On continuable Riemann surfaces (Q1288524)

A Riemann surface \(R\) is called continuable if there exists a conformal mapping of \(R\) onto a proper subregion of some other Riemann surface. \(R\) is call maximal, if it is not continuable. The author proves sufficient conditions for a Riemann surface to be continuable. Let \(R\) be a Riemann surface with Green function \(g_{p_0}(p)\) and let \(B(p_0,\alpha), \alpha >0,\) be the Betti number of \(\{p \in R: g_{p_0}(p)>\alpha \}\). If \(B(p_0,\alpha)\) satisfies some growth restriction as \(\alpha\) tends to zero, then \(R\) is continuable. As a corollary the author proves the continuability of an \(n\)-sheeted unlimited covering surface of the unit disc under certain growth restrictions on the branch points of this covering.