Topological Characterisations of Loewner Traces

The (chordal) Loewner differential equation encodes certain curves in the half-plane (aka traces) by continuous real-valued driving functions. Not all curves are traces; the latter can be defined via a geometric condition called the local growth property. In this paper we give two other equivalent conditions that characterise traces: (1) A continuous curve is a trace if and only if mapping out any initial segment preserves its continuity (which can be seen as an analogue (2) The (not necessarily simple) traces are exactly the uniform limits of simple traces. Moreover, using methods by Lind, Marshall, and Rohde (2010), we infer that uniform convergence of traces implies uniform convergence of their driving functions.

Automated summary

This paper studies the (chordal) Loewner differential equation that encodes curves in the half-plane and provides two equivalent conditions for characterizing traces. The paper also explores the relationship between the uniform limits of traces and the uniform convergence of their driving functions using methods by Lind, Marshall, and Rohde (2010).