Ergodicity of laminations with singularities in K\"ahler surfaces

TL;DR

This paper generalizes a previous result on the uniqueness of directed positive harmonic currents for laminations with hyperbolic singularities, extending it from ^2 to all compact homogeneous Ke4hler surfaces.

Contribution

It extends the ergodicity result for laminations with hyperbolic singularities from ^2 to all compact homogeneous Ke4hler surfaces.

Findings

Unique directed positive harmonic current of mass one exists

No directed closed positive currents in the lamination

Generalization from ^2 to all compact homogeneous Ke4hler surfaces

Abstract

If $\mathcal{L}$ is a laminations with hyperbolic singularities, embedded in a compact homogeneous K\"ahler surface, without directed closed positive currents. Then, $\mathcal{L}$ has a unique directed positive harmonic current of mass one. This result was previously obtained by Fornaess and Sibony for $\mathbb{CP}^2$ and we generalize it for the rest of compact homogeneous K\"ahler surfaces.