Chordal generators and the hydrodynamic normalization for the unit ball

Abstract

Let $c\ge 0$ 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 ${\mathrm{limsup}}_{y\to \infty }y\cdot |G(iy)|\le c.$ 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 $G(z).$