Harmonic currents of finite energy and laminations

By J. Fornaess, N. Sibony. 2004

54 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2f05b814ef31cd17256d474dc1502cc70f8cc9d1

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Riemann Surface Laminations with Singularities (2008, background): “independent of the leaf, a unique ergodicity property, see [32], [33].”
Random Conformal Dynamical Systems (2005, methodology): “This was recently proved by Fornaess and Sibony for a lamination by holomorphic curves of class C 1 contained in the complex projective plane [ FoS ].”
Rigidity of Julia sets for Henon type maps (2013, background): “This point of view provides a strong analogy between equidistribution properties for stable manifolds in discrete holomorphic dynamics and ergodic properties of foliations by Riemann surfaces as developed in [22, 36, 37].”