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.

10 pages, accepted in International Journal of Mathematics