Geometric description of some Loewner chains with infinitely many slits

Abstract

We study the chordal Loewner equation associated with certain driving functions that produce infinitely many slits. Specifically, for a choice of a sequence of positive numbers \((b_n)_{n\ge1}\) and points of the real line \((k_n)_{n\ge1}\), we explicitily solve the Loewner PDE \[ \dfrac{\partial f}{\partial t}(z,t)=-f'(z,t)\sum_{n=1}^{+\infty}\dfrac{2b_n}{z-k_n\sqrt{1-t}}\] in \(\mathbb{H}\times[0,1)\). Using techniques involving the harmonic measure, we analyze the geometric behaviour of its solutions, as \(t\rightarrow1^-\).