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On Teichmüller contraction

Author: Frederick P. Gardiner
Journal: Proc. Amer. Math. Soc. 118 (1993), 865-875
MSC: Primary 30F60; Secondary 32G15
MathSciNet review: 1152277
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Abstract: Universal Teichmüller space is the space of quasi-symmetric homeomorphisms $ QS$ of a circle factored by those Möbius transformations that preserve the circle. Another Teichmüller space, which also has universal properties, is $ QS$ factored by the closed subgroup $ S$ of symmetric homeomorphisms. Teichmüller's metric for $ QS\,\bmod S$ is the boundary dilatation metric. Sullivan's coiling property for Beltrami lines and the Hamilton-Reich-Strebel necessary and sufficient condition for extremality are proved for $ QS\bmod S$. The coiling property implies a contraction principle for certain types of self-mappings of Teichmüller space. It is also shown that the boundary dilatation metric has an infinitesimal form and that this metric is the integral of its infinitesimal form.

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  • [1] L. V. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385-404. MR 0115006 (22:5813)
  • [2] R. Fehlmann, Extremal quasiconformal mappings with free boundary components in domains of arbitrary connectivity, Math. Z. 184 (1983), 109-126. MR 711732 (85i:30042)
  • [3] F. G. Gardiner, Approximation of infinite dimensional Teichmüller spaces, Trans. Amer. Math. Soc. 282 (1984), 367-383. MR 728718 (85f:30082)
  • [4] -, Teichmüller theory and quadratic differentials, Wiley-Interscience, New York, 1987. MR 903027 (88m:32044)
  • [5] F. P. Gardiner and D. P. Sullivan, Symmetric and quasisymmetric structures on a closed curve, Amer. J. Math. 114 (1992), 683-736. MR 1175689 (95h:30020)
  • [6] O. Lehto and K. I. Virtanen, Quasiconformal mappings, Springer-Verlag, Berlin and New York, 1965.
  • [7] B. O'Byrne, On Finsler geometry and applications to Teichmüller spaces, Ann. of Math. Stud., vol. 66, Princeton Univ. Press, Princeton, NJ, 1971, pp. 317-328. MR 0286141 (44:3355)
  • [8] E. Reich, On criteria for unique extremality of Teichmüller mappings, Ann. Acad. Sci. Fenn. AI Math. 6 (1981), 289-302. MR 658931 (83j:30022)
  • [9] E. Reich and K. Strebel, Extremal quasiconformal mappings with given boundary values, Contributions to Analysis, A Collection of Papers Dedicated to Lipman Bers, Academic Press, New York, 1974, pp. 373-391. MR 0361065 (50:13511)
  • [10] H. L. Royden, Automorphisms and isometries of Teichmüller space, Ann. of Math. Stud., vol. 66, Princeton Univ. Press, Princeton, NJ, 1971, pp. 369-383. MR 0288254 (44:5452)
  • [11] D. P. Sullivan, Bounds, quadratic differentials, and renormalization conjectures, Mathematics into the 21st Century, Amer. Math. Soc. Centennial Publication, vol. 2, Amer. Math. Soc., Providence, RI, 1991. MR 1184622 (93k:58194)

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