On Teichmüller contraction
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- by Frederick P. Gardiner PDF
- Proc. Amer. Math. Soc. 118 (1993), 865-875 Request permission
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.References
- Lars Ahlfors and Lipman Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385–404. MR 115006, DOI 10.2307/1970141
- Richard Fehlmann, Extremal quasiconformal mappings with free boundary components in domains of arbitrary connectivity, Math. Z. 184 (1983), no. 1, 109–126. MR 711732, DOI 10.1007/BF01162010
- Frederick P. Gardiner, Approximation of infinite-dimensional Teichmüller spaces, Trans. Amer. Math. Soc. 282 (1984), no. 1, 367–383. MR 728718, DOI 10.1090/S0002-9947-1984-0728718-7
- Frederick P. Gardiner, Teichmüller theory and quadratic differentials, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. A Wiley-Interscience Publication. MR 903027
- Frederick P. Gardiner and Dennis P. Sullivan, Symmetric structures on a closed curve, Amer. J. Math. 114 (1992), no. 4, 683–736. MR 1175689, DOI 10.2307/2374795 O. Lehto and K. I. Virtanen, Quasiconformal mappings, Springer-Verlag, Berlin and New York, 1965.
- Brian O’Byrne, On Finsler geometry and applications to Teichmüller spaces, Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969) Ann. of Math. Studies, No. 66, Princeton Univ. Press, Princeton, N.J., 1971, pp. 317–328. MR 0286141
- Edgar Reich, On criteria for unique extremality of Teichmüller mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), no. 2, 289–301 (1982). MR 658931, DOI 10.5186/aasfm.1981.0617
- Edgar Reich and Kurt 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. 375–391. MR 0361065
- H. L. Royden, Automorphisms and isometries of Teichmüller space, Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969) Ann. of Math. Studies, No. 66, Princeton Univ. Press, Princeton, N.J., 1971, pp. 369–383. MR 0288254
- Dennis Sullivan, Bounds, quadratic differentials, and renormalization conjectures, American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988) Amer. Math. Soc., Providence, RI, 1992, pp. 417–466. MR 1184622
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 118 (1993), 865-875
- MSC: Primary 30F60; Secondary 32G15
- DOI: https://doi.org/10.1090/S0002-9939-1993-1152277-1
- MathSciNet review: 1152277