On maximal Riemann surfaces (Q674610)

A Riemann surface \(\widetilde R\) is called an extension of the Riemann surface \(R\) if there is a conformal embedding of \(R\) into \(\widetilde R\). If \(\widetilde R-R\neq\emptyset\), then we say that the extension is proper. For instance, each Riemann surface is an extension of itself. A Riemann surface is called maximal if it has no proper extensions. An extension \(\widetilde R\) of a Riemann surface \(R\) is called maximal if \(\widetilde R\) is a maximal Riemann surface. It is known that every Riemann surface has a maximal extension [\textit{S. Bochner}, Math. Ann. 98, 406-421 (1928; JFM 53.0322.01)], but it may not be unique, that is, it may have non-conformally equivalent maximal extensions. Maximal Riemann surfaces of course have unique maximal extensions. It is known [\textit{H. Renggli}, Duke Math. J. 42, 211-224 (1975; Zbl 0335.30013)] that \(R\) has a unique maximal extension if and only if \(R\) is conformally equivalent to some \(\widetilde R-E\), where \(\widetilde R\) is a maximal Riemann surface and \(E\) is a closed \(N_D\)-set in \(\widetilde R\). It follows from the above that it is interesting to have some sufficient and necessary conditions for a Riemann surface to be maximal. In this paper, the author obtains two sufficient conditions for a Riemann surface \(R\) to be maximal.