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

## TL;DR

This paper studies foliations of compact complex manifolds with hyperbolic leaves, establishing conditions under which a fiberwise homeomorphism exists, reducing the problem to a Beltrami equation with parameters.

## Contribution

It introduces a new approach to uniformization of foliations with hyperbolic leaves via solving a parameter-dependent Beltrami equation.

## Key findings

- Existence of a finitely smooth fiberwise homeomorphism in generic cases.
- Reduction of the uniformization problem to a Beltrami equation with controlled coefficient growth.
- Application to foliations with negative tangent line bundle on complex manifolds.

## 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 a generic case there exists a finitely smooth homeomophism, holomorphic on the fibers and mapping fiberwise the manifold of universal coverings over the leaves passing through some transversal $B$ onto some domain in $B\times\mathbb{C}$ 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 derivatives of the corresponding coefficient Beltrami grow no faster than some negative power of the distance to the boundary of the disk.