A Komatu-Loewner Equation for Multiple Slits

Abstract

We give a generalization of the Komatu-Loewner equation to multiple slits. Therefore, we consider an $n$-connected circular slit disk $\Omega$ as our initial domain minus $m\in \mathbb{N}$ disjoint, simple and continuous curves that grow from the outer boundary $\partial \mathbb{D}$ of $\Omega$ into the interior. Consequently we get a decreasing family $(\Omega_t)_{t\in[0,T]}$ of domains with $\Omega_0=\Omega$. We will prove that the corresponding Riemann mapping functions $g_t$ from $\Omega_t$ onto a circular slit disk, which are normalized by $g_t(0)=0$ and $g_t'(0)>0$, satisfy a Loewner equation known as the Komatu-Loewner equation.Comment: 26 pages, 8 figure