  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.
