Collisions and Spirals of Loewner Traces

TL;DR

This paper investigates the behavior of Loewner traces driven by functions asymptotic to K√(1−t), revealing stability properties, the special case of K=4, and constructing explicit examples, thereby deepening understanding of trace continuity and hull connectivity.

Contribution

It provides a stability analysis for Loewner traces with asymptotic driving functions, especially highlighting the unique case when K=4 and constructing explicit trace examples.

Findings

K=4 can produce non-locally connected hulls.

For K≠4, the hulls are generated by continuous curves.

The space of continuous trace-driving functions is not convex.

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 \lambda(t) so that the hulls driven by K\lambda(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