Nonuniformizable skew cylinders. A counterexample to the simultaneous uniformization problem (Q2712604)

The author exhibits a nonuniformizable Stein skew cylinder. A holomorphic surjection \(p:M\to D\) is called a Stein skew cylinder if \(M\) is a Stein manifold of dimension \(2\), \(D\) is a simply-connected domain in \({\mathbb C}\), \(p\) has a holomorphic section and the level sets of \(p\) are connected, simply-connected holomorphic curves. It is uniformizable if it is biholomorphically equivalent to a subdomain of \(\overline{\mathbb C}\) equipped with the projection to \(D\). NEWLINENEWLINENEWLINEA consequence of the existence of such a cylinder is that a topologically nontrivial fibration of a smooth algebraic surface \(S\) over \(\overline{\mathbb C}\), given by a rational function on \(S\), need not be uniformizable holomorphically over Teichmüller space. This disproves a tentative conjecture, made by Ilyashenko on the analogy of the situation for topologically trivial fibrations by smooth curves, when the existence of the uniformization is a well-known theorem of Bers.