Open Riemann surfaces and quasiconformal mappings

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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.