Chordal generators and the hydrodynamic normalization for the unit ball

TL;DR

This paper extends the concept of chordal generators and hydrodynamic normalization from the upper half-plane to the unit ball in complex n-dimensional space, providing a broader framework for studying univalent functions and Loewner equations.

Contribution

It generalizes the class of infinitesimal generators and hydrodynamic normalization from the upper half-plane to the Euclidean unit ball in complex space.

Findings

Characterization of generators via hyperbolic length inequalities

Extension of chordal Loewner theory to higher dimensions

New insights into univalent functions in several complex variables

Abstract

Let $c\geq0$ and denote by $\mathcal{K}(\mathbb{H},c)$ the set of all infinitesimal generators $G:\mathbb{H}\to\mathbb{C}$ on the upper half-plane $\mathbb{H}$ such that $\limsup_{y\to\infty}y\cdot |G(iy)|\leq c.$ This class is related to univalent functions $f:\mathbb{H}\to\mathbb{H}$ with hydrodynamic normalization and appears in the so called chordal Loewner equation. In this paper, we generalize the class $\mathcal{K}(\mathbb{H},c)$ and the hydrodynamic normalization to the Euclidean unit ball in $\mathbb{C}^n$. The generalization is based on the observation that $G\in\mathcal{K}(\mathbb{H},c)$ can be characterized by an inequality for the hyperbolic length of $G(z).$