Loewner equations on complete hyperbolic domains

Details Loewner equations on complete hyperbolic domains Loewner equations on complete hyperbolic domains

2011-02-26

arxiv.org/abs/1102.5454v2

Mathematics

Complex Variables

19 pages, revised exposition, improved results, added references

Scientific paper

We prove that, on a complete hyperbolic domain D\subset C^q, 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 C^q) such that the union of the images f_t(D) is the whole C^q. 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.

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