Harmonic currents directed by foliations by Riemann surfaces

TL;DR

This paper investigates positive harmonic currents in foliations by Riemann surfaces near hyperbolic singularities, demonstrating the optimality of Nguy\^en's vanishing Lelong number result and establishing sharp mass estimates.

Contribution

It proves the sharpness of Nguy\^en's theorem on Lelong numbers for harmonic currents near hyperbolic singularities in foliations.

Findings

Lelong number of harmonic currents at the singularity is zero

Mass estimates near the singularity are optimal

No better bounds than Nguy\^en's result are possible

Abstract

We study local positive harmonic currents directed by a foliation by Riemann surfaces near a hyperbolic singularity which have no mass on the separatrices. A theorem of Nguy\^en says that the Lelong number of such a current at the singular point vanishes. We prove that this property is sharp: one cannot have any better mass estimate for this current near the singularity.