Uniformization of Foliations with Hyperbolic Leaves and the Beltrami Equation with Parameters

Abstract

We consider foliations of compact complex manifolds by analytic curves. We suppose that the line bundle tangent to the foliation is negative. We show that, in the generic case, there exists a finitely smooth homomorphism holomorphic on the fibers and mapping fiberwise the manifold of universal coverings over the leaves passing through a given transversal B onto some domain with continuous boundary in B × ℂ depending on the leaves. The problem can be reduced to the study of the Beltrami equation with parameters on the unit disk in the case when the derivatives of the corresponding Beltrami coefficient grow no faster than some negative power of the distance to the boundary of the disk.