Collisions and Spirals of Loewner Traces

Abstract

We analyze Loewner traces driven by functions asymptotic to K\sqrt{1-t}. We prove a stability result when K is not 4 and show that K=4 can lead to non locally connected hulls. As a consequence, we obtain a driving term λ(t) so that the hulls driven by Kλ(t) are generated by a continuous curve for all K > 0 with K not equal to 4 but not when K = 4, so that the space of driving terms with continuous traces is not convex. As a byproduct, we obtain an explicit construction of the traces driven by K\sqrt{1-t} and a conceptual proof of the corresponding results of Kager, Nienhuis and Kadanoff, math-ph/0309006

34 pages, 11 figures

References in corpus (1)

• Exact Solutions for Loewner Evolutions

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