The Loewner equation and Lipschitz graphs

Abstract

The proofs of continuity of Loewner traces in the stochastic and in the deterministic settings employ different techniques. In the former setting of the Schramm–Loewner evolution SLE, H¨older continuity of the conformal maps is shown by estimating the derivatives, whereas the latter setting uses the theory of quasiconformal maps. In this note, we adopt the former method to the deterministic setting and obtain a new and elementary proof that Hölder-1/2 driving functions with norm less than 4 generate simple arcs. We also give a sufficient condition for driving functions to generate curves that are graphs of Lipschitz functions.