Invariant holomorphic foliations on kobayashi hyperbolic homogeneous manifolds

Let M be a Kobayashi hyperbolic homogeneous manifold. Let F be a holomorphic foliation on M invariant under a transitive group G of biholomorphisms. We prove that the leaves of F are the fibers of a holomorphic G-equivariant submersion π: M → N onto a G-homogeneous complex manifold N. We also show that if Q is an automorphism family of a hyperbolic convex (possibly unbounded) domain D in ℂⁿ, then the fixed point set of Q is either empty or a connected complex submanifold of D.