On maximality of two-sheeted unlimited covering surfaces of the unit disc (Q1125252)

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 called maximal, if it is not continuable. The author continues his studies in Hiroshima Math. J. 26, No. 2, 385-404 (1996; Zbl 0868.30038) and Kodai Math. J. 21, No. 3, 318-329 (1998; Zbl 0921.30031). Let \(R\) be a two-sheeted unlimited covering surface of the unit disk \(U\) with projection mapping \(\pi\). This projection mapping continues continuously to the Kuramochi boundary of \(R\). It is proved that \(R\) is maximal if \(\pi^{-1}(e^{i\theta})\) is minimal for every \(e^{i\theta} \in \partial U\). Several extensions and refinements of this result are also obtained.