Chordal generators and the hydrodynamic normalization for the unit ball

Sebastian Schleißinger

Vol. 280 (2016), No. 1, 203–225

DOI: 10.2140/pjm.2016.280.203

Abstract

Let $c\ge 0$ and denote by $\mathcal{K}\left(ℍ,c\right)$ the set of all infinitesimal generators $G:ℍ\to ℂ$ on the upper half-plane $ℍ$ such that ${limsup}_{y\to \infty }y\cdot \left|G\left(iy\right)\right|\le c$. This class is related to univalent functions $f:ℍ\to ℍ$ with hydrodynamic normalization and appears in the so-called chordal Loewner equation.

In this paper, we generalize the class $\mathcal{K}\left(ℍ,c\right)$ and the hydrodynamic normalization to the Euclidean unit ball in ${ℂ}^n$. The generalization is based on the observation that $G\in \mathcal{K}\left(ℍ,c\right)$ can be characterized by an inequality for the hyperbolic length of $G\left(z\right)$.

Keywords

semigroups of holomorphic mappings, infinitesimal generators, hydrodynamic normalization, chordal Loewner equation

Mathematical Subject Classification 2010

Primary: 20M20, 32A40, 37L05

Milestones

Received: 16 January 2015

Revised: 4 May 2015

Accepted: 5 May 2015

Published: 28 December 2015