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 \((b_n)_{n\ge 1}\) and points of the real line \((k_n)_{n\ge 1}\), we explicitily solve the Loewner PDE

$$\begin{aligned} \dfrac{\partial f}{\partial t}(z,t)=-f'(z,t)\sum _{n=1}^{+\infty }\dfrac{2b_n}{z-k_n\sqrt{1-t}} \end{aligned}$$

in \(\mathbb {H}\times [0,1)\). Using techniques involving the harmonic measure, we analyze the geometric behaviour of its solutions, as \(t\rightarrow 1^-\).