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Some metrics on Teichmüller spaces of surfaces of infinite type


Authors: Lixin Liu and Athanase Papadopoulos
Journal: Trans. Amer. Math. Soc. 363 (2011), 4109-4134
MSC (2000): Primary 32G15, 30F30, 30F60
DOI: https://doi.org/10.1090/S0002-9947-2011-05090-7
Published electronically: March 23, 2011
MathSciNet review: 2792982
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Abstract: Unlike the case of surfaces of topologically finite type, there are several different Teichmüller spaces that are associated to a surface of topologically infinite type. These Teichmüller spaces first depend (set-theoretically) on whether we work in the hyperbolic category or in the conformal category. They also depend, given the choice of a point of view (hyperbolic or conformal), on the choice of a distance function on Teichmüller space. Examples of distance functions that appear naturally in the hyperbolic setting are the length spectrum distance and the bi-Lipschitz distance, and there are other useful distance functions. The Teichmüller spaces also depend on the choice of a basepoint. The aim of this paper is to present some examples, results and questions on the Teichmüller theory of surfaces of infinite topological type that do not appear in the setting of the Teichmüller theory of surfaces of finite type. In particular, we point out relations and differences between the various Teichmüller spaces associated to a given surface of topologically infinite type.


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  • 1. W. Abikoff, The real analytic theory of Teichmüller space, Lecture Notes in Mathematics 820, Springer-Verlag (1980). MR 590044 (82a:32028)
  • 2. L. V. Ahlfors, Lectures on quasiconformal maps, Van Nostrand-Reinhold, Princeton, New Jersey, 1966. New edition: AMS University Lecture Series, vol. 38, 2006. MR 0200442 (34:336)
  • 3. D. Alessandrini, L. Liu, A. Papadopoulos and W. Su, On various Teichmüller spaces of a surface of infinite topological type, to appear in the Proceedings of the Amer. Math. Soc.
  • 4. P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Birkhäuser (1992). MR 1183224 (93g:58149)
  • 5. A. Basmajian, Hyperbolic structures for surfaces of infinite type, Trans. Amer. Math. Soc. 336, No. 1 (1993) 421-444. MR 1087051 (93e:30087)
  • 6. M. Gromov, Structures métriques pour les variétés riemanniennes, Textes Mathématiques, 1. Paris: Cedic/Fernand Nathan (1981). MR 682063 (85e:53051)
  • 7. A. Douady & C. J. Earle, Conformally natural extensions of homeomorphisms of the circle, Acta Math. 157 (1986) 23-48. MR 857678 (87j:30041)
  • 8. C. Earle & I. Kra, On holomorphic mappings between Teichmüller spaces, Contributions to Analysis, Academic Press, New York (1974), 107-124. MR 0430319 (55:3324)
  • 9. C. Earle, F. Gardiner & N. Lakic, Teichmüller spaces with asymptotic conformal equivalence, Preprint, IHES, 1995.
  • 10. A. Epstein, Effectiveness of Teichmüller modular groups, In the Tradition of Ahlfors and Bers, Contemporary Math. 256, AMS, 2000, 69-74. MR 1759670 (2001a:30059)
  • 11. A. Fathi, F. Laudenbach & V. Poénaru, Travaux de Thurston sur les surfaces, Astérisque 66-67 (1979). MR 568308 (82m:57003)
  • 12. A. Fletcher, Local rigidity of infinite dimensional Teichmüller spaces, J. London Math. Soc., 74 (1) (2006), 26-40. MR 2254550 (2007g:30066)
  • 13. E. Fujikawa, Limit sets and regions of discontinuity of Teichmüller modular groups, Proc. Amer. Math. Soc. 132 (2004) 117-126. MR 2021254 (2004h:30057)
  • 14. E. Fujikawa, H. Shiga & M. Taniguchi, On the action of the mapping class group for Riemann surfaces of infinite type, J. Math. Soc. Japan 56, No. 4 (2004) 1069-1086. MR 2091417 (2005e:30075)
  • 15. Y. Imayoshi & M. Taniguchi, An introduction to Teichmüller spaces, Springer-Verlag, Berlin - Heidelberg - New York, 1992. MR 1215481 (94b:32031)
  • 16. B. V. Kerékjártó, Vorlesungen über Topologie. I, Varlag von Julius Springer, Berlin, 1923.
  • 17. N. Lakic, An isometry theorem for quadratic differentials on Riemann surfaces of finite genus, Trans. Amer. Math. Soc., 349, No. 7 (1997) 2951-2967. MR 1390043 (97i:30062)
  • 18. Z. Li, Teichmüller metric and length spectrum of Riemann surfaces, Sci. China Ser. A 3, (1986) 802-810. MR 855233 (87k:32040)
  • 19. L. Liu, On the length spectrums of non-compact Riemann surfaces, Ann. Acad. Sci. Fenn. Math. 24 (1999) 11-22. MR 1678001 (2001a:32020)
  • 20. L. Liu, On the metrics of length spectrum in Teichmüller space, Chinese J. Cont. Math. 22 (1) (2001) 23-34. MR 1826852 (2002f:30056)
  • 21. L. Liu, Z. Sun and H. Wei, Topological equivalence of metrics in Teichmüller space, Ann. Acad. Sci. Fenn. Math. 33, No. 1 (2008) 159-170. MR 2386845 (2008k:32034)
  • 22. V. Markovic, Biholomorphic maps between Teichmüller spaces, Duke Math. J., 120, No. 2 (2003) 405-431. MR 2019982 (2004h:30058)
  • 23. K. Matsuzaki, The infinite product of Dehn twists acting on infinite dimensional Teichmüller spaces, Kodai Math. J. 26 (2003) 279-287. MR 2018722 (2004k:30110)
  • 24. K. Matsuzaki, Inclusion relations between the Bers embeddings of Teichmüller spaces, Israel J. Math., 140 (2004) 113-123. MR 2054840 (2005e:30077)
  • 25. S. Nag, The complex analytic theory of Teichmüller spaces, John Wiley, Canadian Mathematical Society Series of Monographs and Advanced Texts, 1988. MR 927291 (89f:32040)
  • 26. A. O. Prishlyak & K. I. Mischenko, Classification of noncompact surfaces with boundary, Methods of Functional Analysis and Topology Vol. 13 (2007), no. 1, pp. 62-66. MR 2308580 (2008f:57026)
  • 27. I. Richards, On the classification of noncompact surfaces, Trans. Amer. Math. Soc. 106 (1963) 159-269. MR 0143186 (26:746)
  • 28. H. Royden, Automorphisms and isometries of Teichmüller space, in: Advances of the Theory of Riemann Surfaces, Ann. Math. Stud., 66 (1971) 369-384. MR 0288254 (44:5452)
  • 29. H. Shiga, On a distance defined by the length spectrum on Teichmüller space, Ann. Acad. Sci. Fenn. Math. 28, No. 2 (2003) 315-326. MR 1996441 (2004i:30043)
  • 30. W. P. Thurston, Three-dimensional geometry and topology, Vol. 1, Princeton mathematical series 35, Princeton University Press, 1997. MR 1435975 (97m:57016)

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Additional Information

Lixin Liu
Affiliation: Department of Mathematics, Sun Yat-sen (Zongshan) University, 510275, Guangzhou, People’s Republic of China
Email: mcsllx@mail.sysu.edu.cn

Athanase Papadopoulos
Affiliation: Institut de Recherche Mathématique Avancée, Université de Strasbourg and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France
Email: papadopoulos@math.u-strasbg.fr

DOI: https://doi.org/10.1090/S0002-9947-2011-05090-7
Keywords: Teichmüller space, infinite-type surface, Teichmüller metric, quasiconformal metric, length spectrum metric, bi-Lipschitz metric
Received by editor(s): June 23, 2008
Received by editor(s) in revised form: March 16, 2009, and April 18, 2009
Published electronically: March 23, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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