Loewner equations on complete hyperbolic domains

We prove that, on a complete hyperbolic domain $Dsubset mathbb{C}^q$, any Loewner PDE associated with a Herglotz vector field of the form $H(z,t)=Lambda(z)+O(|z|^2)$, where the eigenvalues of $Lambda$ have strictly negative real part, admits a solution given by a family of univalent mappings $(f_tcolon D o mathbb{C}^q)$ which satisfies $cup_{tgeq 0}f_t(D)=mathbb{C}^q$. If no real resonance occurs among the eigenvalues of $Lambda$, then the family $(e^{Lambda t}circ f_t)$ is uniformly bounded in a neighborhood of the origin. We also give a generalization of Pommerenke's univalence criterion on complete hyperbolic domains.