HARMONIC CURRENTS DIRECTED BY FOLIATIONS BY RIEMANN SURFACES

The paper’s main theorem (Theorem 1.1) states exactly the claim under audit: for a foliation by Riemann surfaces with a hyperbolic singularity at 0 in C^2 and any continuous ε with ε(0)=0, there exists a positive dd^c-closed directed current T without mass on the separatrices such that ‖T‖_{δD^2} ≥ ε(δ) δ^2 for all 0<δ<1 . The authors linearize the vector field near the hyperbolic singularity (F = η z1 ∂/∂z1 + z2 ∂/∂z2) using Poincaré–Dulac, parametrize leaves explicitly, and set up a sector S and a conformal map Φ(ζ)=ζ^γ from S to the upper half-plane; they construct a positive harmonic weight via a Poisson extension of a carefully chosen boundary function tied to ε, and define T as a push-forward current supported on a single leaf L1 . They then prove T is a positive dd^c-closed (1,1)-current (Proposition 3.2) and establish the sharp lower mass bound (Proposition 3.3) with detailed estimates; T has no mass on the separatrices since L1⊂(D*)^2, and they also note that averaging over α produces a smooth form on (D*)^2 satisfying the same bound (Remark 3.4) . The candidate’s solution adopts the same architecture: linearization, logarithmic/conformal parametrization of leaves, Poisson extension of a boundary datum dominated by ε(e^{-t}), and construction of a positive directed current from harmonic weights on leaves. The only substantive gaps are (i) an imprecise choice of the leaf coordinate (they use w=log z2 and call it an ‘upper half-plane’ coordinate; the paper correctly uses z2=e^{iζ} with ζ∈H), and (ii) omission of the conformal power map Φ(ζ)=ζ^γ needed to normalize sector geometry; both are easily rectified by replacing w with ζ:=-i log z2 and inserting the Φ-step as in the paper. With these clarifications, the candidate’s proof matches the paper’s approach and conclusions.