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On the boundary behavior of holomorphic mappings of plane domains into Riemann surfaces. (English) Zbl 0707.30037

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.
Reviewer: A.Baernstein II

MSC:

30F99 Riemann surfaces
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