On simultaneous uniformization and local nonuniformizability
The author shows that there exists an affine algebraic foliation whose universal covering manifold (in short u.c.m.) is not uniformizable. Moreover, each u.c.m. of such a foliation is nowhere locally uniformizable. Furthermore, the above foliation may be chosen such that it is transversally affine and admits a Liouvillian first integral, with dense and hyperbolic leaves and an attractive cycle. Thus, the conjectures of Ilyashenko (stating that each u.c.m. is uniformizable) and of Shcherbakov (stating that each u.c.m. of an algebraic foliation with hyperbolic leaves is locally uniformizable) are given a negative answer.
scientific article; zbMATH DE number 1754782