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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)
##### Keywords:
holomorphic motions
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##### References:
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