Geometric description of some Loewner chains with infinitely many slits

Abstract

We study the chordal Loewner equation associated with certain driving functions that produce infinitely many slits. Specifically, for a choice of a sequence of positive numbers $(b_n{)}_{n\ge 1}$ and points of the real line $(k_n{)}_{n\ge 1}$, we explicitily solve the Loewner PDE $$\frac{\partial f}{\partial t}(z,t)=-f^{\prime }(z,t)\sum _{n=1}^{+\infty }\frac{2b_n}{z-k_n\sqrt{1-t}}$$ in $\mathbb{H}\times [0,1)$. Using techniques involving the harmonic measure, we analyze the geometric behaviour of its solutions, as $t\to 1^{-}$.