Osaka Journal of Mathematics

Extensions of holomorphic motions and holomorphic families of Möbius groups

Sudeb Mitra and Hiroshige Shiga

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Abstract

A normalized holomorphic motion of a closed set in the Riemann sphere, defined over a simply connected complex Banach manifold, can be extended to a normalized quasiconformal motion of the sphere, in the sense of Sullivan and Thurston. In this paper, we show that if the given holomorphic motion, defined over a simply connected complex Banach manifold, has a group equivariance property, then the extended (normalized) quasiconformal motion will have the same property. We then deduce a generalization of a theorem of Bers on holomorphic families of isomorphisms of Möbius groups. We also obtain some new results on extensions of holomorphic motions. The intimate relationship between holomorphic motions and Teichmüller spaces is exploited throughout the paper.

Article information

Source
Osaka J. Math., Volume 47, Number 4 (2010), 1167-1187.

Dates
First available in Project Euclid: 20 December 2010

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1292854320

Mathematical Reviews number (MathSciNet)
MR2791561

Zentralblatt MATH identifier
1222.30037

Subjects
Primary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]
Secondary: 37F30: Quasiconformal methods and Teichmüller theory; Fuchsian and Kleinian groups as dynamical systems 37F45: Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations

Citation

Mitra, Sudeb; Shiga, Hiroshige. Extensions of holomorphic motions and holomorphic families of Möbius groups. Osaka J. Math. 47 (2010), no. 4, 1167--1187. https://projecteuclid.org/euclid.ojm/1292854320


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