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 $g_t^{\prime }(0)>0$, satisfy a Loewner equation known as the Komatu-Loewner equation.

26 pages, 8 figures