Riemann surfaces in fibered polynomial hulls

Marshall A. Whittlesey Department of Mathematics, Texas A&M University

TBD mathscidoc:1701.332925

Arkiv for Matematik, 37, (2), 409-423, 1998.1
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 C^{2}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.
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  title={Riemann surfaces in fibered polynomial hulls},
  author={Marshall A. Whittlesey},
  booktitle={Arkiv for Matematik},
Marshall A. Whittlesey. Riemann surfaces in fibered polynomial hulls. 1998. Vol. 37. In Arkiv for Matematik. pp.409-423. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203557142558734.
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