Open Riemann surfaces and quasiconformal mappings

(1)在对开Riemann曲面的研究中，我们证明了：在有限维约化的TeichMuller空间中，Denjoy域不能共形嵌入到其补集为正测度的域上，因此这种嵌入是唯一的。给定的黎曼曲面可以共形嵌入的黎曼曲面集成为约化的TeichMuller空间中的实解析子流形。我们研究了黎曼曲面之间的一些嵌入条件。(2)在同胚映射群的研究中，对于紧致子流形上相同的非紧连通二维PL-流形的同胚映射拓扑群，确定了其连通分支的同伦型，并且在某些例外情况下是压缩的。(3)在曲面的代数研究中，对于具有穿孔的黎曼曲面基本群的任一有限指数子群，都存在一个有限指数的特征子群，其中的商是无心的。(4)在小波研究中，M阶尺度函数的光滑性关于S指数。考虑了一个与秩为M的小波尺度函数密切相关的函数方程，在解只有一个消失点的情况下，研究了消失点的位置和非负解的存在性。(5)在马尔可夫过程的研究中，给出了从属对称马尔可夫过程的Dirichlet形式的显式公式。特别地，给出了系统的尖端常返性和暂时性准则以及全局容量不等式。

Our main subject is to analyze phenomenon which is caused on open Riemann surface.(1) In the study of open Riemann surfaces, we show the following : Denjoy domain can not be conformally embedded to a domain whose complement in the complex plane is of positive measure, hence the embedding is unique, In a finite dimensional reduced Teichmuller space. the set of Riemann surfaces, which a given Riemann surface can be conformally embedded, become a real analytic submanifold in the reduced Teichmuller space. We investigate some embeddable condition between Riemann surfaces. Beside we give an extension of Ahlfors Rauch variational formula.(2) In the study of homeomorphic mappings group, for the topological group of homeomorphic mappings of non compact, connected 2 dimensional PL-manifold whose is identical on a compact submanifold, the homotopy type of the connected component is determined, and it is contractive some exceptional case. In.the exceptional case, the homotopy type is that of circle.(3) In the algebraic study of surface, For any subgroup with finite index of the fundamental group of a Riemann surface with puncture, there exists a characteristic subgroup with finite index, by which quotient is centerfree.(4) In the study of wavelet, the smoothness of rank M scaling function in terms of the s_p-exponent. a functional equation, which is closely related to scaling function of rank M wavelet, is considered, the case of the solution has only one vanishing point is precisely investigated about the position of vanishing point and the existence of non negative solution.(5) In the study of Markov process, An explicit formula for the Dirichlet form of a subordinated symmetric Markov process is given. Especially, sharp recurrence and transience criteria and global capacitary inequalities are given.