Loewner equations on complete hyperbolic domains

TL;DR

This paper proves existence and properties of solutions to Loewner PDEs on complete hyperbolic domains in complex space, generalizing univalence criteria and analyzing the behavior of associated univalent mappings.

Contribution

It establishes existence of solutions to Loewner PDEs with specific Herglotz vector fields on complete hyperbolic domains and generalizes Pommerenke's univalence criterion.

Findings

Solutions are given by univalent mappings covering the whole space.

Under no resonance, the family (e^{At} irc f_t) is uniformly bounded.

Generalization of Pommerenke's univalence criterion for hyperbolic domains.

Abstract

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.