Tohoku Mathematical Journal

Teichmüller spaces and tame quasiconformal motions

Yunping Jiang, Sudeb Mitra, Hiroshige Shiga, and Zhe Wang

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The concept of “quasiconformal motion” was first introduced by Sullivan and Thurston (in [24]). Theorem 3 of that paper asserted that any quasiconformal motion of a set in the sphere over an interval can be extended to the sphere. In this paper, we give a counter-example to that assertion. We introduce a new concept called “tame quasiconformal motion” and show that their assertion is true for tame quasiconformal motions. We prove a much more general result that, any tame quasiconformal motion of a closed set in the sphere, over a simply connected Hausdorff space, can be extended as a quasiconformal motion of the sphere. Furthermore, we show that this extension can be done in a conformally natural way. The fundamental idea is to show that the Teichmüller space of a closed set in the sphere is a “universal parameter space” for tame quasiconformal motions of that set over a simply connected Hausdorff space.

Article information

Tohoku Math. J. (2), Volume 70, Number 4 (2018), 607-631.

First available in Project Euclid: 4 January 2019

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Zentralblatt MATH identifier

Primary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]
Secondary: 30C99: None of the above, but in this section 30F99: None of the above, but in this section 37F30: Quasiconformal methods and Teichmüller theory; Fuchsian and Kleinian groups as dynamical systems

quasiconformal motions tame quasiconformal motions holomorphic motions continuous motions Teichmüller spaces


Jiang, Yunping; Mitra, Sudeb; Shiga, Hiroshige; Wang, Zhe. Teichmüller spaces and tame quasiconformal motions. Tohoku Math. J. (2) 70 (2018), no. 4, 607--631. doi:10.2748/tmj/1546570827.

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  • L. V. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics, Ann. of Math. 72 (1960), 385–404.
  • L. Bers and H. Royden, Holomorphic families of injections, Acta Math. 157 (1986), 259–286.
  • A. Douady and C. J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), 23–48.
  • C. J. Earle, I. Kra and S. L. Krushkaĺ, 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 the tradition of Ahlfors and Bers (Stony Brook, NY, 1998), 39–67, Contemp. Math., 256, Amer. Math. Soc., Providence, RI, 2000.
  • F. P. Gardiner, Teichmüller Theory and Quadratic Differentials, Wiley, New York, 1987.
  • F. P. Gardiner and N. Lakic, Quasiconformal Teichmüller Theory, Math. Surveys and Monogr. 76, American Mathematical Society, Providence, RI, 2000.
  • J. H. Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Volume 1, Matrix Editions, Ithaca, NY, 2006.
  • Y. Jiang and S. Mitra, Douady-Earle section, holomorphic motions, and some applications, Quasiconformal mappings, Riemann surfaces, and Teichmüller spaces, 219–251, Contemp. Math., 575, Amer. Math. Soc., Providence, RI, 2012.
  • Y. Jiang, S. Mitra and H. Shiga, Quasiconformal motions and isomorphisms of continuous families of Möbius groups, Israel J. Math. 188 (2012), 177–194.
  • I. Kra, On Teichmüller's theorem on the quasi-invariance of cross ratios, Israel J. Math. 30 (1978), no. 1-2, 152–158.
  • G. S. Lieb, Holomorphic Motions and Teichmüller Space, Ph.D. dissertation, Cornell University, New York, January 1990.
  • W. S. Massey, Algebraic Topology: An Introduction, Springer-Verlag, 1977.
  • R. Mañé, P. Sad and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193–217.
  • S. Mitra, Teichmüller spaces and holomorphic motions, J. Anal. Math. 81 (2000), 1–33.
  • S. Mitra, Extensions of holomorphic motions to quasiconformal motions, In the tradition of Ahlfors-Bers. IV, 199–208, Contemp. Math. 432 American Mathematical Society, Providence, RI, 2007.
  • S. Mitra and H. Shiga, Extensions of holomorphic motions and holomorphic families of Möbius groups, Osaka J. Math. 47 (2010), no. 4, 1167–1187.
  • J. R. Munkres, Topology: Second Edition, Prentice Hall, 2000.
  • S. Nag, The Torelli spaces of punctured tori and spheres, Duke Math. J. 48 (1981), 359–388.
  • S. Nag, The Complex Analytic Theory of Teichmüller Spaces, Canadian Math. Soc. Monographs and Advanced Texts, Wiley-Interscience, 1988.
  • R. Narasimhan and Yves Nievergelt, Complex Analysis in One Variable, Second edition, Birkhäuser, 2001.
  • H. Shiga, On the hyperbolic length and quasiconformal mappings, Complex Var. Theory Appl. 50 (2005), no. 2, 123–130.
  • T. Sorvali, The boundary mapping induced by an isomorphism of covering groups, Ann. Acad. Sci. Fenn. Series A, I Math. 526 (1972), 1–31.
  • D. Sullivan and W. P. Thurston, Extending holomorphic motions, Acta Math. 157 (1986), 243–257.
  • S. Wolpert, The length spectrum as moduli for compact Riemann surfaces, Ann. of Math. 109 (1979), no. 3-4, 323–351.