The Loewner Equation for Multiple Slits, Multiply Connected Domains and Branch Points

TL;DR

This paper investigates the differentiability properties of conformal mappings related to multiple slits and multiply connected domains, extending Loewner's theorem and exploring the complexities when slits originate from the same point.

Contribution

It compares the differentiability of conformal maps for multiple slits with that for single slits, especially when the slits start at the same point, and extends the analysis to multiply connected domains using the Komatu-Loewner equation.

Findings

Differentiability behavior differs when slits start at the same point.

The case t_0=0 with coinciding initial points is more complex.

Extension of results to multiply connected domains with Komatu-Loewner equation.

Abstract

Let $\gamma_1,\gamma_2:[0,T]\to \overline{\mathbb{D}}\setminus\{0\}$ be parametrizations of two slits $\Gamma_1:=\gamma(0,T], \Gamma_2=\gamma_2(0,T]$ such that $\Gamma_1$ and $\Gamma_2$ are disjoint. \\ Let $g_t$ to be the unique normalized conformal mapping from $\mathbb{D}\setminus (\gamma_1[0,t]\cup \gamma_2[0,t])$ onto $\mathbb{D}$ with $g_t(0)=0,$ $g'_t(0)>0$. Furthermore, for $k=1,2$, denote by $h_{k;t}$ the unique normalized conformal mapping from $\mathbb{D}\setminus \gamma_k[0,t]$ onto $\mathbb{D}$ with $h_{k;t}(0)=0,$ ${h'_{k;t}(0)}>0$.\\ Loewner's famous theorem (\cite{Loewner:1923}) can be stated in the following way: The function $t\mapsto h_{k;t}$ is differentiable at $t_0$ if and only if $t\mapsto \log(h_{k;t}'(0))$ is differentiable at $t_0$.\\ In this paper we compare the differentiability of $t\mapsto h_{k;t}$ with that of $t\mapsto g_t.$ We show that the situation is more complicated in the case $t_0=0$ with $\gamma_1(0)=\gamma_2(0).$\\ Furthermore, we also look at this problem in the case of a multiply connected domain with its corresponding Komatu-Loewner equation.