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Holomorphic families of injections. (English) Zbl 0619.30027
This paper contains new proofs and extensions of some results by R. Mañé, P. Sad and D. Sullivan [Ann. Sci. Éc. Norm. Supér., IV. Ser. 16, 193-217 (1983; Zbl 0524.58025)] and D. P. Sullivan and W. P. Thurston [Acta Math. 157, 243-257 (1986; reviewed above)]. Let E be a subset of the Riemann sphere $${\hat {\mathbb{C}}}={\mathbb{C}}\cup \{\infty \}$$ containing at least 4 points. Let $$\Delta_ r$$ denote the open disc $$| z| <r$$ in $${\mathbb{C}}$$. A map $f:\Delta_ r\times E\to {\mathbb{C}}$ will be called admissible if $$f(0,z)=z$$ for all $$z\in E$$, for every fixed $$\lambda \in \Delta_ r$$ the map f($$\lambda$$,$$\cdot):E\to {\hat {\mathbb{C}}}$$ is an injection, and for every fixed $$z\in E$$ the map $$f(\cdot,z):\Delta_ r\to {\hat {\mathbb{C}}}$$ is holomorphic.
Theorem 1. If $$f:\Delta_ 1\times E\to {\hat {\mathbb{C}}}$$ is admissible, then every f($$\lambda$$,$$\cdot)$$ is the restriction to E of a quasiconformal self-map $$F_{\lambda}$$ of $${\hat {\mathbb{C}}}$$, of dilatation not exceeding $K=(1+| \lambda |)/(1-| \lambda |).$ Theorem 2. If $$f:\Delta_ 1\times E\to {\mathbb{C}}$$ is admissible and E has a nonempty interior $$\omega$$, then for each $$\lambda \in \Delta_ 1$$ the map $$f(\lambda,\cdot)|_{\omega}$$ is a K- quasiconformal homeomorphism of $$\omega$$ into $${\hat {\mathbb{C}}}$$ with $$K=(1+| \lambda |)/(1-| \lambda |)$$. The Beltrami coefficient of $$f(\lambda,\cdot)|_{\omega}$$ given by $\mu (\lambda,z)=\frac{\partial f(\lambda,z)| \omega}{\partial \bar z}/\frac{\partial f(\lambda,z)| \omega}{\partial z}$ is a holomorphic function of $$\lambda \in \Delta_ 1$$, and an element of the Banach space $$L_{\infty}(\omega)$$. The author’s proofs make essential use of the theory of quasiconformal maps and of Teichmüller spaces.
Reviewer: N.A.Gusevskij

##### MSC:
 30C62 Quasiconformal mappings in the complex plane 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
##### Keywords:
K-quasiconformal homeomorphism; Teichmüller spaces
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##### References:
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