The Beltrami equation with parameters and uniformization of holomorphic foliations with hyperbolic leaves

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 a generic case there exists a finitely smooth homeomorphism, holomorphic on the fibers and mapping fiberwise the manifold of universal coverings over the leaves passing through some transversal B onto some domain in B�C with continuous boundary depending on the leaves. The problem can be reduced to a study of the Beltrami equation with parameters on the unit disk in the case, when the derivatives of the corresponding Beltrami coefficient grows no faster than some negative power of the distance to the boundary of the disk. We give here the proof of the main result modulo some technical lemmas. These lemmas will be proved in the papers to come.