Geometric Description of Some Loewner Chains with Infinitely Many Slits

Abstract

AbstractWe 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 $$(b_n)_{n\ge 1}$$ ( b n ) n ≥ 1 and points of the real line $$(k_n)_{n\ge 1}$$ ( k n ) n ≥ 1 , we explicitily solve the Loewner PDE $$\begin{aligned} \dfrac{\partial f}{\partial t}(z,t)=-f'(z,t)\sum _{n=1}^{+\infty }\dfrac{2b_n}{z-k_n\sqrt{1-t}} \end{aligned}$$ ∂ f ∂ t ( z , t ) = - f ′ ( z , t ) ∑ n = 1 + ∞ 2 b n z - k n 1 - t in $$\mathbb {H}\times [0,1)$$ H × [ 0 , 1 ) . Using techniques involving the harmonic measure, we analyze the geometric behaviour of its solutions, as $$t\rightarrow 1^-$$ t → 1 - .