We show that for every open Riemann surface with non-abelian fundamental group there is a multiple-valued function on such that the fiberwise convex hull of the graph of fails to contain the graph of a single-valued holomorphic function on .
We prove that a foliation on with hyperbolic singularities and with “many" parabolic leaves (i.e. leaves without Green functions) is in fact a linear foliation. This is done in two steps: first we prove that there exists an algebraic leaf, using the technique of harmonic measures, then we show that the holonomy of this leaf is linearizable, from which the result follows easily.