Harmonic Currents of Finite Energy and Laminations

2004/02/26 by John-Erik Fornaess, John-Erik Fornæss, Nessim Sibony +2
Earth and Planetary Sciences · Mathematics · #Complex Variables (math.CV) #Dynamical Systems (math.DS) #Earthquake Detection and Analysis #FOS: Mathematics #math.CV #math.DS

paper · pdf · doi:10.48550/arxiv.math/0402432

29 pages

arxiv created 2004/02/26 · openalex publication_date 2004/02/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce, on a complex Kahler manifold (M,ω), a notion of energy for harmonic currents of bidegree (1,1). This allows us to define ∫ T \wedge T \wedge ω^k-2, for positive harmonic currents. We then show that for a lamination with singularities of a compact set in P² there is a unique positive harmonic current which minimizes energy. If X is a compact laminated set in P² of class C¹ it carries a unique positive harmonic current T of mass 1. The current T can be obtained by an Ahlfors type construction starting with a arbitrary leaf of X.

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