A Komatu-Loewner Equation for Multiple Slits

We give a generalization of the Komatu-Loewner equation to multiple slits. Therefore, we consider an $n$-connected circular slit disk $Ω$ as our initial domain minus $m\in \mathbb{N}$ disjoint, simple and continuous curves that grow from the outer boundary $\partial \mathbb{D}$ of $Ω$ into the interior. Consequently we get a decreasing family $(Ω_t)_{t\in[0,T]}$ of domains with $Ω_0=Ω$. We will prove that the corresponding Riemann mapping functions $g_t$ from $Ω_t$ 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.

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