Directed harmonic currents for laminations on certain compact complex surfaces

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.