Thurston proposed that conformal mappings can be approximated by circle packing isomorphisms and the approach can be implemented efficiently. Based on the circle packing methods the rate of convergence of approximatin...Thurston proposed that conformal mappings can be approximated by circle packing isomorphisms and the approach can be implemented efficiently. Based on the circle packing methods the rate of convergence of approximating solutions for quasiconformal mappings in the plane is discussed.展开更多
This paper gets the Beltrami equations satisfied by a 1-quasiconformal mapping, which are exactly CR or anti-CR equations on （2,2）-type quadric Q0. This means a 1-quasiconformal mapping on Q0 is CR or anti-CR. This ...This paper gets the Beltrami equations satisfied by a 1-quasiconformal mapping, which are exactly CR or anti-CR equations on （2,2）-type quadric Q0. This means a 1-quasiconformal mapping on Q0 is CR or anti-CR. This reduces the determination of 1- quasiconformal mappings to a problem on the theory of several complex analysis. The result about the group of CR automorphisms is used to determine the unit component of group of 1-quasiconformal mappings.展开更多
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 d...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.展开更多
The Beurling Ahlfors extension was generalized to improve the bound estimate of a constant about extremal quasiconformal deformations, which is closely related to the extremal quasiconformal mapping theory.
In this article,we first give two simple examples to illustrate that two types of parametric representation of the family ofΣ0 K have some gaps.Then we also find that the area derivative formula(1.6),which is used to...In this article,we first give two simple examples to illustrate that two types of parametric representation of the family ofΣ0 K have some gaps.Then we also find that the area derivative formula(1.6),which is used to estimate the area distortion ofΣ0 K,cannot be derived from[6],but that formula still holds forΣ0 K through our amendatory parametric representation for the one obtained by Eremenko and Hamilton.展开更多
Rodin and Sullivan （1987） proved Thurston＇s conjecture that a scheme based on the Circle Packing Theorem converges to the Riemann mapping, thereby proved a refreshing geometric view of the Riemann Mapping Theorem. ...Rodin and Sullivan （1987） proved Thurston＇s conjecture that a scheme based on the Circle Packing Theorem converges to the Riemann mapping, thereby proved a refreshing geometric view of the Riemann Mapping Theorem. Naturally, we consider to use the ellipses to pack the bounded simply connected domain and obtain similarly a sequence simplicial homeomorphism between the ellipse packing and the circle packing. In this paper, we prove that these simplicial homeomorphism approximate a quasiconformal mapping from the bounded simply connected domain onto the unit disk with the modulus of their complex dilatations tending to 1 almost everywhere in the domain when the ratio of the longer axis and shorter axis of the ellipse tending to ∞.展开更多
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 intrig...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.展开更多
Quasiconformal mappings between hyperbolic triangles are considered.We give an explicit estimate of the dilation of the quasiconformal mappings,which generalizes Bishop's results.
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 m...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.展开更多
An explicit example of a Reich sequence for a uniquely extremal quasiconformal mapping in a borderline case between uniqueness and non-uniqueness is given.
LET D be the unit disk in the complex plane C and f be a sense preserving quasisymmetrichomeomorphism of D onto itself. Denote by Q a quadrilateral D （z<sub>1</sub>, z<sub>2</sub>, z<sub>...LET D be the unit disk in the complex plane C and f be a sense preserving quasisymmetrichomeomorphism of D onto itself. Denote by Q a quadrilateral D （z<sub>1</sub>, z<sub>2</sub>, z<sub>3</sub>, z<sub>4</sub> ） with do-main D and vertices z<sub>1</sub>, z<sub>2</sub>, z<sub>3</sub>, z<sub>4</sub> ∈D, and by M（Q） its conformal modulus. We are in-展开更多
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 e...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.展开更多
We obtain some convergence properties concerning Faber polynomials and apply them to studying univalent functions with quasiconformal extensions. In particular, by introducing an operator on the usual l2 space, we obt...We obtain some convergence properties concerning Faber polynomials and apply them to studying univalent functions with quasiconformal extensions. In particular, by introducing an operator on the usual l2 space, we obtain some new characterizations of quasiconformal extendablity and asymptotic conformality for univalent functions.展开更多
A real-valued function f(x)on belongs to Zygmund class if its Zygmund norm It is proved that when f ∈, there exists an extension F(z)of f to H = {Imz > 0} such that It is also proved that if f(0)=f(1)= 0,then
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 coeff...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 ?(z) in its equivalent class and a compact subset E ? △ with positive measure such that the essential upper bound of ?(z) on E is less than the norm of [μ].展开更多
An open problem is to characterize the Fourier coefficients of Zygmund functions.This problem was also explicitly suggested by Nag and later by Teo and Takhtajan-Teo in the course of study of the universal Teichmu...An open problem is to characterize the Fourier coefficients of Zygmund functions.This problem was also explicitly suggested by Nag and later by Teo and Takhtajan-Teo in the course of study of the universal Teichmu¨ller space.By a complex analysis approach,we give a characterization for the Fourier coefficients of a Zygmund function by a quadratic form.Some related topics are also discussed,including those analytic functions with quasiconformal deformation extensions.展开更多
For every torsion free Fuchsian group F with Poincaré’s -operator norm ║г║=1, it is proved that there exists an extremal Beltrami differential of F which is also extremal under its own boundary correspondence...For every torsion free Fuchsian group F with Poincaré’s -operator norm ║г║=1, it is proved that there exists an extremal Beltrami differential of F which is also extremal under its own boundary correspondence. It is also proved that the imbedding of the Teichmüller space T(Γ) into the universal Teichmüller space T is not a global isometry unless Γ is an elementary group.展开更多
In this paper,it is proved that the Mori constant,the Hlder coeflicient of the family QC<sub>K</sub>（B）of K-quasiconformal self-mappings of the unit disk B with the origin fixed,is at most 46<sup>1...In this paper,it is proved that the Mori constant,the Hlder coeflicient of the family QC<sub>K</sub>（B）of K-quasiconformal self-mappings of the unit disk B with the origin fixed,is at most 46<sup>1-1/K</sup>.It is also shown that the Hlder coefficient of the restriction of a map f∈QC<sub>K</sub>（B）to the disk|z|≤sin 25.7°is at most 16<sup>1-1/K</sup>.展开更多
M. Fait, J. Krzyz and J. Zygmunt proved that a strongly starlike function of order α on the unit disk can be extended to a k-quasiconformal mapping with k ≤ sin（απ/2） on the whole complex plane C which fixes the...M. Fait, J. Krzyz and J. Zygmunt proved that a strongly starlike function of order α on the unit disk can be extended to a k-quasiconformal mapping with k ≤ sin（απ/2） on the whole complex plane C which fixes the point at infinity. An open question is whether such a function can be extended to a k-quasiconformal mapping with k 〈α to the whole plane C. In this paper we will give a negative approach to the question.展开更多
基金This project is supported in part by NSF of China(60575004, 10231040)NSF of GuangDong, Grants from the Ministry of Education of China(NCET-04-0791)Grants from Sun Yat-Sen University
文摘Thurston proposed that conformal mappings can be approximated by circle packing isomorphisms and the approach can be implemented efficiently. Based on the circle packing methods the rate of convergence of approximating solutions for quasiconformal mappings in the plane is discussed.
基金Supported by the National Natural Science Foundation of China (10571155)
文摘This paper gets the Beltrami equations satisfied by a 1-quasiconformal mapping, which are exactly CR or anti-CR equations on （2,2）-type quadric Q0. This means a 1-quasiconformal mapping on Q0 is CR or anti-CR. This reduces the determination of 1- quasiconformal mappings to a problem on the theory of several complex analysis. The result about the group of CR automorphisms is used to determine the unit component of group of 1-quasiconformal mappings.
基金Supported by the Natural Science Foundation of Huaqiao University(02HZR12)Supported by the Natural Science Foundation of Overseas Chinese Affairs Office under the State Council(01QZR01)
文摘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.
文摘The Beurling Ahlfors extension was generalized to improve the bound estimate of a constant about extremal quasiconformal deformations, which is closely related to the extremal quasiconformal mapping theory.
基金National Natural Science Foundation of China(11971182)the Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University(ZQN-PY402)+1 种基金Research projects of Young and Middle-aged Teacher's Education of Fujian Province(JAT190508)Scientific research project of Quanzhou Normal University(H19009).
文摘In this article,we first give two simple examples to illustrate that two types of parametric representation of the family ofΣ0 K have some gaps.Then we also find that the area derivative formula(1.6),which is used to estimate the area distortion ofΣ0 K,cannot be derived from[6],but that formula still holds forΣ0 K through our amendatory parametric representation for the one obtained by Eremenko and Hamilton.
基金supported by the National Natural Science Foundation of China(10701084)Chongqing Natural Science Foundation (2008BB0151)
文摘Rodin and Sullivan （1987） proved Thurston＇s conjecture that a scheme based on the Circle Packing Theorem converges to the Riemann mapping, thereby proved a refreshing geometric view of the Riemann Mapping Theorem. Naturally, we consider to use the ellipses to pack the bounded simply connected domain and obtain similarly a sequence simplicial homeomorphism between the ellipse packing and the circle packing. In this paper, we prove that these simplicial homeomorphism approximate a quasiconformal mapping from the bounded simply connected domain onto the unit disk with the modulus of their complex dilatations tending to 1 almost everywhere in the domain when the ratio of the longer axis and shorter axis of the ellipse tending to ∞.
基金Supported by the National Natural Science Foundation of China(10671174, 10401036)a Foundation for the Author of National Excellent Doctoral Dissertation of China(200518)
文摘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.
基金Partially Supported by NSFC(Grant No.12071047)Fundamental Research Funds for the Central Universities(Grant No.500421126).
文摘Quasiconformal mappings between hyperbolic triangles are considered.We give an explicit estimate of the dilation of the quasiconformal mappings,which generalizes Bishop's results.
基金Supported by National Natural Science Foundation of China (Grant No 10971030)
文摘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.
文摘An explicit example of a Reich sequence for a uniquely extremal quasiconformal mapping in a borderline case between uniqueness and non-uniqueness is given.
文摘LET D be the unit disk in the complex plane C and f be a sense preserving quasisymmetrichomeomorphism of D onto itself. Denote by Q a quadrilateral D （z<sub>1</sub>, z<sub>2</sub>, z<sub>3</sub>, z<sub>4</sub> ） with do-main D and vertices z<sub>1</sub>, z<sub>2</sub>, z<sub>3</sub>, z<sub>4</sub> ∈D, and by M（Q） its conformal modulus. We are in-
基金Project supported by the National Natural Science Foundation of China (Grant No. 19871002).
文摘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.
基金supported by the Program for New Century Excellent Talents in University (Grant No. 06-0504)National Natural Science Foundation of China (Grant No. 10771153)
文摘We obtain some convergence properties concerning Faber polynomials and apply them to studying univalent functions with quasiconformal extensions. In particular, by introducing an operator on the usual l2 space, we obtain some new characterizations of quasiconformal extendablity and asymptotic conformality for univalent functions.
文摘A real-valued function f(x)on belongs to Zygmund class if its Zygmund norm It is proved that when f ∈, there exists an extension F(z)of f to H = {Imz > 0} such that It is also proved that if f(0)=f(1)= 0,then
基金This work was supported by the National Natural Science Foundation of China(Grant No,10271029).
文摘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 ?(z) in its equivalent class and a compact subset E ? △ with positive measure such that the essential upper bound of ?(z) on E is less than the norm of [μ].
基金supported by National Natural Science Foundation of China (Grant No.11071179)
文摘An open problem is to characterize the Fourier coefficients of Zygmund functions.This problem was also explicitly suggested by Nag and later by Teo and Takhtajan-Teo in the course of study of the universal Teichmu¨ller space.By a complex analysis approach,we give a characterization for the Fourier coefficients of a Zygmund function by a quadratic form.Some related topics are also discussed,including those analytic functions with quasiconformal deformation extensions.
文摘For every torsion free Fuchsian group F with Poincaré’s -operator norm ║г║=1, it is proved that there exists an extremal Beltrami differential of F which is also extremal under its own boundary correspondence. It is also proved that the imbedding of the Teichmüller space T(Γ) into the universal Teichmüller space T is not a global isometry unless Γ is an elementary group.
文摘In this paper,it is proved that the Mori constant,the Hlder coeflicient of the family QC<sub>K</sub>（B）of K-quasiconformal self-mappings of the unit disk B with the origin fixed,is at most 46<sup>1-1/K</sup>.It is also shown that the Hlder coefficient of the restriction of a map f∈QC<sub>K</sub>（B）to the disk|z|≤sin 25.7°is at most 16<sup>1-1/K</sup>.
基金supported by National Natural Science Foundation of China (Grant Nos.10671004,10831004)The Doctoral Education Program Foundation (Grant No.20060001003)
文摘We show that the extremal polygonal quasiconformal mappings are biLipschitz with respect to the hyperbolic metric in the unit disk.
基金This research was supported by NNSF of China(Grant No.10231040)NCET(06-0504)
文摘M. Fait, J. Krzyz and J. Zygmunt proved that a strongly starlike function of order α on the unit disk can be extended to a k-quasiconformal mapping with k ≤ sin（απ/2） on the whole complex plane C which fixes the point at infinity. An open question is whether such a function can be extended to a k-quasiconformal mapping with k 〈α to the whole plane C. In this paper we will give a negative approach to the question.
