Weil-Petersson万有Teichmüller空间与Dirichlet能量相关问题研究

This research is mainly about three related problems in quasiconformal mapping and Teichmüller space theory..The first one is the characterization of the Weil-Petersson universal Teichmüller space. Universal Teichmüller space has several equal models. We hope to find out whether a known condition about an operator induced by quasisymmetric homeomorphism is sufficient for the homeomorphism to belong to Weil-Petersson class. Meanwhile we try to find some characterizations with respect to quasidisk. .The second problem is the property of a functional of Weil-Petersson universal Teichmüller space defined by the minimal energy.Quasisymmetric homeomorphism in Weil-Petersson class has exact one quasiconformal extension which is harmonic in Poincare metric, which also minimizes the modified Dirichlet energy integral in all the quasiconformal extensions of the homeomorphism..The third problem is the existence of harmonic mapping and the extremal mapping of finite distortion under some functionals. The existence of harmonic mapping and Dirichlet energy are closely related to the extremal problem of mapping of finite distortion with smallest mean distortion.

本项目主要研究拟共形映射与Teichmüller空间理论中三个相互关联的问题。.问题之一是Weil-Petersson 万有Teichmüller空间的刻画。研究主要希望进一步判定拟对称同胚所诱导的某算子条件的充分性，同时考虑从拟圆角度的刻画；.问题之二是研究Weil-Petersson万有Teichmüller空间上的某一泛函的性质。Weil-Petersson万有Teichmüller空间中的元素存在唯一的双曲度量下的调和拟共形扩张，并且其修正的Dirichlet能量积分为这一拟对称同胚的所有拟共形扩张中最小的，因而由最小的能量积分可以确定Weil-Petersson万有Teichmüller空间上的一个泛函；.问题之三是讨论区域间调和映照的存在性以及有限伸缩映射在某些泛函下的极值问题。调和映照的存在性、Dirichlet能量和有限伸缩映射的平均伸缩极值问题联系密切。

研究主要关于万有Teichmuller空间子空间的特征，尤其是Weil-Petersson类子空间，这一子空间与万有Teichmuller空间上的Weil-Petersson度量密切联系，是无穷可微同胚子集在Weil-Petersson度量下的完备空间，同时，该子空间包含于另几类广受关注的子空间：对称同胚的子空间、BMO-Teichmuller空间、VMO-Teichmuller空间。.研究通过单叶函数拟共形延拓的伸缩商得到了单叶函数的Grunsky算子本性模的一个估计，并由此给出算子的紧性条件，即单叶函数可以渐进共形延拓到单位圆，从而其对应的单位圆周的同胚为对称同胚，由此对于对称同胚对应的Grunsky算子为紧算子这一结论给出了新的证明。.基于交比这一有力工具，以及在以往研究中已经通过交比给出了Weil-Petersson类拟对称同胚的必要特性，因此研究主要的思路是通过单位圆周拟对称同胚的交比给出判断拟对称同胚是Weil-Petersson类拟对称同胚的充分条件。以拟对称同胚的交比线性控制拟共形同胚的伸缩商，以及关于导数具有Holder连续性的微分同胚的有关研究，对于Poincare度量下伸缩商的估计和本研究的进一步进行有重要意义。.