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We consider foliations of compact complex manifolds by analytic curves. Wesuppose that the line bundle tangent to the foliation is negative. We show thatin a generic case there exists a finitely smooth homeomophism, holomorphic onthe fibers and mapping fiberwise the manifold of universal coverings over theleaves passing through some transversal $B$ onto some domain in$B\times\mathbb{C}$ with continuous boundary. depending on the leaves. Theproblem can be reduced to a study of the Beltrami equation with parameters onthe unit disk in the case, when derivatives of the corresponding coefficientBeltrami grow no faster than some negative power of the distance to theboundary of the disk.

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