HOLOMORPHIC MOTIONS AND POLYNOMIAL HULLS

A holomorphic motions of E subset-of C over the unit disc D is a f:D x C --> C such that f(O, w) = w, w is-an-element-of E, the function f(z, w) = f(z)(w) is holomorphic in z, and f(z):E --> C is an injection for all z is-an-element-of 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 x C --> C. As a main step a "holomorphic axiom of choice" is obtained (concerning selections from the sets C/f(z)(E), z is-an-element-of D). The proof uses earlier results on the existence of analytic discs in the polynomial hulls of some subsets of C2.