极值拟共形映射与渐近Teichmuller空间相关问题

Research of extremal quasiconformal mappings and Teichmuller spaces is one of the mainstream directions of modern mathematical research, which have close ties and mutual influence with other areas of mathematics such as complex dynamical systems, the Klein group theory and low-dimensional topology, and also have a wide range of applications in theoretical physics, statistical physics,thermodynamics, and other disciplines. There are a large number of challenging open problems to be resolved in extremal quasiconformal mappings and Teichmuller spacestheory. This project will focus on the important problems of extremal quasiconformalmappings and asymptotic Teichmuller spaces, which have already attracted a wide range of attention, such as uniquely extremal quasiconformal mappings, uniquely infinitesimal extremal quasiconformal mappings, non-decreasabledilatation problems and a characterization of points in the closure of the image of the asymptotic Bers map from the asymptotic Teichmuller space of the unit disk, and explore their application prospects.

拟共形映射与Teichmuller空间理论的研究属于现代数学研究的主流方向之一，它们与复动力系统、Klein群理论、低维拓扑等数学领域有密切联系和相互影响，并在理论物理、统计物理、热力学等其他学科有广泛的应用。拟共形映射和Teichmuller空间理论中有大量具有挑战性的问题有待解决，本申请项目将着重研究极值拟共形映射和渐近Teichmuller空间中受到广泛关注的重要问题，如唯一极值与无限小唯一极值拟共形映射不可缩小伸缩商问题，渐近Teichumller空间的嵌入模型和其边界点的刻画等问题，并探讨它们的应用前景。

本申请项目着重研究了极值拟共形映射和渐近Teichmuller空间中受到广泛关注的重要问题，主要包括.1.唯一极值拟共形映射delta序列的构造问题，申请人对一类经典的唯一极值映射构造出了其对应的具体的delta序列；.2.渐近Teichumller空间的嵌入模型和其边界点的刻画问题，申请人利用实分析的方法研究一类具有Lipschitz性质的有界解析函数，并利用所得的结果刻画出了渐近Teichmuller 空间嵌入模型的边界点的一个必要条件,并探讨它们的应用前景。