On the boundary behavior of holomorphic mappings of plane domains into Riemann surfaces (Q919486)

Let D be a plane domain and E be a compact subset of D of class \(N_ B\); that is, each bounded holomorphic function in \(D\setminus E\) has a holomorphic extension to D. The author proves that each holomorphic mapping f from \(D\setminus E\) into a Riemann surface W which carries non- constant bounded holomorphic functions has a holomorphic extension to a function which maps D into a slightly larger Riemann surface \(W'\). Under certain hypotheses one may take \(W'=W\), and this enables the author to recapture a recent result of Shiga. The author proves also an analogue of his main theorem in which ``bounded'' is replaced by ``finite Dirichlet integral''. He raises two questions which remain open. One of them asks whether a holomorphic f of bounded valence from \(D\setminus E\), where E is of class \(N_ D\), into an arbitrary Riemann surface always has such an extension.