Nonuniformizable skew cylinders. A counterexample to the simultaneous uniformization problem

Abstract

At the end of 1960-ths Yu.S.Ilyashenko stated the problem: is it true that for any one-dimensional holomorphic foliation with singularities on a Stein manifold leaves intersecting a transversal disc can be uniformized so that the uniformization function would depend holomorphically on the transversal parameter? In the present paper we construct a two-dimensional counterexample. This together with the previous result of Yu.S.Ilyashenko (Lemma 1) implies existence of a counterexample given by a foliation on affine (projective) algebraic surface by level curves of a rational function with singularities deleted. This implies also that Bers' simultaneous uniformization theorem for topologically trivial holomorphic foliations by compact Riemann surfaces does not extend to the general fibrations by compact Riemann surfaces with singularities.