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Holomorphic motions and polynomial hulls


Author: Zbigniew Slodkowski
Journal: Proc. Amer. Math. Soc. 111 (1991), 347-355
MSC: Primary 58F23; Secondary 30C35, 30C62, 32E20
DOI: https://doi.org/10.1090/S0002-9939-1991-1037218-8
MathSciNet review: 1037218
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Abstract: A holomorphic motion of $ E \subset \mathbb{C}$ over the unit disc $ D$ is a map $ f:D \times \mathbb{C} \to \mathbb{C}$ such that $ f(0,w) = w,w \in E$, the function $ f(z,w) = {f_z}(w)$ is holomorphic in $ z$, and $ {f_z}:E \to \mathbb{C}$ is an injection for all $ z \in D$. Answering a question posed by Sullivan and Thurston [13], we show that every such $ f$ can be extended to a holomorphic motion $ F:D \times \mathbb{C} \to \mathbb{C}$. As a main step a "holomorphic axiom of choice" is obtained (concerning selections from the sets $ \mathbb{C}\backslash {f_z}(E),z \in D)$. The proof uses earlier results on the existence of analytic discs in the polynomial hulls of some subsets of $ {\mathbb{C}^2}$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1037218-8
Keywords: Holomorphic motion, isotopy, polynomially convex hull, analytic disc, hyperbolic domain
Article copyright: © Copyright 1991 American Mathematical Society

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