度量空间上的拟共形映射及其相关研究

The main aim of this project is to study the following problems:..(1) Quasiconformal mappings in R^n: In 1989, when Heinonen discussed the quasi conformal mappings on John domains, he raised an open problem regarding the relationship between weakly quasisymmetric mappings and quasisymmetric mappings. We plan to work on this problem by using the conformal modulus method of path families etc. Some applications of the planned results will be given. This will reveal the close relationship among quasiconformal mappings, quasisymmetric mappings and John domains...(2) Gehring-Hayman's inequlity in Banach spaces: How to establish the Gehring-Hayman inequality in Banach spaces attracts much attention. In 1993 and in 2005, respectively, Heinonen, Rohde and Vaisala put forward this as an open problem. We plan to devote to this problem by establishing a new method which is the combination of geometry and analysis. As an application, we will study the related problems raised by Gehring, Hag and Martio in 1989 etc...(3) The quasiconformal theory of the Gromov hyperbolic boundaries of uniform domains in metric spaces: For uniform domains in metric spaces, by using quasisymmetric mappings with respet to boundaries, quasimobius transformations with respect to the boundaries etc between their metric boundaries and their Gromov hyperbolic boundaries, we will characterize their roughly quasiisometric equivalence. As an application, we will discuss the relationship between the Lipschitz equivalence and the quasisymmetric equivalence of two homeomorphic uniform domains in metric spaces...This study has the very important theoretical significance.

本项目计划主要研究以下内容。（1）R^n中的拟共形映射：1989年，Heinonen在研究John域上的拟共形映射时提出了关于弱拟对称映射与拟对称映射关系的猜测。我们将利用曲线族的共形模等来讨论这个问题，从而揭示拟对称映射、拟共形映射与John域之间的紧密联系。（2）Banach空间上的Gehring-Hayman不等式：如何建立Banach空间中的Gehring-Hayman不等式被Heinonen和Rohde于1993年、Vaisala于2005年分别作为公开问题提出。我们将建立一种几何与分析相结合的办法来攻克此问题，并给出相关应用。（3）度量空间中一致域Gromov双曲边界上的拟共形理论：我们将利用度量空间中一致域的度量边界和Gromov双曲边界之间的（相对边界）拟对称对应和(相对边界)拟Mobius对应等来刻画它们之间的rough拟等距等价性，并给出应用。此研究具有重要的理论意义。

此项目研究期间，我们按原计划对拟共形映射、Gromov双曲性、调和映射、调和映射与拟共形映射的联系、调和映射与偏微分方程的联系等展开了研究，得到了系列结果，其中肯定回答公开问题和猜测6个。共发表论文（标注本项目资助）39篇，其中在SCIE刊物发表论文37篇。具体如下：.（一）在前一个项目资助期间所得到研究工作的基础上，进一步讨论了欧几里德空间上的拟共形映射、拟对称映射和弱拟对称映射三者的关系；研究了度量空间上的Semisolidity和局部弱拟对称性的关系。这些研究均是围绕欧几里德空间上的拟共形映射是否具有拟对称性这一极具重要性的问题而展开的。这些研究解决了由美国Heinonen教授等提出的公开问题。主要结果发表在Adv. Math.、Studia Math.等国际权威刊物。.（二）提出了roughly Apollonian bilipschitz同胚以及φ-距离比同胚两个映射类，得到了域的Gromov双曲性等性质关于这些映射类的不变性。同时开始了拟度量空间上拟共形映射的研究：提出了弱Quasi-Mobius映射，得到了此类映射、以及与Quasi-Mobius映射相关的系列结果，为将来的继续研究奠定了基础。相关结果发表在Ann. Acad. Sci. Fenn. Math. 等国际权威刊物。.（三）找到了双曲Poisson方程的通解，同时还研究了Poisson方程和双曲Poisson方程解的Lipschitz连续性。这些研究推广了已有关于调和映射的相关研究。主要结果发表在 Calc. Var. Partial Differential Equations、Israel J. Math.、Ann. Acad. Sci. Fenn. Math.等国际权威刊物。