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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)
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