A Komatu-Loewner Equation for Multiple Slits

TL;DR

This paper extends the Komatu-Loewner equation to handle multiple slits in an n-connected domain, providing a framework for understanding the evolution of complex shapes in conformal mapping.

Contribution

The authors generalize the Komatu-Loewner equation to multiple slits in n-connected domains, broadening its applicability in complex analysis and conformal mapping.

Findings

Derived a Loewner equation for multiple slits in n-connected domains

Proved the existence of a decreasing family of domains with conformal maps satisfying the generalized equation

Established normalization conditions for the Riemann mapping functions

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.