The Loewner equation and Lipschitz graphs
Steffen RohdeUniversity of Washington, Seattle, USA
Huy TranTechnische Universität Berlin, Germany
Michel ZinsmeisterUniversité d'Orléans, France
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.
Cite this article
Steffen Rohde, Huy Tran, Michel Zinsmeister, The Loewner equation and Lipschitz graphs. Rev. Mat. Iberoam. 34 (2018), no. 2, pp. 937–948DOI 10.4171/RMI/1010