Loewner equations on complete hyperbolic domains

2011/02/26 by Arosio, Leandro
#Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1102.5454

Abstract

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)=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→ 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∘ 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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