Riemann surfaces in fibered polynomial hulls

Abstract

Let Δ be the closed unit disk in C, let Γ be the circle, let Π: Δ×C→Δ be projection, and let A(Δ) be the algebra of complex functions continuous on Δ and analytic in int Δ. Let K be a compact set in C² such that Π(K)=Γ, and let K_λ≠{w∈C|(λ,w)∈K}. Suppose further that (a) for every λ∈Γ, K_λ is the union of two nonempty disjoint connected compact sets with connected complement, (b) there exists a function Q(λ,w)≠(w-R(λ))²-S(λ) quadratic in w with R,S∈A(Δ) such that for all λ∈Γ, {w∈C|Q(λ,w)=0}υ int K_λ, where S has only one zero in int Δ, counting multiplicity, and (c) for every λ∈Γ, the map ω→Q(λ,ω) is injective on each component of K_λ. Then we prove that К/K is the union of analytic disks 2-sheeted over int Δ, where К is the polynomial convex hull of K. Furthermore, we show that БК/K is the disjoint union of such disks.