Problems Of Extremal Quasiconformal Mappings And Teichm(u|")ller Spaces

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.