CONVEXITY OF COMPLEMENTS OF LIMIT SETS FOR HOLOMORPHIC FOLIATIONS ON SURFACES

The paper proves that for a holomorphic foliation F on a compact Kähler surface satisfying (*), if the limit set L is thin (in particular if it has Lebesgue measure zero), then S\L is a modification of a Stein domain. It does so by constructing a Hermitian metric on the normal bundle with positive curvature near L (positivity along leaves, then near singularities, then in all directions on L using thinness), and then a proper strictly plurisubharmonic exhaustion on S\L, completing the argument via standard 1‑convex/Remmert reduction machinery. The candidate solution reproduces this strategy: uniqueness of a directed harmonic current (hence a well-defined limit set), potential-theoretic input from thinness to build transverse weights, positivity of the normal bundle near L, and a global psh exhaustion leading to 1‑convexity and modification of a Stein domain. Minor differences are cosmetic (e.g., emphasis on Lyapunov exponent negativity as an input rather than a byproduct, and referencing Brunella’s 2003 Inventiones paper rather than the specific Brunella result cited in the paper). Substantively, they match.