A research of moving open Riemann surfaces

我们的主要课题是分析在运动的黎曼曲面上产生的现象，黎曼曲面的解析结构、几何结构和代数结构是密切相关的。我们对这一主题进行了多边研究，并取得了一些成果。在分析方面，研究了黎曼曲面共形焊接下解析结构的变化，以及黎曼曲面全纯拟共形变形下Bergman度规的行为。研究了在无穷远处衰减的电势扰动下二维和三维泡利算子在本质谱底部附近的离散特征值的渐近分布，以及具有冯诺依曼和维格纳电位的薛定谔算子正特征值的不存在域。在概率论中，给出了从属马尔可夫过程的狄利克雷形式的显式公式，并证明了从属马尔可夫过程保留狄利克雷形式的核。研究了隶属马尔可夫过程的一些全局性质。特别给出了递归准则和全局容量不等式。在多分辨率分析中，给出了膨胀因子为n>2的小波的显式构造。从几何的角度考虑了一类Riemann曲面上的同胚及其子类拟共形映射，并证明了它们是一个（s-SIGMA）流形。用新方法计算了复射影空间上线束的解析扭转。从代数的角度，研究了沿边界粘贴的两个曲面映射类群过滤的相容性。

Our main subject is to analyze phenomenon which is caused on a moving Riemann surface, where analytic, geometric and algebraic structure are closely related. We study the theme multilateraly and get some results. In the field of analysis, the change of analytic structure under conformal welding of Riemann surface, and the behavior of Bergman metric under holomorphically quasiconformal deformation of Riemann surface are investigated, In the physical aspect, the asymptotic distribution of discrete eigenvalues near the bottom of the essential spectrum for 2-and 3-dimensional Pauli operators perturbed by electric potentials falling off at infinity and the non existence domain of positive eigenvalues of Schrodinger operator with von Neumann and Wigner potentials are studied. In the probability theory, an explicit formula for the Dirichlet form of a subordinated Markov process is given and it is shown that subordination preserves cores of Dirichlet forms. Some global properties for subordinated Markov processes are also studied. Especially, recurrence criteria and global capacitary inequalities are given. In multi resolution analysis, explicit construction of wavelet with a dilation factor n>2 is given. From a geometric point of view, the class of homeomorphism on a Riemann surface and its subclass of quasiconformal mappings are considered and it is shown that the pair is an (s-SIGMA)-manifold. Analytic torsion of line bundle on complex projective space is calculated by new approach. From an algebraic point of view, the compatibility of the filtration of mapping class groups of two surfaces pasted along the boundaries is studied.