Moduli of quadrilaterals and extremal quasiconformal extensions of quasisymmetric functions

We establish a relationship between Strebel boundary dilatation of a quasisymmetric function of the unit circle and indicated by the change in the module of the quadrilaterals with vertices on the circle. By using general theory of universal Teichmüller space, we show that there are many quasisymmetric functions of the circle have the property that the smallest dilatation for a quasiconformal extension of a quasisymmetric function of the unit circle is larger than indicated by the change in the module of quadrilaterals with vertices on the circle. Received: December 8, 1996     

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.     

Estimation for the Dilatation of the Generalized Beurling-Ahlfors' Extension

51.IntroductionLethbeahomeomorphismontherealaxisR,andsatisfyh(lco)=loo.Wecallaquasisymmetricfunction.lfthereexistsa'constantMsuchthatM-1Sp(x,t)SM,x,teR,(l.2)then(1.2)iscalledtheM-condition-Byvirtueoftheclassicresultofpaper[lj,ifp(x,t)sat-isfiestheM-condition,thenthereexistsaK(M')-quasiconformaIextensiononthel1alfplaneH,whichisthewell-knownBeurling-Ahlfors'extension:ytz)=u(x,y) iv(x,y),whereAfterpaper[lj,manyworkshavebeendoneforestimatingK-Particularly,itshouldbere-markedheretl1atM.Leht…     

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].     

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。     

单位圆到凸区域上的调和拟共形映照

设F（x）=p（x）e^ir（x）为单位圆周到约当凸曲线Γ上的保向同胚映照.本文证明:若ess inf|F’（x）|>0且对于一切的φ∈R有|F（φ+x）+F（φ-x）-2F（φ）|≤M|x|^α,这里α>1,M为正常数,则ω=P[F]（z）为单位圆到凸区域Ω=int（Γ）上为调和拟共形映照.     

Approximation of fixed points of strictly pseudocontractive mappings on arbitrary closed, convex sets in a Banach space

We show that any fixed point of a Lipschitzian, strictly pseudocontractive mapping on a closed, convex subset of a Banach space is necessarily unique, and may be norm approximated by an iterative procedure. Our argument provides a convergence rate estimate and removes the boundedness assumption on , generalizing theorems of Liu.

     

On farthest points of sets

For a convex closed bounded set in a Banach space, we study the existence and uniqueness problem for a point of this set that is the farthest point from a given point in space. In terms of the existence and uniqueness of the farthest point, as well as the Lipschitzian dependence of this point on a point in space, we obtain necessary and su.cient conditions for the strong convexity of a set in several infinite-dimensional spaces, in particular, in a Hilbert space. A set representable as the intersection of closed balls of a fixed radius is called a strongly convex set. We show that the condition “for each point in space that is sufficiently far from a set, there exists a unique farthest point of the set” is a criterion for the strong convexity of a set in a finite-dimensional normed space, where the norm ball is a strongly convex set and a generating set.     

A note on point shift differentials

LetT be the universal Teichmüller space viewed as the set of all normalized quasisymmetric homeomorphism of the unit circleS ¹=∂Δ. Denote byV _h [z ₀] the variability set ofz ₀ with respect toh∈T. The following is proved: Leth ₀ be a point ofT. Suppose thatμ ₀ is an arbitrarily given extremal Beltrami differential ofh ₀ andf ₀: μ→μ is a quasiconformal mapping with the Beltrami coefficientμ ₀ andf ₀1_s=h ₀. Then there are a sequenceh _n of points inT and a sequencew _n of points in Δ withh _n ∈(Δ−V _h [z ₀]) andw _n →f ₀(z ₀) andh _n →h ₀ andn∞ such that the point shift differentials determined byh _n asw _n form a Hamilton sequence ofμ ₀. Project supported by the National Natural Science Foundation of China (Grant No. 19531060) and the Doctoral Education Program Foundation of China.     

Local boundary dilatation of quasiconformal maps

By studying the existence problem of locally extremal Beltrami differential posed by F. Gardiner and N. Lakic, we introduce a new definition of the local boundary dilatations of points in Teichmuller spaces of simply connected plane domains, which is the same as the usual ones in the case of Jordan domains. Then we show that the answer to the problem of F. Gardiner and N. Lakic is affirmative according to this new definition. Comparing with the usual definitions, the problem of F. Gardiner and N. Lakic is partly answered and the results of G. Cui and Y. Qi is generalized.     

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.     

度量空间中拟对称映射与拟双曲一致域的研究

本文主要考虑度量空间中拟双曲一致域与拟对称映射之间的关系,并证明了度量空间中拟双曲一致域在拟对称映射下仍然是保持不变的.     

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.

     

Metric Density and Quasimobius Mappings

We study the notion of -density of metric spaces which was introduced by V. Aseev and D. Trotsenko. Interrelation between -density and homogeneous density is established. We also characterize -dense spaces as arcwise connected metric spaces in which arcs are the quasimobius images of the middle-third Cantor set. Finally, we characterize quasiconformal self-mappings of ⁿ in terms of -density.     

Finding squares and rectangles in sets of points

A number of problems concerning sets of points in the plane are studied, e.g. establishing whether it contains a subset of size 4, which are the vertices of a square or rectangle. Both the problems of finding axis-parallel squares and rectangles, and arbitrarily oriented squares and rectangles are studied. Efficient algorithms are obtained for all of them. Furthermore, we investigate the generalizations tod-dimensional space, where the problem is to find hyperrectangles and hypercubes. Also, upper and lower bounds are given for combinatorial problems on the maxium number of subsets of size 4, of which the points are the vertices of a square or rectangle. Then we state an equivalence between the problem of finding rectangles, and the problem of findingK _{2, 2} subgraphs in bipartite graphs. Thus we immediately have an efficient algorithm for this graph problem.This work was partially supported by the ESPRIT Basic Research Action No. 3075 (project ALCOM). Work of the second author was also supported by the Dutch Organisation for Scientific Research (N.W.O.).     

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．     

拟共形映射和弱Cigar域

设f是R~n到R~n上的同胚,本文证明了f是拟共形映射的充要条件是f将R~n中的任一弱Cigar域映成R~n中的弱Cigar域。     

拟共形映射与John域

本证明了有界一致域必定是Hohn域和Jphn域的拟共不变性．