Uniformization of compact foliated spaces by surfaces of hyperbolic type

TL;DR

This paper presents a new proof of the uniformization theorem for hyperbolic foliated surfaces using a laminated Ricci flow to establish a conformally equivalent metric with constant negative curvature.

Contribution

It introduces a laminated Ricci flow approach to prove the uniformization theorem for foliated spaces with hyperbolic surface leaves.

Findings

Existence of a laminated Riemannian metric with constant Gaussian curvature -1

The metric is smooth on leaves and transversally continuous

Provides a new proof of the uniformization theorem for hyperbolic foliations

Abstract

We give a new proof of the uniformization theorem of the leaves of a lamination by surfaces of hyperbolic conformal type. We use a laminated version of the Ricci flow to prove the existence of a laminated Riemannian metric (smooth on the leaves, transversaly continuous) with leaves of constant Gaussian curvature equal to -1, which is conformally equivalent to the original metric.