A Loewner Equation for Infinitely Many Slits

It is well-known that the growth of a slit in the upper half-plane can be encoded via the chordal Loewner equation, which is a differential equation for schlicht functions with a certain normalisation. We prove that a multiple slit Loewner equation can be used to encode the growth of the union Γ of multiple slits in the upper half-plane if the slits have pairwise disjoint closures. Under certain assumptions on the geometry of Γ , our approach allows us to derive a Loewner equation for infinitely many slits as well.