The Loewner equation and Lipschitz graphs

  • Steffen Rohde

    University of Washington, Seattle, USA
  • Huy Tran

    Technische Universität Berlin, Germany
  • Michel Zinsmeister

    Université 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. 937–948

DOI 10.4171/RMI/1010