Dipolar Komatu–Loewner differential equation

Abstract

By constructing a suitable holomorphic vector field for a finitely connected domain we derive that any Jordan arc in a strip standard slit domain D starting on the real axis can be described through dipolar Komatu–Loewner equation. The Komatu–Loewner equations for the slits of strip standard domain are established. We prove that conversely, the dipolar Komatu–Loewner equation generates a family of conformal maps from sub-domains of D onto another strip standard slit domains.