Holomorphic motions and polynomial hulls

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 ${\mathbb {C}^2}$.