Directed harmonic currents for laminations on certain compact complex surfaces

Abstract

Let ℒ be a Lipschitz lamination by Riemann surfaces embedded in M. If M is a complex torus, ℂℙ1 × ℂℙ1 or 𝕋1 × ℂℙ1 and there is no directed closed current then there exists a unique directed harmonic current of mass one. Moreover, if ℒ is embedded in M = ℂℙ1 × ℂℙ1 and has no compact leaves, then there is no directed closed current. If ℒ is not Lipschitz, then slightly weaker results are obtained.