A remark on universal coverings of holomorphic families of Riemann surfaces (Q2574388)

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.