A research of moving open Riemann surfaces

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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.