Some remarks on meromorphic functions on open Riemann surfaces (Q1080985)

In this work R denotes an arbitrary open Riemann surface of genus g (\(\geq \infty)\). \(\{\Omega_ n\}\) \((n=1,2,...)\) is a special canonical exhaustion of R. \(D=\{dw_ i| dw_ i\) \((i=1,2,...)\) semiexact holomorphic differential on R, \(\int_{A_ j}dw_ j=\delta_{ij}\) (Kronecker's \(\delta)\), \(\int_{B_ j}dw_ j\in C\}\). \(D_ 0=\{dw_ i\in D|\) \(dw_ i\) square integrable\(\}\), \(M=\{f|\) f meromorphic function on R, (i) there exist an integer \(n_ 0\) such that for all \(n\geq n_ 0\int_{\gamma_ n^{(i)}}d \log f=0\) \((i=1,2,...,m_ n)\) where \(\gamma_ n^{(i)}\) is a component of \(\partial \Omega_ n\) not containing zeros and poles of f, \((ii)\quad \lim_{n\to \infty}\int_{\partial \Omega_ n}w_ id \log f=0,\quad dw_ i\in D,\quad w_ i=\int^{p}_{p_ 0}dw_ i=w_ i(p)\}.\) \(M_ 0\) is defined for \(D_ 0\) as M is defined for D. By replacing the second period condition in M, the author considers the classes M' and \(M_ 0'\). Also \(M_{\chi}\) corresponds to \(\Gamma_{\chi}\)-behaviour differentials in Kusunoki sense. With this notation the author gives some necessary and sufficient conditions for the existence of a single valued meromorphic function f, which has a given divisor, belonging to M, \(M_ 0\), M', \(M_ 0'\) and \(M_{\chi}\) respectively.