zbMATH — the first resource for mathematics

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

30C62 Quasiconformal mappings in the complex plane
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
Full Text: DOI
[1] Agard, S. B. &Gehring, F. W., Angles and quasiconformal mappings.Proc. London Math. Soc. (3), 14 (1965), 1–21. · Zbl 0131.07902 · doi:10.1112/plms/s3-14A.1.1
[2] Ahlfors, L. V. &Bers, L., Riemann’s mapping theorem for variable metrices.Ann. of Math. (2), 72 (1960), 385–404. · Zbl 0104.29902 · doi:10.2307/1970141
[3] Ahlfors, L. V. &Weill, G., A uniqueness theorem for Beltrami equations.Proc. Amer. Math. Soc., 13 (1962), 975–978. · Zbl 0106.28504 · doi:10.1090/S0002-9939-1962-0148896-1
[4] Bers, L., Extremal quasiconformal mappings.Advances in the Theory for Riemann Surfaces, Ann. of Math. Studies, 66 (1970), 27–52.
[5] –, Finite dimensional Teichmüller spaces and generalizations.Bull. Amer. Math. Soc. (N.S.), 5 (1981), 131–172. · Zbl 0485.30002 · doi:10.1090/S0273-0979-1981-14933-8
[6] Earle, C. J. &Kra, I., On holomorphic mappings between Teichmüller spaces.Contributions to Analysis, Academic Press, New York, (1974), 107–124. · Zbl 0307.32016
[7] Harvey, W. J. (Ed.),Discrete groups and automorphic functions. Academic Press, New York, 1977. · Zbl 0411.30033
[8] Hille, E.,Analytic Function Theory, Vol. II. Ginn and Co. (1962). · Zbl 0102.29401
[9] Hubbard, J. H., Sur les sections analytique de la courbe universelle de Teichmüller.Mem. Amer. Math. Soc., 166 (1976), 1–137. · Zbl 0318.32020
[10] Lehto, O. &Virtanen, K. I.,Quasiconformal mappings in the plane. Springer-Verlag, Berlin, 1973. · Zbl 0267.30016
[11] Mañé, R., Sad, P. &Sullivan, D., On the dynamics of rational maps.Ann. Sci. Ecole Norm. Sup., 16 (1983), 193–217. · Zbl 0524.58025
[12] Reich, E. &Strebel, K., Extremal quasiconformal mappings with given boundary values.Contributions to Analysis, Academic Press, New York, (1974), 375–391. · Zbl 0318.30022
[13] Royden, H. L., Automorphisms and isometries of Teichmüller space.Advances in the Theory of Riemann Surfaces, Ann. of Math. Studies, 66 (1971), 369–383.
[14] Strebel, K., On quasiconformal mappings of open Riemann surfaces.Comment. Math. Helv., 52 (1978), 301–321. · Zbl 0421.30017 · doi:10.1007/BF02566081
[15] Sullivan, D. &Thurston, W. P., Extending holomorphic motions.Acta Math., 157 (1986), 243–257. · Zbl 0619.30026 · doi:10.1007/BF02392594
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.