(with Steffen Rohde and Michel Zinsmeister) The Loewner equation and Lipschitz graphs. Rev. Mat. Iberoamericana 34 (2018), 937-948, preprint.
The proofs of continuity of Loewner traces in the stochastic \cite{RS} and in the deterministic settings \cite{MR},\cite{Li} 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 H\"older-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.
Convergence of an algorithm simulating Loewner curves. Ann. Acad. Sci. Fenn. Math. 40 (2015), 601-616, arXiv, update.
We give a condition for driving functions so that Loewner curves simulated by a variant of the zipper algorithm converge in the sup norm to the actual Loewner curves. In particular, we can apply this theorem for algorithms that simulate SLE_\kappa with \kappa \neq 8. This was conjectured by S. Rohde and O. Schramm, and mentioned in arxiv:0909.2438
(with Joan Lind) The regularity of Loewner curves, Indiana Univ. Math. J. 65 No. 5 (2016), 1675–1712, arXiv.
We show that if the driving function is in C^beta then the Loewner curve is in C^{beta+1/2} for all real beta >1. This extends the result of Carto Wong. This problem is related to the papers of C. Earle & A. Epstein and D. Marshall & S. Rohde. We also prove that if the driving function is real analytic then so is the curve. This is the converse of a result in Earle and Epstein’s paper.
(with Mario Bonk) The continuum self-similar tree, arXiv.
(with Mario Bonk) Quasi-symmetries between dendrite Julia sets, in preparation, draft available.
(with Peter Friz) On the regularity of SLE trace, Forum of Mathematics, Sigma, 5, E19, 2017, arxiv.
Loewner equation driven by complex-valued functions, arXiv.
(with Atul Shekhar, and Yilin Wang) Remarks on Loewner Chains Driven by Finite Variation Functions, arXiv., published in Annales Academiæ Scientiarum Fennicæ.
(with Yizheng Yuan) A support theorem for SLE curves. arXiv, published paper in Electronic Journal of Probability.