Loewner equations on complete hyperbolic domains

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