Resonances in Loewner equations

TL;DR

This paper proves the existence of Loewner chains and solutions to the Loewner-Kufarev PDE for certain Herglotz vector fields on the unit ball in complex space, highlighting the role of resonances in normality.

Contribution

It establishes conditions under which Loewner chains exist and are normal, extending the theory to higher dimensions with specific vector field structures.

Findings

Loewner chains exist for the given Herglotz vector fields.

Normality of the Loewner chain depends on absence of real resonances.

The Loewner-Kufarev PDE admits solutions for all positive times under these conditions.

Abstract

We prove that given a Herglotz vector field on the unit ball of $\mathbb{C}^n$ of the form $H(z,t)=(a_1 z_1,...,a_n z_n)+O(|z|^2)$ with $\Re a_j<0$ for all $j$, its evolution family admits an associated Loewner chain, which is normal if no real resonances occur. Hence the Loewner-Kufarev PDE admits a solution defined for all positive times.