Almost complex structures on universal coverings of foliations

We consider foliations of compact complex manifolds by analytic curves. It is well known that if the line bundle tangent to the foliation is negative, then, in general position, all leaves are hyperbolic. The manifold of universal coverings over the leaves passing through some transversal has a natural complex structure. We show that in a typical case this structure can be defined as a smooth almost complex structure on the product of the base by the unit disk. We prove that this structure is quasiconformal on the leaves and that the corresponding $ (1,0)$-forms and their derivatives with respect to the coordinates on the base and in the leaves admit uniform estimates. The derivatives grow no faster than some negative power of the distance to the boundary of the disk.