Geometric description of some Loewner chains with infinitely many slits

TL;DR

This paper provides an explicit solution to a Loewner PDE with infinitely many slits driven by a specific sum, and analyzes the geometric behavior of these solutions as time approaches a critical value.

Contribution

It explicitly solves a Loewner PDE with infinitely many slits and studies the geometric evolution of the solutions using harmonic measure techniques.

Findings

Explicit solution to the Loewner PDE with infinitely many slits

Analysis of geometric behavior as time approaches 1

Use of harmonic measure to understand slit evolution

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 $t\rightarrow1^-$.