Osaka Journal of Mathematics

Extensions of holomorphic motions and holomorphic families of Möbius groups

Sudeb Mitra and Hiroshige Shiga

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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.

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

First available in Project Euclid: 20 December 2010

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Zentralblatt MATH identifier

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


Mitra, Sudeb; Shiga, Hiroshige. Extensions of holomorphic motions and holomorphic families of Möbius groups. Osaka J. Math. 47 (2010), no. 4, 1167--1187.

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  • L.V. Ahlfors and L. Bers: Riemann's mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385--404.
  • R. Arens: Topologies for homeomorphism groups, Amer. J. Math. 68 (1946), 593--610.
  • K. Astala and G.J. Martin: Holomorphic motions; in Papers on Analysis, Report Univ. Jyväskylä 83, Jyväskylä, 27--40, 2001.
  • L. Bers: Holomorphic families of isomorphisms of Möbius groups, J. Math. Kyoto Univ. 26 (1986), 73--76.
  • L. Bers and H.L. Royden: Holomorphic families of injections, Acta Math. 157 (1986), 259--286.
  • E.M. Chirka: On the propagation of holomorphic motions, Dokl. Math. 70 (2004), 516--519.
  • A. Douady: Prolongement de mouvements holomorphes (d'après Słodkowski et autres), Séminaire Bourbaki 1993/94, Astérisque 227 (1995), Exp. No. 775, 3, 7--20.
  • A. Douady and C.J. Earle: Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), 23--48.
  • C.J. Earle: On the Carathéodory metric in Teichmüller spaces; in Discontinuous Groups and Riemann Surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), Ann. of Math. Studies 79, Princeton Univ. Press, Princeton, N.J., 99--103, 1974.
  • C.J. Earle: Some maximal holomorphic motions; in Lipa's Legacy (New York, 1995), Contemp. Math. 211, Amer. Math. Soc., Providence, RI, 183--192, 1997.
  • C.J. Earle, F.P. Gardiner and N. Lakic: Isomorphisms between generalized Teichmüller spaces; in Complex Geometry of Groups (Olmué, 1998), Contemp. Math. 240, Amer. Math. Soc., Providence, RI, 97--110, 1999.
  • C.J. Earle, I. Kra and S.L. Krushkal$'$: Holomorphic motions and Teichmüller spaces, Trans. Amer. Math. Soc. 343 (1994), 927--948.
  • C.J. Earle and S. Mitra: Variation of moduli under holomorphic motions; in In the Tradition of Ahlfors and Bers (Stony Brook, NY, 1998), Contemp. Math. 256, Amer. Math. Soc., Providence, RI, 39--67, 2000.
  • F.P. Gardiner: Teichmüller Theory and Quadratic Differentials, Wiley, New York, 1987.
  • F.P. Gardiner and N. Lakic: Quasiconformal Teichmüller Theory, Mathematical Surveys and Monographs 76, Amer. Math. Soc., Providence, RI, 2000.
  • J.H. Hubbard: Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, vol. 1, Matrix Editions, Ithaca, NY, 2006.
  • Y. Imayoshi and M. Taniguchi: An Introduction to Teichmüller Spaces, Springer, Tokyo, 1992.
  • Y. Jiang and S. Mitra: Some applications of universal holomorphic motions, Kodai Math. J. 30 (2007), 85--96.
  • G.S. Lieb: Holomorphic motions and Teichmüller space, Ph.D. dissertation, Cornell University (1990).
  • R. Mañé, P. Sad and D. Sullivan: On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), 193--217.
  • S. Mitra: Teichmüller spaces and holomorphic motions, J. Anal. Math. 81 (2000), 1--33.
  • S. Mitra: Extensions of holomorphic motions, Israel J. Math. 159 (2007), 277--288.
  • S. Mitra: Extensions of holomorphic motions to quasiconformal motions; in In the Tradition of Ahlfors--Bers, IV, Contemp. Math. 432, Amer. Math. Soc., Providence, RI, 199--208, 2007.
  • S. Nag: The Complex Analytic Theory of Teichmüller Spaces, Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley, New York, 1988.
  • H. Shiga: On analytic and geometric properties of Teichmüller spaces, J. Math. Kyoto Univ. 24 (1984), 441--452.
  • Z. Slodkowski: Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc. 111 (1991), 347--355.
  • T. Sugawa: The Bers projection and the $\lambda$-lemma, J. Math. Kyoto Univ. 32 (1992), 701--713.
  • D.P. Sullivan and W.P. Thurston: Extending holomorphic motions, Acta Math. 157 (1986), 243--257.