Loewner Curvature

TL;DR

This paper explores the concept of Loewner curvature, establishing a geometric framework for understanding phase transitions and curve properties in conformal mappings, with implications for curve simplicity and approximation.

Contribution

It introduces a new interpretation of phase transitions in Loewner theory through constant Loewner curvature and characterizes curves with conformal self-similarity.

Findings

Curves with bounded Loewner curvature are simple.

Every smooth curve has a best-approximating constant-curvature curve.

A geometric comparison principle is established.

Abstract

The purpose of this paper is to interpret the phase transition in the Loewner theory as an analog of the hyperbolic variant of the Schur theorem about curves of bounded curvature. We define a family of curves that have a certain conformal self-similarity property. They are characterized by a deterministic version of the domain Markov property, and have constant Loewner curvature. We show that every sufficiently smooth curve in a simply connected plane domain has a best-approximating curve of constant Loewner curvature, establish a geometric comparison principle, and show that curves of Loewner curvature bounded by 8 are simple curves.