Harmonic currents of finite energy and laminations

Abstract.

We introduce a notion of energy for harmonic currents of bidegree (1, 1) on a complex Kähler manifold (M, ω). This allows us to define $\int {T\Lambda T\Lambda \omega ^{k - 2} ,}$ for positive harmonic currents. We then show that for a lamination with singularities of a compact set in $\mathbb{P}^2,$ without directed positive closed currents, there is a unique positive harmonic current which minimizes energy. If X is a compact laminated set in $\mathbb{P}^2$ of class $\mathcal{C}^{1}$ it carries a unique positive harmonic current T of mass 1. The current T can be obtained by an Ahlfors type construction starting with an arbitrary leaf of X. When X has a totally disconnected set of singularities, contained in a countable union of analytic sets, the above construction still gives positive harmonic currents.