A geometric interpretation of the Ahlfors-Weill mappings and an induced foliation of H₃

Abstract

In a 1962 paper [1] Ahlfors and Weill showed that a large class of univalent functions F _∞: Δ → $\hat{C}$= ℂ∪∞ can be extended to quasiconformal self-maps A_∞:$\hat{C}$→ $\hat{C}$ as homeomorphisms of the entire sphere. The extension was obtained by illustrating quasiconformal homeomorphisms F ₀: Δ → $\hat{C}$whose images are precisely$\hat{C}$\cl(F _∞ (Δ)). Both F ₀ and F _∞ extend to ∣z∣ = 1 and agree there. Herein it will be shown that both F _∞ and F ₀ are special cases of a more general collection of maps F _t : Δ → cl(H³) =H³ ∪ $\hat{C}$where $\hat{C}$ = ∂H³. The maps F _t occur in a study of the ODE g″ + Sg = 0 on Δ and give rise to a foliation of cl(H³) by 2-disks all agreeing on [z] = 1.