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.