摘要
本文主要目的在于研究拟共形映射极值问题以及与之相关的Teichmüller空间性质.拟共形映射的概念诞生于上世纪30年代,1940年左右,德国数学家Teichmüller利用极值拟共形映射理论来研究Riemann模问题,对这一经典的几何问题给予了完美的解答.此后,对拟共形映射与Teichmüller空间理论的研究就一直倍受数学家们所关注,Ahlfors,Bets,Gehring,Earle,Gardiner,Reich,Strebel,李忠,伍胜健,沈玉良和陈纪修等数学家都对该理论进行了深刻的研究.如今,拟共形映射与Teichmüller空间理论已交叉渗透到微分几何、偏微分方程、拓扑学等其它数学分支.
拟共形映射的极值理论主要研究给定边界对应的拟共形映射族中极值映射的存在性、唯一性、以及极值映射的性质、特征的刻画等问题.本文的第二章和第三章将研究这些问题并得到一系列结果.
对数导数在判定共形映射能否拟共形映射扩张、估计区域的单叶性内径以及描述万有Teichmüller空间的性质方面都起到非常重要的作用,对数导数的研究将对拟共形映射理论的发展起到积极的作用.本文的第四章我们将研究万有Teichmüller空间对数导数模型的一些几何性质.
Teichmüiller空间的切空间(也称无限小Teichmüller空间)对研究Teichmüller空间的性质以及刻画极值拟共形映射的特征都有重要的意义.因此本文中我们也将讨论无限小Teichmüller空间中的一些未知的问题.全文共分为五章.
第一章,绪论.我们简要的介绍拟共形映射与Teichmüller空间理论的历史背景和研究意义,并阐述本文所研究问题的由来和现状以及主要结果.
第二章,二次抛物区域上拟共形映射的极值性.在给定所有边界点对应的前提下,我们已经知道了很多类区域上极值拟共形映射的刻画和性质,但是若降低边界对应要求,同样区域上极值拟共形映射的情况还不清楚.Strebel([94][95])曾经对几种不同的区域研究过这个问题,我们将在二次抛物区域上研究这个问题.
第三章,Teichmüller空间与其切空间的一些非等价性.刻画一个极值或者唯一极值拟共形映射的特征一直是拟共形极映射值理论研究的热点.Hamilton([41]),Krushkal([45]),Reich和Strebel([76])共同研究得到了极值拟共形映射的特征刻画.1998年,Bo(?)in,Lakic,Markovi(?)和Mateljevi(?)([9])研究得到了唯一极值拟共形映射的特征刻画.从这些论文中我们发现Teichmüller空间与其切空间具有许多等价的极值性质.在本章中,我们将研究Teichmüller空间与其切空间在Strebel点、极值Teichmüller Beltrami系数的存在性、常数模极值Beltrami系数存在性之间的等价性等问题.
第四章,万有Teichmüller空间对数导数模型的几何性质.对数导数和单叶函数的拟共形扩张具有深刻的联系.万有Teichmüller空间对数导数模型也具有许多奇特的几何性质,Zhuravlev([116])证明了万有Teichmüller空间对数导数模型是由无穷多个互不相交的分支组成的.在这一章,我们研究万有Teichmüller空间对数导数模型每个分支内测地线和球的几何性质.
第五章,无限小Teichmüller空间中的问题.从本文第三章,我们知道Teichmüller空间与无限小Teichmüller空间具有许多相似和不相似的性质.在这一章,我们主要研究无限小Teichmüller空间中测地圆盘的个数以及无限小极值Beltrami系数的Hamilton序列问题.
The purpose of this thesis is to study the problem of extremal quasiconformal mappings and the associated properties of Teichmüller space. The concept of quasiconformal mappings was born in the 1930s, around 1940, German mathematician Teichmüller applied extremal quasiconformal mappings theory to study the modular problem of Reimann surfaces, and gave a perfect answer to this classical geometry problem. Since then, the theory of quasiconformal mappings and Teichmüller space has been greatly concerned by mathematicians, such as Ahlfors, Bers, Gehring, Earle, Gardiner, Reich, Strebel, Li Zhong, Wu Shengjian, Shen Yuliang and Chen Jixiu etc. Today, the theory of quasiconformal mappings and Teichmüller space has been cross-infiltrated into differential geometry, partial differential equations, topology and other branches of mathematics.
The main research contents in the theory of extremal quasiconformal mappings are the existence, uniqueness of extremal quasiconformal mappings for given boundary correspondence, and the properties and characteristics of extremal mappings, and so on. In chaptersⅡandⅢof this thesis, we will study these issues and obtain a series of results.
Logarithmic derivative plays an important role in determining whether a con-formal mapping can be quaisconformally extended, in estimating the inner radius of univalency of some domains, and in the description of universal Teichmüller space, the study of logarithmic derivative will play an active role in the development of quasiconfoaml mappings theory. In chaptersⅣof this thesis, we will study some geometrical properties of the universal Teichmüller space by the model of logarithmic derivative.
The tangent space of Teichmüller space (also called the infinitesimal Teichmüller space) plays an important role in the study of the Teichmüller space and in the description of extremal quaisiconformal mappings. Therefore, in this thesis we will also discuss some unknown problems in the infinitesimal Teichmüller space. The thesis is divided into five chapters.
ChapterⅠ, Introduction. We briefly introduce historical background and significance of the theory of quasiconformal mappings and Teichmüller space, and then describe the origin and the development of the problems which are studied in this dissertation, and state the results we obtained.
ChapterⅡ, On the extremality of quasiconformal mappings on quadratic parabolic domain. We already know a lot about the extremal quasiconformal mappings for given correspondence of all boundary points, but it remains unclear for the extremal quasiconformal mappings if we reduce the correspondence condition to a subset of the boundary. Strebel ([94] [95]) had studied this issue on several different domains, we will discuss this issue on quadratic parabolic domains.
ChapterⅢ, Some unequivalence between Teichmüller space and its tangent space. The description of the characteristics of extremality and unique extremality of quasiconformal mappings is a hot point in the theory of extremal quasiconformal mappings. Characterizations of extremal quasiconformal mappings were proved by Hamilton([41]), Krushkal([45]), Reich and Strebel([76]). Later in 1998, Bo(z|ˇ)in, Lakic, Markovi(?) and Mateljevi(?) ([9]) obtained some important characteristics of uniquely extremal quasiconformal mappings. From all these papers, we find that there are many equivalent properties between Teichmüller space and its tangent space. In this chapter, we will study the equivalence problem on Strebel points, the existence of extremal Teichmüller Beltrami coefficients, the existence of extremal Beltrami coefficients with constant modulus between Teichmüller space and its tangent space.
ChapterⅣ, Geometrical properties of the universal Teichmüller space by logrith-mic derivative. Logarithmic derivative is closely connected with quasiconformal extensions of univalent functions. The model of universal Teichmüller space by logarithmic derivative has many special geometric properties, Zhuravelv([116]) proved that the universal Teichmüller space by logarithmic derivative consists of infinitely many disjoint components. In this chapter, we study the geometric properties of geodesics and balls in each component of the universal Teichmüller space by logarithmic derivative.
ChapterⅤ, Some problems on infinitesimal Teichmüller space. From Chapter III, we know that there are many similar and non-similar properties between Teichmüller space and its tangent space. In this chapter, we study the non-uniqueness of geodesic disks in infinitesimal Teichmüller space, and also the Hamilton sequences of infinites-imally extremal Beltrami coefficients.
引文
[1] Ahlfors L.V., On quasiconformal mappings, J. Anal. Math. 3 (1954), 1-58 and 207-208.
[2] Ahlfors L.V., Lecture on quasiconformal mappings, Nostrand company, New York, 1966.
[3] Ahlfors L.V., Quasiconformal reflections, Acta. Math. 109 (1963), 291-301.
[4] Agard S. and Fehlmann R., On the extremality and unique extremality of affine mappings in space, Ann. Acad. Sci. Fenn. Ser. A I Math. 11 (1986), no. 1, 87-110.
[5] Agard S. and Fehlmann R., An addendum: "On the extremality and unique extremality of affine mappings in space" , Ann. Acad. Sci. Fenn. Ser. A I Math. 12 (1987), no. 2, 299-301.
[6] Astala K. and Gehring F. W., Injectivity, the BMO norm and the universal Teichmuller space, J. Anal. Math. 46 (1986), 16-57.
[7] Beardon A.F., The geometry of discrete groups, Spring-Verlag, New York Heidelberg Berlin, 1982.
[8] Beurling A. and Ahlfors L.V., The boundary correspondence under quasiconformal mappings, Acta. Math. 96 (1956), 125-142.
[9] Bozin V., Lakic N., Markovic V. and Mateljevic M., Unique extremality, J. Anal. Math. 75 (1998), 299-338.
[10] Chen Jixiu and Wei Hanbai, On some constants of quasiconformal deformation and Zygmund class, Chinese Ann. Math. Ser. B 16 (1995), no. 3, 325-330.
[11] Chen Jixiu, Chen Zhiguo and He Chengqi, Boundary correspondence under μ(z)-homeomorphisms, Michigan Math. J. 43 (1996), no. 2, 211-220.
[12] Chen Jixiu and Wei Hanbai, Some geometric properties on a model of universal Teichmuller spaces, Chinese Ann. Math. Ser. B 18 (1997), no. 3, 309-314.
[13] Chen Jixiu and Chen Zhiguo, A remark on: "An approximation condition and extremal quasiconformal extension" , Chinese Sci. Bull. 42(1997), no. 21, 1765-1767.
[14] Chen Jixiu, Zhu Huacheng and Liang Xiangqian, Moduli of quadrilaterals and substantial boundary points, Chinese Ann. Math. Ser. A 23 (2002), no. 2, 197-204.
[15] Chen Zhiguo, Chen Jixiu and He Chengqi, Functions with unbounded d-derivative and their boundary functions, J. Austral. Math. Soc. Ser. A 63 (1997), no. 1, 100-109.
[16] Chen Zhiguo, Chen Jixiu and He Chengqi, Teichmiiller extremal mappings on hyperbolic domains, Chinese Ann. Math. Ser. A 19 (1998), no. 3, 333-338.
[17] Chen Zhiguo, Geometric characterization for affine mappings and Teichmiiller mappings, Studia Math. 157 (2003), no. 1, 71-82.
[18] Cheng Tao and Chen Jixiu, On the inner radius of univalency by pre-Schwarzian derivative, Sci. China Ser. A 50 (2007), no. 7, 987-996.
[19] Cui Guizhen and Qi Yi, Local boundary dilatation of quasiconformal maps in the disk, Proc. Amer. Math. Soc. 130 (2002), no. 5, 1383-1389.
[20] Cui Guizhen and Zinsmeister Michel, BMO-Teichmiiller spaces, Illinois J. Math. 48 (2004), no. 4, 1223-1233.
[21] Douady A. and Earle C. J., Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), no. 1-2, 23-48.
[22] Earle C. J., The Teichmiiller distance is differentiable, Duke Math. J. 44 (1977), no. 2, 389-397.
[23] Earle C. J., Kra I. and Krushkall S. L., Holomorphic motions and Teichmiiller spaces, Trans. Amer. Math. Soc. 343 (1994), no. 2, 927-948.
[24] Earle C. J. and Li Zhong, Extremal quasiconformal mappings in plane domains, In: Quasiconformal mappings and analysis, New York: Springer, 1998, 141-157.
[25] Earle C. J. and Li Zhong, Isometrically embedded polydisks in infinite dimensional Teichmiiller spaces, J. Geom. Anal. 9 (1999), no. 1, 51-71.
[26] Earle C. J., Gardiner F. P. and Lakic N., Asymptotic Teichmiiller space. I. The complex structure, In: Contemp. Math. 256, 2000, 17-38.
[27] Earle C. J. and Lakic N., Variability sets on Riemann surfaces and forgetful maps between Teichmiiller spaces, Ann. Acad. Sci. Fenn. Math. 27(2002), no. 2, 307-324.
[28] Earle C. J., Gardiner Frederick P. and Lakic N., Asymptotic Teichmiiller. IL The metric structure, In: Contemp. Math. 355, 2004, 187-219.
[29] Fehlmann R., Quasiconformal mappings with free boundary components, Ann. Acad. Sci. Fenn. Ser. A I Math. 7 (1982), no. 2, 337-347.
[30] Fehlmann R., Extremal quasiconformal mappings with free boundary components in domains of arbitrary connectivity, Math. Z. 184 (1983), no. 1, 109-126.
[31] Fehlmann R. and Sakan K., On the set of substantial boundary points for extremal quasiconformal mappings, Complex Variables Theory Appl. 6 (1986), no. 2-4, 323-335.
[32] Fehlmann R. and Sakan K., On extremal quasiconformal mappings with varying dilatation bounds, Osaka J. Math. 23 (1986), no. 4, 751-764.
[33] Fehlmann R., Extremal problems for quasiconformal mappings in space, J. Analyse Math. 48 (1987), 179-215.
[34] Gardiner Frederick P., On the variation of Teichmuller's metric, Proc. Roy. Soc. Edinburgh Sect. A 85 (1980), no. 1-2, 143-152.
[35] Gardiner Frederick P., Approximation of infinite-dimensional Teichmiiller spaces, Trans. Amer. Math. Soc. 282 (1984), no. 1, 367-383.
[36] Gardiner Frederick P., Teichmiiller theory and quadratic differentials, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, 1987.
[37] Gardiner Frederick P., On completing triangles in infinite-dimensional Teichmuller spaces, Complex Variables Theory Appl. 10 (1988), no. 2-3, 237-247.
[38] Gardiner Frederick P. and Lakic N., Local boundary dilatation, In: Contemp. Math. 256, 2000, 131-141.
[39] Gardiner Frederick P. and Lakic N., Quasiconformal Teichmiiller theory, Mathematical Surveys and Monographs, 76. American Mathematical Society, 2000.
[40] Goldberg Lisa R., On the shape of the unit sphere in Q(△), Proc. Amer. Math. Soc. 118 (1993), no. 4, 1179-1185.
[41]Hamilton R.S.,Extremal quasiconformal mappings with prescribed boundary values,Trans.Amer.Math.Soc.138(1969),399-406.
[42]Heinonen J.and Koskela P.,Definition of quasiconformality,Invent.Math.120(1995),61-79.
[43]Huang Xin Zhong,On the extremality for Teichmiiller mappings,J.Math.Kyoto Univ.35(1995),no.1,115-132.
[44]Kravetz S.,On the geometry of Teichmuller spaces and the structure of their modular groups,Ann.Acad.Sci.Fenn.Set.A.I.278(1959),1-35.
[45]Krushkal S.,Extremal quasiconformal mappings,Siberian Math.J.10(1969),411-418.
[46]Lai Wancai and Wu Zemin,On extremality and unique extremality of Teichmuller mappings,Chinese Ann.Math.Set.B 16(1995),no.3,399-406.
[47]Lakic N.,Strebel points,In:Contemp.Math.211,1997,417-431.
[48]Lakic N.,Substantial boundary points for plane domains and Gardiner's conjecture,Ann.Acad.Sci.Fenn.Math.25(2000),no.2,285-306.
[49]Lehto O.,Univalent Functions and Teichmiiller Space,New York:SpringerVerlag,1986.
[50]Li Zhong,On the Existence of Extremal Teichmiiller Mappings,Comment.Math.Helv 57(1982),511-517.
[51]李忠,拟共形映射及其在黎曼曲面论忠的应用,北京:科学出版社,1988.
[52]Li Zhong,Nonuniqueness of geodesics in infinite-dimensional Teichmiiller spaces,Complex Variables Theory Appl.16(1991),no.4,261-272.
[53]Li Zhong,A problem on the convexity of the Teichmuller metric,Sci.China Ser.A 36(1993),no.10,1178-1185.
[54]Li Zhong,Non-convexity of spheres in infinite-dimensional Teichmuller spaces,Sci.China Ser.A 37(1994),no.8,924-932.
[55]Li Zhong,A note on geodesics in infinite-dimensional Teichmiiller spaces,Ann.Acad.Sci.Fenn.Ser.A I Math.20(1995),no.2,301-313.
[56]Li Zhong,Closed geodesics and non-differentiability of the metric in infinitedimensional Teichmuller spaces,Proc.Amer.Math.Soc.124(1996),no.5,1459-1465.
[57] Li Zhong and Qi Yi, A note on point shift differentials, Sci. China Ser. A 42 (1999), no. 5, 449-455.
[58] Li Zhong, Wu Shengjian and Qi Yi, A class of quasisymmetric mappings with substantial points, Chinese Sci. Bull. 45 (2000), no. 4, 313-316.
[59] Li Zhong, Strebel differentials and Hamilton sequences, Sci. China Ser. A 44 (2001), no. 8, 969-979.
[60] Li Zhong, Geodesic discs in Teichmiiller space, Sci. China Ser. A 48 (2005), no. 8, 1075-1082.
[61] Marden A. and Strebel K., A characterization of Teichmiiller differentials, J. Differential Geom. 37 (1993), no. 1, 1-29.
[62] Markovic V. and Mateljevic M., The unique extremal QC mapping and uniqueness of Hahn-Banach extensions, Mat. Vesnik 48 (1996), no. 3-4, 107-112.
[63] Reich E., On the relation between local and global properties of boundary values for extremal quasi-conformal mappings, In: Discontinuous groups and Riemann surfaces, Ann. of Math. Studies 79, New York: Princeton Univ. Press, 1974, 391-407.
[64] Reich E., On extremality and unique extremality of affine mappings, In: Topics in analysis, Lecture Notes in Math. 419, Berlin: Springer, 1974, 294-304
[65] Reich E., On the uniqueness problem for extremal quasiconformal mappings with prescribed boundary values, In: Complex analysis Joensuu, Lecture Notes in Math. 747, Berlin: Springer, 1979, 314-320.
[66] Reich E., On criteria for unique extremality of Teichmiiller mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), no. 2, 289-301 (1982).
[67] Reich E., A criterion for unique extremality of Teichmiiller mappings, Indiana Univ. Math. J. 30 (1981), no. 3, 441-447.
[68] Reich E., Nonuniqueness of Teichmiiller extremal mappings, Proc. Amer. Math. Soc. 88 (1983), no. 3, 513-516.
[69] Reich E., On the structure of the family of extremal quasiconformal mappings of parabolic regions, Complex Variables Theory Appl. 5 (1986), no. 2-4, 289-300.
[70] Reich E., Construction of Hamilton sequences for certain Teichmiiller mappings, Proc. Amer. Math. Soc. 103 (1988), no. 3, 789-796.
[71] Reich E., Extremal extensions from the circle to the disk, In: Quasiconformal mappings and analysis, New York: Springer, 1998, 321-335.
[72] Reich E., The unique extremality counterexample, J. Anal. Math. 75 (1998), 339-347.
[73] Reich E., Extremal quasiconformal mappings of the disk, Handbook of complex analysis: geometric function theory, Vol. 1, 75-136, North-Holland, Amsterdam, 2002.
[74] Reich E. and Strebel K., On quasiconformal mappings which keep the boundary points fixed, Trans. Amer. Math. Soc. 138(1969), 211-222.
[75] Reich E. and Strebel K., On the extremality of certain Teichmiiller mappings, Comment. Math. Helv. 45(1970), 353-362.
[76] Reich E. and Strebel K., Extremal quasiconformal mappings with given boundary values, In: Contributions to analysis, New York: Academic Press, 1974, 375-391.
[77] Royden H. L., Automorphisms and isometries of Teichmiiller space, In: Advances in the Theory of Riemann Surfaces, Ann. of Math. Studies 66. New York: Princeton Univ. Press, 1971, 369-383,
[78] Rudin W., Real and complex analysis, New York: McGraw-Hill, 1987.
[79] Shen Yuliang, On the unique extremality of quasiconformal mappings with dilatation bounds, Tohoku Math. J. (2) 56 (2004), no. 1, 105-123.
[80] Shen Yuliang and Chen Jixiu, Quasiconformal mappings with non-decreasable dilatations, Chinese Ann. Math. Ser. A 23 (2002), no. 4, 459-466.
[81] Shen Yuliang, On Teichmiiller geometry, Complex Variables Theory Appl. 44 (2001), no. 1, 73-83.
[82] Shen Yuliang, A note on Hamilton sequences for extremal Beltrami coefficients, Proc. Amer. Math. Soc. 129 (2001), no. 1, 105-109.
[83] Shen Yuliang, A counterexample theorem in quasiconformal mapping theory, Sci. China Ser. A 43 (2000), no. 9, 929-936.
[84] Shen Yuliang, On the extremality of quasiconformal mappings and quasiconformal deformations, Proc. Amer. Math. Soc. 128 (2000), no. 1, 135-139.
[85] Shen Yuliang, On unique extremality, Complex Variables Theory Appl. 40 (1999), no. 2, 149-162.
[86] Shen Yuliang, On the geometry of infinite-dimensional Teichmiiller spaces, Acta Math. Sinica (N.S.) 13 (1997), no. 3, 413-420.
[87] Shen Yuhang, Some remarks on the geodesics in infinite-dimensional Teichmiiller spaces, Acta Math. Sinica (N.S.) 13 (1997), no. 4, 497-502.
[88] Shen Yuliang, Extremal quasiconformal mappings compatible with Fuchsian groups, Acta Math. Sinica (N.S.) 12 (1996), no. 3, 285-291.
[89] Shen Yuliang, On the weak uniform convexity of Q(R), Proc. Amer. Math. Soc. 124 (1996), no. 6, 1879-1882.
[90] Shen Yuliang, Nonuniqueness of geodesics in the universal Teichmuller space, Adv. in Math. (China) 24 (1995), no. 3, 237-243.
[91] Strebel K., On the dilatation of extremal quasiconformal mappings of polygons, Comment. Math. Helv. 74 (1999), no. 1, 143-149.
[92] Strebel K., Point shift differentials and extremal quasiconformal mappings, Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 2, 475-494.
[93] Strebel K., On certain extremal quasiconformal mappings of parabolic regions, Mitt. Math. Sem. Giessen 228 (1996), 39-50.
[94] Strebel K., On the extremality and unique extremality of certain Teichmuller mappings, In: Complex analysis, Basel: Birkhauser, 1988, 225-238.
[95] Strebel K., On the extremality and unique extremality of quasiconformal mappings of a parallel strip, Rev. Roumaine Math. Pures Appl. 32 (1987), no. 10, 923-928.
[96] Strebel K., Is there always a unique extremal Teichmuller mapping?, Proc. Amer. Math. Soc. 90 (1984), no. 2, 240-242.
[97] Strebel K., On lifts of extremal quasiconformal mappings, J. Analyse Math. 31 (1977), 191-203.
[98] Strebel K., On the existence of extremal Teichmuller mappings, J. Analyse Math. 30 (1976), 464-480.
[99] Strebel K., Zur Frage der Eindeutigkeit extremaler quasikonformer Abbildungen des Einheitskreises, Comment. Math. Helv. 36(1961/1962), 306-323.
[100]Tanigawa H.,Holomorphic families of geodesic discs in infinite-dimensional Teichmuller spaces,Nagoya Math.J.127(1992),117-128.
[101]Vaisala J.,Lectures on n-dimensional quasiconformal mappings,Lecture Notes in Mathematics 229.Berlin-New York:Springer-Verlag,1971.
[102]Wu Shengjian,Hamilton sequences for extremal quasiconformal mappings in the unit disk,Sci.China Set.A 42(1999),no.10,1032-1038.
[103]Wu Shengjian,Moduli of quadrilaterals and extremal quasiconformal extensions of quasisymmetric functions,Comment.Math.Helv.72(1997),no.4,593-604.
[104]Yao Guowu,Exsistence of extremal Beltrami coefficents with non-constant modulus,preprint.
[105]Yao Guowu,Local extremality and extremal non-decreasable dilatation,Bull.Aust.Math.Soc.75(2007),321-329.
[106]Yao Guowu and Qi Yi,The modulus of extremal Beltrami coefficients,J.Math.Kyoto Univ.46(2006),235-247.
[107]Yao Guowu,Is there always an extremal Teichmuller mapping,J.Anal.Math.94(2004),363-375
[108]Yao Guowu,A proof of Sethares' conjecture,Sci.China Ser.A 47(2004),no.2,236-244.
[109]Yao Guowu,On criteria for extremality of Teichmuller mappings,Proc.Amer.Math.Soc.132(2004),no.9,2647-2654.
[110]Yao Guowu,Hamilton sequences and extremality for certain Teichmfiller mappings,Ann.Acad.Sci.Fenn.Math.29(2004),no.1,185-194.
[111]Wang Zhe,The distance between different components of the universal Teichmuller space,Chin.Ann.Math.26(2005),537-542.
[112]Wei Hanbai,关于万有Teichmuller空间T_1的分支,数学年刊25(2004),425-428.
[113]Zhou Zemin,Chen Jixiu and Yang Zongxin,On the extremal sets of extremal quasiconformal mappings,Sci.China Set.A 46(2003),no.4,552-561.
[114]Zhou Zemin and Chen Jixiu,Quasiconformal mappings with non-decreasable Beltrami coefficients,Acta Math.Sinica(Chin.Ser.) 46(2003),no.2,379-384.
[115]Zhu Huacheng and Chen Jixiu,On characterizations of unique extremality,J.Math.Anal.Appl.273(2002),no.2,283-293.
[116]Zhuravelv I V.,Mordel of the universal Teichmuller space,Siberian Mathematical Journal 27(1986),691-697.