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Two-sheeted coverings of the disc. (English) Zbl 0663.30042

Let \(X\) be a Riemann surface, let \(D\) denote the unit disk in the complex plane, and let \(\phi: X\to D\) be a holomorphic mapping of constant finite valence \(m\). Let \(N\) be the number of points in \(X\) of branch order \(m-1\), where \(0\leq N\leq \infty\). The author proves that if \(\phi_ 1: X\to D\) is a proper holomorphic mapping which vanishes \(m_ 1\) times, counting multiplicities, then both (1) \(m\) divides \(m_ 1\), and (2) \(\phi_ 1=g(\phi)\), where \(g\) is a finite Blaschke product with \(m_ 1/m\) zeros, counting multiplicities. Specializing this result to the case \(m=2\), it is proved that if \(N>2\), and if \(\phi_ 1\) is also of valence 2 then \(\phi_ 1=A(\phi)\), where \(A\in \operatorname{Aut} D\), the automorphism group of \(D\). Let \(\text{Prop}\;X\) denote the proper holomorphic mappings of \(X\) into itself. It is proved that if \(X\) is a two-sheeted covering of \(D\), that is, there exists a \(\phi\) as above with \(m=2\), and if \(f\in \text{Prop}\;X-\operatorname{Aut} X\), and if \(f\) fixes a point of \(X\), then that point is the only point fixed by \(X\) and that point is a branch point of \(X\).
Reviewer: P.Lappan

MSC:

30F20 Classification theory of Riemann surfaces
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