摘要
本论文主要讨论了极值拟共形映射与Teichmiiller空间中的若干个问题,主要包括了：
1.极值Beltrami系数的Hamilton序列的构造问题.
2.具有弱不可缩伸缩商的极值拟共形映射在每个Teichmuller等价类中的存在性问题.
3.万有Teichmuller空间对数导数嵌入模型及Schwarz导数嵌入模型的几何性质；平面区域的Schwarz导数单叶性内径问题.
4.渐近Teichmuller空间的几何性质,主要包括闭测地线存在性问题及关于球的非凸性问题.
全文共分为五章.
第一章,引言.我们简要介绍了拟共形映射与Teichmuller空间理论的历史背景,研究意义及现状，并阐述本文所研究的问题以及主要结果.
第二章,极值Beltrami系数的Hamilton序列.本章考虑了Strebel点与Hamil-ton序列之间的关系F. P. Gardiner最早研究了这个问题(见[42]),我们讨论了在无限小Teichmiiller空间中的对应情况,证明了范金华在[35]中得到的使{φn}成为Hamilton序列的充分条件不是必要的.
第三章,具有弱不可缩伸缩商的极值拟共形映射.具有不可缩伸缩商的拟共形映射的概念是由Edgar Reich引进的,在极值拟共形映射理论中起到了重要的作用.这其中有一个至今未解决的问题,即在每个Teichmuller等价类中，是否一定存在一个极值的具有不可缩伸缩商的拟共形映射?在本章中,我们部分地解决了这个问题.证明了在每个Teichmuller等价类中,一定存在一个极值的具有弱不可缩伸缩商的拟共形映射.
第四章,万有Teichmuller空间的嵌入模型及区域的单叶性内径.在本章中,我们证明了在万有Teichmuller空间的对数导数嵌入模型T1(△)中,存在无穷多个点[h]∈L(?)T1(△),h(△)相互不Mobius等价,它们到边界的距离均为1,而在万有Teichmuller空间的Schwarz导数嵌入模型T(△)中,只有一个点Sid具有类似性质.另外还获得了万有Teichmuller空间两类嵌入模型的测地线的一些新的性质以及一类正规三角形外部区域的Schwarz导数单叶性内径.
第五章,渐近Teichmuller空间的闭测地线及球的非凸性.在本章中,我们研究渐近Teichmuller空间的几何性质.在无限维渐近Teichmuller空间上构造了闭的测地线,并证明了渐近Teichmuller空间关于球的非凸性.
In this thesis, some problems of extremal quasiconformal mappings and Te-ichmiiller spaces are discussed, especially including:
1. The construction problem of the Hamilton sequences for extremal Beltrami coefficients.
2. The existence problem of extremal quasiconformal mappings with weakly non-decreasable dilatations in every Teichmuller equivalence class.
3. The geometric property of the Schwarzian derivative embedding model of Universal Teichmuller Space and pre-Schwarzian derivative embedding model of Uni-versal Teichmuller Space; the problem of the inner radius of univalency by the Schwarzian derivative.
4. The geometric property of asymptotic Teichmuller space, especially the existence problem of closed geodesies and the non-convexity problem of spheres.
There are five chapters in this thesis.
Chapter Ⅰ, Introduction. In this chapter, the background, significance and status of the research of quasiconformal mappings and Teichmuller spaces are intro-duced, and the problems and main results of this thesis are presented.
Chapter Ⅱ, Hamilton sequences for extremal Beltrami coefficients. In this chap-ter, the relationship between Hamilton sequence and Strebel points is discussed, which was first studied by F. P. Gardiner in [42]. We prove that in the case of infinitesimal Teichmuller space, the sufficient condition for(φn} to be a Hamilton sequence obtained by Fan in [35] is not necessary.
Chapter Ⅲ, Extremal quasiconformal mappings with weakly non-decreasable dilatations. The notion of non-decreasable dilatation for quasiconformal mappings, which was introduced by Edgar Reich, plays an important role in the theory of extremal quasiconformal mappings. It is an interesting open problem so far whether an extremal quasiconformal mapping with non-decreasable dilatation exists in every Teichmuller equivalence class. In this chapter, we have partially solved this problem. It is proved that for every Teichmuller equivalence class, there exists an extremal quasiconformal mapping with weakly non-decreasable dilatation.
Chapter Ⅳ, The embedding models of Universal Teichmuller Space and the inner radius of univalency of plane domains. In this chapter, we find that in pre-Schwarzian derivative embedding model of Universal Teichmuller Space T1(△), there exist infinitely many [h]∈L(?)T1(△) such that h(△) are not Mobius equivalent to each other, and the distance from each point [h] to the boundary of T1(△) equals to1, while in Schwarzian derivative embedding model of Universal Teichmuller Space, only Sid has the analogous property. Some other properties of the Schwarzian deriva-tive embedding model of Universal Teichmuller Space and pre-Schwarzian derivative embedding model of Universal Teichmuller Space are concerned, and the inner ra-dius of univalency for the outer domain of a class of normal circular triangles by the Schwarzian derivative is also obtained.
Chapter V, Closed geodesies and non-convexity of spheres in asymptotic Te-ichmuller spaces. In this chapter, the geometric property of asymptotic Teichmuller space is studied. Closed geodesies in any infinitely dimensional asymptotic Te-ichmuller space are constructed, and the non-convexity of spheres in asymptotic Teichmuller space is proved.
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