本项目主要研究以下内容：（1）欧氏空间中的特殊域与拟共形映射：计划以拟共形映射与拟对称映射的关系为主要工具，寻求不同的量来刻画欧氏空间中拟共形等价于一致域的John域的特征，此研究将完善Heinonen的相关讨论；同时通过域的度量边界和Gromov双曲边界之间的对应关系来讨论域的球可分离性和Gehring-Hayman不等式，得到Gromov双曲性的新刻画，从而攻克Balogh和Buckley于2003年在Invent. Math.上提出的公开问题。（2）Banach空间中特殊域上自由拟共形映射：Vaisala在研究相对边界的拟Mobius变换和CQH映射时提出了“Banach空间中的nature域是否具有相对边界拟Mobius变换的不变性”这一猜想。我们计划利用几何构造等方法来研究此问题。作为应用，将研究关于自由拟共形映射Bilipschitz扩张的另一猜想。此研究具有重要的理论意义。

The main aim of this project is to study the following problems:.(1) Special domains and quasiconformal mappings in R^n: With the aid of the relations between quasiconformalm mappings and quasisymmetric mappings, we shall find conditions for the domains in R^n to be John, which are quasiconformally equivalent to uniform domains. This research will make the corresponding discussions due to Heinonen to be more perfect. Also we shall discuss the ball separation property and Gehring-Hayman’s inequality in the domains in R^n and then get new characteristics of Gromov hyperbolicity by use of the relations between the metric boundary and the Gromov hyperbolic boundary of domains. These will completely solve an open problem raised by Balogh and Buckley in Invent. Math. in 2003. (2) Freely quasiconformal mappings on special domains in Banch spaces: When Vaisala studied quasi-mobius mappings and CQH mappings in Banach spaces, he proposed the following open problem: Whether are the natural domains in Banach spaces invariant under the quasi-Mobius mappings? We plan to study this problem by using the geometric construction method etc. As applications, we shall study another problem on the Bilipschitz extention theorem of freely quasiconformal mappings. This research is very significant in theory.