Topological characterisations of Loewner traces
Abstract
The (chordal) Loewner differential equation encodes certain curves in the halfplane (aka traces) by continuous realvalued 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 of the domain Markov property of SLE). 2. The (not necessarily simple) traces are exactly the uniform limits of simple traces. Moreover, using methods by Lind, Marshall, Rohde (2010), we infer that uniform convergence of traces imply uniform convergence of their driving functions.
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 DOI:
 10.48550/arXiv.2003.05535
 arXiv:
 arXiv:2003.05535
 Bibcode:
 2020arXiv200305535Y
 Keywords:

 Mathematics  Complex Variables;
 30C20
 EPrint:
 Indiana Univ. Math. J. 71 No. 3 (2022), 10271046