Convexity of complements of limit sets for holomorphic foliations on surfaces

Let F be a holomorphic foliation on a compact Kähler surface with hyperbolic singularities and no foliation cycle. We prove that if the limit set of F has zero Lebesgue measure, then its complement is a modification of a Stein domain. This applies for the case of suspensions of Kleinian representations, answering a question asked by Brunella. The proof consists in building, in several steps, a metric of positive curvature for the normal bundle of F near the limit set. Then we construct a proper strictly plurisubharmonic exhaustion function for the complement of the limit set by extending Brunella’s method to our singular context. The arguments hold more generally when the limit set is thin, a property relying on Brownian motion.