Directed harmonic currents for laminations on certain compact complex surfaces

TL;DR

This paper investigates the existence and uniqueness of directed harmonic currents for Lipschitz laminations by Riemann surfaces on certain compact complex surfaces, revealing conditions under which such currents exist or are absent.

Contribution

It establishes the existence and uniqueness of directed harmonic currents in specific complex surfaces and explores the impact of Lipschitz regularity on these currents.

Findings

Unique directed harmonic current exists on certain surfaces without directed closed currents.

No directed closed current exists if the lamination has no compact leaves in specific cases.

Weaker results are obtained for non-Lipschitz laminations.

Abstract

Let $\mathcal{L}$ be a Lipschitz lamination by Riemann surfaces embedded in $M$. If $M$ is a complex torus, $\mathbb{CP}^1\times\mathbb{CP}^1$ or $\mathbb{T}^1\times\mathbb{CP}^1$ and there is no directed closed current then there exists a unique directed harmonic current of mass one. Moreover if $\mathcal{L}$ is embedded in $M=\mathbb{CP}^1\times\mathbb{CP}^1$ and has no compact leaves, then there is no directed closed current. If $\mathcal{L}$ is not Lipschitz, then slightly weaker results are obtained.