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A remark on universal coverings of holomorphic families of Riemann surfaces. (English) Zbl 1082.30032
The authors study the universal covering space $$\widehat{M}$$ of a holomorphic family $$(M, \pi, R)$$ of Riemann surfaces over a Riemann surface $$R.$$ It is well-known as Koebe’s uniformization theorem for a Riemann surface that the universal covering space $$\widetilde{R}$$ of a complex manifold $$R$$ of dimension one is given as follows: $$\overline{{\mathbb C}}, {\mathbb C}$$ and the unit disk. However, universal coverings and fundamental groups of complex manifolds of higher dimension are very complicated.
Theorem 1. The universal covering space $$\widehat{M}$$ of a holomorphic family of Riemann surfaces $$(M, \pi, R)$$ of type $$(g, n)$$ is not biholomorphically equivalent to the two-dimensional unit ball $${\mathbb B}_{2}$$ provided that $$(M, \pi, R)$$ is locally trivial, $$n > 0,$$ or $$R$$ is not compact.
Theorem 2. The universal covering space $$\widehat{M}$$ of a holomorphic family of Riemann surfaces $$(M, \pi, R)$$ is biholomorphically equivalent to the two-dimensional polydisc $$\Delta^{2}$$ if and only if all the fibers $$S_{t} = \pi^{-1}(t)$$ are biholomorphically equivalent.

##### MSC:
 30F10 Compact Riemann surfaces and uniformization 30F60 Teichmüller theory for Riemann surfaces
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