拟共形Teichmuller理论及其相关研究

利用经典的和最近新发展的研究工具和方法，本项目拟将进一步研究万有Teichmuller空间及其子空间的相关性质. 为了更好的了解弦弧曲线空间的结构，我们利用Muckenhoupt权和渐进权定义一些拟对称映射子类，通过FKP延拓与共形粘和，经Bers嵌入与预Bers嵌入，刻画这类拟对称映射所对应的Teichmuller空间理论， 特别的，给实直线上强对称同胚一个新的刻画. Markovic证明了著名的Schoen猜想，我们进一步研究一些经典和新引进的拟对称映射子类的调和拟共形延拓以及所对应Teichmuller空间的调和嵌入. 我们将刻画导数属于BMO、VMO以及Besov的向量场的流曲线在BMO、VMO以及可积Teichmuller空间中是连续可微的. 建立单叶函数的拟共形延拓、拟圆周的拟共形反射与Jones \beta 数的关系并应用到万有Teichmulelr空间及其子空间的研究中.

By means of some classical and recently developed research tools and methods, this project shall give some further study of the universal Teichmuller space and its subspaces. In order to better understand the structure of the chord arc curves space, we introduce some quasisymmetric subclasses in terms of Muckenhoupt weights and asymptotically weights. Using FKP extension and conformal welding, via Bers embedding and pre-Bers embedding, we investigate their Teichmuller theory, and in particular, give the strong symmetric homeomorphism a new characterization. Markovic proved the famous Schoen conjecture, we shall study the harmonic quasiconformal extensions of some classical and new quasisymmetric subclasses and the harmonic embedding of the corresponding Teichmuller spaces. We shall prove the flow curves of the vector fields which derivatives belong to BMO, VMO and Besov spaces are continuous differentiable in BMO, VMO, and integrable Teichmuller spaces. We shall also establish the relationship between quasiconformal extension of univalent functions, quasiconformal reflection of quasicircle and the Jones \beta number and apply it to the study of the universal Teichmuller space and its subspaces.