A remark on universal coverings of holomorphic families of Riemann surfaces

Abstract

We study the universal covering space ˜{M} of a holomorphic family (M, π, R) of Riemann surfaces over a Riemann surface R. The main result is that (1) ˜{M} is topologically equivalent to a two-dimensional cell, (2) ˜{M} is analytically equivalent to a bounded domain in C², (3) ˜{M} is not analytically equivalent to the two-dimensional unit ball C₂ under a certain condition, and (4) ˜{M} is analytically equivalent to the two-dimensional polydisc Δ² if and only if the homotopic monodoromy group of (M, π, R) is finite.