Hyperbolic entropy for harmonic measures on singular holomorphic foliations

Let F=(M,L,E) be a Brody-hyperbolic singular holomorphic foliation on a compact complex manifold M. Suppose that F has isolated singularities and that its Poincar\'e metric is complete. This is the case for a very large class of singularities, namely, non-degenerate and saddle-nodes in dimension $2$. Let μ be an ergodic harmonic measure on F. We show that the upper and lower local hyperbolic entropies of μ are leafwise constant almost everywhere. Moreover, we show that the entropy of μ is at least $2$.