Loewner equations on complete hyperbolic domains

We prove that, on a complete hyperbolic domain D⊂CqD⊂Cq, any Loewner PDE associated with a Herglotz vector field of the form H(z,t)=Λ(z)+O(|z|2)H(z,t)=Λ(z)+O(|z|2), where the eigenvalues of ΛΛ have strictly negative real part, admits a solution given by a family of univalent mappings (ft:D→Cq)(ft:D→Cq) which satisfies ∪t≥0ft(D)=Cq∪t≥0ft(D)=Cq. If no real resonance occurs among the eigenvalues of ΛΛ, then the family (eΛt∘ft)(eΛt∘ft) is uniformly bounded in a neighborhood of the origin. We also give a generalization of Pommerenke’s univalence criterion on complete hyperbolic domains.