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Extensions of holomorphic motions. (English) Zbl 0835.30012
Holomorphic motions are families of injections depending holomorphically on a complex parameter. Let \(D(p, R)= \{z: |z-p |< R\}\) and \(D= D( 0,1 )\). Precisely, a map \((z, w)\to f_z (w): D\times E\to \overline {C}\), where \(\overline {C}= C\cup \{\infty\}\) is the extended complex plane, is a holomorphic motion of \(E\) in \(\overline {C}\) over \(D\) if (i) \(f_0= \text{id}_E\), (ii) \(f_z\) is an injection for every \(z\in D\) and (iii) the map \(z\to f_z (w): D\to \overline {C}\) is holomorphic to every \(w\in E\). In 1991 the author proved that every holomorphic motion extends to a holomorphic motion \((z, w)\to F_z (w): D\times \overline {C}\to \overline {C}\) of the whole sphere such that \(F_z\mid E= f_z\) for \(z\in D\). Now, the author obtains invariant extensions of holomorphic motions, a result conjectured by C. McMullen. More precisely, let \(f: D\times E\to \overline {C}\), where \(E\) is closed and contains at least three points, be a holomorphic motion. Assume the sets \(\overline {C}\setminus f_z (E)\), \(z\in D\), admit holomorphically varying groups of conformal automorphisms. Then a holomorphic motion \(F: D\times \overline {C}\to \overline {C}\) extending \(f\) can be chosen so that it commutes with these automorphisms. A less general result has been obtained independently by C. J. Earle, I. Kra and S. L. Krushkal, Holomorphic motions and Teichmüller spaces, Trans. Am. Math. Soc. (to appear).

MSC:
30C62 Quasiconformal mappings in the complex plane
30F60 Teichmüller theory for Riemann surfaces
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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