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

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.