion we explicitly estimate in terms of the data. This condition, uniformly separated in modulus, allows the set where the distortion is large to accumulate on the entire boundary S, but it does not allow a component to run out to the boundary - a necessary restriction. The lift of a Jordan domain in a Riemann surface to its universal cover D is always uniformly separated in modulus, and this allows us to apply these results in the theory of Riemann surfaces to identify an interesting link between the support of the high distortion of a map between surfaces and their geometry - again with explicit estimates. As part of our investigations, we study mappings ?: S → S which are the germs of a conformal mapping and give good bounds on the distortion of a quasiconformal extension of ? to the disk D. We then extend these results to the germs of quasisymmetric mappings. These appear of independent interest and identify new geometric invariants.     

Best mean-quasiconformal mappings

We look for best mean-quasiconformal mappings as extremals of the functional equal to the integral of the square of the functional of the conformality distortion multiplied by a special weight. The mapping inverse to an extremal is an extremal of the same functional. We obtain necessary and sufficient conditions for the Petrovskii ellipticity of the system of Euler equations for an extremal. We prove the local unique solvability of boundary values problems for this system in the 2-dimensional case. In the general case we prove the unique solvability of boundary value problems for the system linearized at the identity mapping.     

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.