The Loewner equation for multiple slits, multiply connected domains and branch points

Abstract

Let $\mathbb{D}\subset \mathbb{C}$ be the unit disk and let ${\gamma }_1,{\gamma }_2:[0,T]\to \bar{\mathbb{D}}\setminus \{0\}$ be parametrizations of two slits ${\mathrm{\Gamma }}_1:=\gamma (0,T],{\mathrm{\Gamma }}_2:={\gamma }_2(0,T]$ such that ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ are disjoint.

Let $g_t$ be the unique normalized conformal mapping from $\mathbb{D}\setminus ({\gamma }_1[0,t]\cup {\gamma }_2[0,t])$ onto $\mathbb{D}$ with $g_t(0)=0$, $g_t^{\prime }(0)>0$. 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 $h_{k;t}(0)=0$, $h_{k;t}^{\prime }(0)>0$.

Loewner’s famous theorem (1923) can be stated in the following way: The function $t\mapsto h_{k;t}$ is differentiable at $t_0$ if and only if $t\mapsto log(h_{k;t}^{\prime }(0))$ 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.