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Loewner equations on complete hyperbolic domains (1102.5454v2)
Published 26 Feb 2011 in math.CV and math.DS
Abstract: We prove that, on a complete hyperbolic domain D\subset Cq, any Loewner PDE associated with a Herglotz vector field of the form H(z,t)=A(z)+O(|z|2), where the eigenvalues of A have strictly negative real part, admits a solution given by a family of univalent mappings (f_t: D\to Cq) such that the union of the images f_t(D) is the whole Cq. If no real resonance occurs among the eigenvalues of A, then the family (e{At}\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.