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Extending holomorphic motions. (English) Zbl 0619.30026
Let X be a subset of \({\mathbb{C}}\), a holomorphic motion of X in \({\mathbb{C}}\) is a map f:T\(\times X\to {\mathbb{C}}\) defined for some connected open subset \(T\subset {\mathbb{C}}\) containing 0 such that a) for any fixed \(x\in X\), \(f_ t(x)=f(t,x)\) is a holomorphic mapping of T to \({\mathbb{C}}\), b) for any fixed \(t\in T\), \(f_ t\) is an embedding, and c) \(f_ 0\) is the identity map of X.
The main result of this paper is the following theorem:
Theorem 1. There is a universal constant \(a>0\) such that any holomorphic motion of any set \(X\subset {\mathbb{C}}\) parametrized by the disk \(T=D_ 1\) of radius 1 about 0 can be extended to a holomorphic motion of \({\mathbb{C}}\) with time parameter in the disk \(D_ a\) of radius a about 0.
Corollary. If f is a holomorphic motion of a set X, then for each time the map \(f_ t\) extends to a quasiconformal map of \({\mathbb{C}}\). Also the authors obtain some results about quasiconformal motions.
Reviewer: N.A.Gusevskij

MSC:
30C62 Quasiconformal mappings in the complex plane
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[1] Bers, L. &Royden, H. L., Holomorphic families of injections.Acta Math., 157 (1986), 259–286. · Zbl 0619.30027 · doi:10.1007/BF02392595
[2] Mañé, R., Sad, P. &Sullivan, D., On the dynamics of rational mapsAnn. Sci. Ecole Norm. Sup., 16 (1983), 193–217. · Zbl 0524.58025
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