Loewner Curvature

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.