Chordal generators and the hydrodynamic normalization for the unit ball

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 $G(z).$