Open Riemann surfaces and quasiconformal mappings

我们的主要课题是分析在开黎曼曲面上产生的现象。(1)在开Riemann曲面的研究中，我们证明了：在有限维约化Teichmuller空间中，Denjoy域不能共形嵌入到复平面上的补为正测度的域，因此嵌入是唯一的.可以保形嵌入给定Riemann曲面的Riemann曲面集成为约化Teichmuller空间中的真实的解析子流形。研究了Riemann曲面之间的可嵌入条件。此外，我们还推广了Ahlfors-Rauch变分公式. (2)在同胚映射群的研究中，对于在紧致子流形上全同的非紧连通二维PL-流形的同胚映射所构成的拓扑群，确定了其连通分支的同伦型，并在某些例外情况下证明了它是压缩的.在例外情况下，同伦类型是圆的同伦类型。(3)在曲面的代数研究中，对带穿刺的Riemann曲面的基本群的任意有限指标子群，都存在一个有限指标特征子群，它的商是无中心的。(4)在小波的研究中，研究了M阶尺度函数的s_p指数光滑性。考虑一个与M阶小波尺度函数密切相关的函数方程，精确地研究了其解只有一个消失点的情况下消失点的位置及非负解的存在性. (5)在马氏过程的研究中，给出了从属对称马氏过程的Dirichlet型的一个显式公式。特别地，给出了一类非线性系统的强常返性和强瞬时性的判别准则以及整体的容性不等式。

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.