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Holomorphic motions and polynomial hulls. (English) Zbl 0741.32009
Let \(E\) be a subset of the complex line \(\mathbb{C}\). A holomorphic motion of \(E\) in \(\mathbb{C}\), parametrized by the unit disc \(D\), is a map \(f:D\times E\to\mathbb{C}\) such that (a) for any fixed \(\omega\in E\), the map \(z\to f(z,w)\) is holomorphic on \(D\); (b) for any fixed \(z\in D\), the map \(w\to f(z,w)\) is an injection; (c) \(w\to f(0,w)\) is an identity on \(E\).
The author gives an affirmative answer of the two questions posed by Sullivan and Thurston:
Theorem 1.3. Every holomorphic motion \(f:D\times E\to\mathbb{C}\) of an arbitrary subset \(E\) of \(\mathbb{C}\) can be extended to a holomorphic motion \(F:D\times\mathbb{C}\to\mathbb{C}\) of the whole of \(\mathbb{C}\), parametrized by the same unit disc.
Theorem 1.4. Let \(f(z,w)=f_ z(w)\) be a holomorphic motion of a subset \(E\subset\mathbb{C}\), parametrized by \(z\in D\). Then, for every point \(a\) outside \(E\), there is a holomorphic map \(g:D\to\mathbb{C}\) such that
(i) \(g(0)=a\) and
(ii) \(g(z)\notin f_ z(E)\) for every \(z\in D\).
In his proofs the author uses the relation between holomorphic motions and polynomial hulls.

32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
30E25 Boundary value problems in the complex plane
30D45 Normal functions of one complex variable, normal families
30C62 Quasiconformal mappings in the complex plane
Full Text: DOI
[1] Herbert Alexander and John Wermer, Polynomial hulls with convex fibers, Math. Ann. 271 (1985), no. 1, 99 – 109. · Zbl 0538.32011 · doi:10.1007/BF01455798 · doi.org
[2] Lipman Bers and H. L. Royden, Holomorphic families of injections, Acta Math. 157 (1986), no. 3-4, 259 – 286. · Zbl 0619.30027 · doi:10.1007/BF02392595 · doi.org
[3] Franc Forstnerič, Polynomial hulls of sets fibered over the circle, Indiana Univ. Math. J. 37 (1988), no. 4, 869 – 889. · Zbl 0647.32017 · doi:10.1512/iumj.1988.37.37042 · doi.org
[4] John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. · Zbl 0469.30024
[5] Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. · Zbl 0123.21502
[6] Donna Kumagai, Variation of fibers and polynomially convex hulls, Complex Variables Theory Appl. 11 (1989), no. 3-4, 261 – 267. · Zbl 0683.32010
[7] Serge Lang, Introduction to complex hyperbolic spaces, Springer-Verlag, New York, 1987. · Zbl 0628.32001
[8] R. Mañé, P. Sad, and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193 – 217. · Zbl 0524.58025
[9] Raghavan Narasimhan, Analysis on real and complex manifolds, Advanced Studies in Pure Mathematics, Vol. 1, Masson & Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1968. · Zbl 0188.25803
[10] Zbigniew Słodkowski, Analytic set-valued functions and spectra, Math. Ann. 256 (1981), no. 3, 363 – 386. · Zbl 0452.46028 · doi:10.1007/BF01679703 · doi.org
[11] Zbigniew Slodkowski, Polynomial hulls with convex sections and interpolating spaces, Proc. Amer. Math. Soc. 96 (1986), no. 2, 255 – 260. · Zbl 0588.32017
[12] Zbigniew Slodkowski, Polynomial hulls in \?² and quasicircles, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 16 (1989), no. 3, 367 – 391 (1990). · Zbl 0703.32007
[13] Dennis P. Sullivan and William P. Thurston, Extending holomorphic motions, Acta Math. 157 (1986), no. 3-4, 243 – 257. · Zbl 0619.30026 · doi:10.1007/BF02392594 · doi.org
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