Siberian Mathematical Journal

, Volume 45, Issue 4, pp 646–668 | Cite as

Complex Geometry of the Universal Teichmuller Space

  • S. L. Krushkal


We prove that all invariant distances on the universal Teichmuller space agree and are determined by the Grunsky coefficients of the naturally related conformal maps. This fact yields various important consequences; in particular, we obtain solutions of certain well-known geometric problems in complex analysis and related fields.

Teichmuller space Teichmuller metric invariant distances Kobayashi metric Caratheodory metric Grunsky coefficients Green's function plurisubharmonic function 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kobayashi S., Hyperbolic Complex Spaces, Springer-Verlag, New York (1998).Google Scholar
  2. 2.
    Dineen S., The Schwarz Lemma, Clarendon Press, Oxford (1989).Google Scholar
  3. 3.
    Klimek M., Pluripotential Theory, Clarendon Press, Oxford (1991).Google Scholar
  4. 4.
    Royden H. L., "Automorphisms and isometries of Teichmüller space," in: Advances in the Theory of Riemann Surfaces, Princeton Univ. Press, Princeton, 1971, pp. 369–383. (Ann. of Math. Stud.; 66.)Google Scholar
  5. 5.
    Gardiner F. P., Quadratic Differentials and Teichmüller Spaces, Wiley-Interscience, New York (1987).Google Scholar
  6. 6.
    Earle C. J., Kra I., and Krushkal S. L., "Holomorphic motions and Teichmüller spaces," Trans. Amer. Math. Soc., 343, 927–948 (1994).Google Scholar
  7. 7.
    Krushkal S. L., "The Green function of Teichmüller spaces with applications," Bull. Amer. Math. Soc., 27, 143–147 (1992).Google Scholar
  8. 8.
    Krushkal S. L., "Grunsky inequalities of higher rank with applications to complex geometry and function theory," in: Israel Mathematical Conference Proceedings, Amer. Math. Soc., Providence RI, 2003, 16, pp. 127–153.Google Scholar
  9. 9.
    Lelong P., "Fonction de Green pluricomplexe et lemmes de Schwarz dans le espace de Banach," J. Math. Pures Appl., 69, 319–347 (1989).Google Scholar
  10. 10.
    Bedford E. and Demailly J.-P., "Two counterexamples concerning the pluri-complex Green function in Cn," Indiana Univ. Math. J., 37, 865–867 (1988).Google Scholar
  11. 11.
    Poletskii E. A. and Shabat B. V., "Several complex variables. III: Geometric function theory," in: Ed. G. M. Henkin. Encyclopedia of Mathematical Sciences. Vol. 9, Springer-Verlag, Berlin, 1989, pp. 63–111.Google Scholar
  12. 12.
    Klimek M., "Infinitesimal pseudo-metrics and the Schwarz lemma," Proc. Amer. Math. Soc., 105, 134–140 (1989).Google Scholar
  13. 13.
    Earle C. J., "On the Carathéodory metric in Teichmüller spaces," in: Discontinuous Groups and Riemann Surfaces, Princeton Univ. Press, Princeton, 1984, pp. 99-103. (Ann. of Math. Stud.; 79.)Google Scholar
  14. 14.
    . Krushkal S. L., "Two theorems on a Teichmüller space," Dokl. Akad. Nauk SSSR, 228, No. 2, 290–292 (1976).Google Scholar
  15. 15.
    Krushkal S. L., "Invariant metrics in Teichmüller spaces," Sibirsk. Mat. Zh., 22, No. 2, 209–213 (1981).Google Scholar
  16. 16.
    Krushkal S. L., "Hyperbolic metrics on finite-dimensional Teichmüller spaces," Ann. Acad. Sci. Fenn. Ser. AI. Math., 15, 125–132 (1990).Google Scholar
  17. 17.
    Earle C. J., "Schwarz's lemma and Teichmüller contraction," in: Complex Manifolds and Hyperbolic Geometry (Guanajuato, 2001), Amer. Math. Soc., Providence, RI, 2002, pp. 79–85. (Contemp. Math.; 311.)Google Scholar
  18. 18.
    Earle C. J., Harris L. A., Hubbard J. H., and Mitra S., Schwarz's Lemma and the Kobayashi and Carathéodory Metrics on Complex Banach Manifolds [Preprint] (2001).Google Scholar
  19. 19.
    Grunsky H., "Koeffizientenbediengungen für schlicht abbildende meromorphe Funktionen," Math. Z., 45, 29–61 (1939).Google Scholar
  20. 20.
    Kühnau R., "Verzerrungssatze und Koeffizientenbedingungen vom Grunskyschen Typ für quasikonforme Abbildungen," Math. Nachr., 48, 77–105 (1971).Google Scholar
  21. 21.
    Kühnau R., "Zu den Grunskyschen Coeffizientenbedingungen," Ann. Acad. Sci. Fenn. Ser. AI. Math., 6, 125–130 (1981).Google Scholar
  22. 22.
    Krushkal S. L., "Grunsky coefficient inequalities, Carathéodory metric and extremal quasiconformal mappings," Comment. Math. Helv., 64, 650–660 (1989).Google Scholar
  23. 23.
    Pommerenke Ch., Univalent Functions, Vandenhoeck & Ruprecht, Göttingen (1975).Google Scholar
  24. 24.
    Krushkal S. L. and Kühnau R., Quasikonforme Abbildungen-neue Methode und Anwendungen, Teubner, Leipzig (1983). (Teubner-Texte zur Math.; 54.)Google Scholar
  25. 25.
    Krushkal S. L., "On the Grunsky coefficient conditions," Sibirsk. Mat. Zh., 28, No. 1, 104–110 (1987).Google Scholar
  26. 26.
    Federer H., Geometric Measure Theory, Springer-Verlag, Berlin (1969).Google Scholar
  27. 27.
    Reshetnyak Yu. G., Space Mappings with Bounded Distortion [in Russian], Nauka, Novosibirsk (1982).Google Scholar
  28. 28.
    Kühnau R., "Wann sind die Grunskyschen Koeffizientenbedingungen hinreichend für Q-quasikonforme Fortsetzbarkeit?" Comment. Math. Helv., 61, 290–307 (1986).Google Scholar
  29. 29.
    Krushkal S. L., "Polygonal quasiconformal maps and Grunsky inequalities," J. Anal. Math., 90, 175–196 (2003).Google Scholar
  30. 30.
    Earle C. J. and Kra I., "On sections of some holomorphic families of closed Riemann surfaces," Acta Math., 137, 49–79 (1976).Google Scholar
  31. 31.
    Farkas H. M. and Kra I., Riemann Surfaces, Springer-Verlag, New York (1992).Google Scholar
  32. 32.
    Bers L., "Holomorphic differentials as functions of moduli," Bull. Amer. Math. Soc., 67, 206–210 (1961).CrossRefGoogle Scholar
  33. 33.
    Krushkal S. L., Quasiconformal Mappings and Riemann Surfaces, Wiley, New York (1979).Google Scholar
  34. 34.
    Hamilton R., "Extremal quasiconformal mappings with prescribed boundary values," Trans. Amer. Math. Soc., 138, 399–406 (1969).Google Scholar
  35. 35.
    Reich E. and Strebel K., "Extremal quasiconformal mappings with given boundary values," in: Ed. L. V. Ahlfors} et al. Contribution to Analysis, Acad. Press, New York, 1974, pp. 375–391.Google Scholar
  36. 36.
    Bers L., "An approximation theorem," J. Anal. Math., 14, 1–4 (1965).Google Scholar
  37. 37.
    Li Zhong, "Nonuniqueness of geodesics in infinite dimensional Teichmüller spaces," Complex Variables Theory Appl., 16, No. 4, 261–271 (1991).Google Scholar
  38. 38.
    Tanigawa H., "Holomorphic families of geodesic discs in infinite dimensional Teichmüller spaces," Nagoya Math. J., 127, 117–128 (1992).Google Scholar
  39. 39.
    Royden H. L., "Complex Finsler metrics," Contemp. Math., 49, 119–124 (1986).Google Scholar
  40. 40.
    Wong B., "On the holomorphic curvature of some intrinsic metrics," Proc. Amer. Math. Soc., 65, 57–61 (1977).Google Scholar
  41. 41.
    Lehto O., Univalent Functions and Teichmüller spaces, Springer-Verlag, New York (1987).Google Scholar
  42. 42.
    Krushkal S. L., "Strengthening pseudoconvexity of finite-dimensional Teichmüller spaces," Math. Ann., 290, 681–687 (1991).Google Scholar
  43. 43.
    McMullen C. T., "The moduli space of Riemann surfaces is Kähler hyperbolic," Ann. of Math., 151, 327–357 (2000).Google Scholar
  44. 44.
    Mañé R., Sad P., and Sullivan D., "On dynamics of rational maps," Ann. Sci. Ecol. Norm. Sup., 16, 193–216 (1983).Google Scholar
  45. 45.
    Bers L. and Royden H. L., "Holomorphic families of injections," Acta Math., 157, 259–286 (1986).Google Scholar
  46. 46.
    Slodkowski Z., "Holomorphic motions and polynomial hulls," Proc. Amer. Math. Soc., 111, 347–355 (1991).Google Scholar
  47. 47.
    Sullivan D. and Thurston W. P., "Extending holomorphic motions," Acta Math., 157, 243–257 (1986).Google Scholar
  48. 48.
    Kühnau R., "Möglichst konforme Spiegelung an einer Jordankurve," Jber. Deutsch. Math. Verein., 90, 90–109 (1988).Google Scholar
  49. 49.
    Krushkal S. L., "Extension of conformal mappings and hyperbolic metrics," Siberian Math. J., 30, No. 5, 730–744 (1989).Google Scholar
  50. 50.
    Krushkal S. L., "Quasiconformal extremals of non-regular functionals," Ann. Acad. Sci. Fenn. Ser. AI. Math., 17, 295–306 (1992).Google Scholar
  51. 51.
    Sibony N., "A class of hyperbolic manifolds," in: Recent Developments in Several Complex Variables / J. E. Forness, Princeton Univ. Press, Princeton, 1981, pp. 357–372. (Ann. of Math. Stud.; 100.)Google Scholar
  52. 52.
    Strebel K., "On the existence of extremal Teichmüller mappings," J. Anal. Math., 30, 464–480 (1976).Google Scholar
  53. 53.
    Ahlfors L.V. and Weill G., "A uniqueness theorem for Beltrami equations," Proc. Amer. Math. Soc., 13, 975–978 (1962).Google Scholar
  54. 54.
    Krushkal S. L. and Kühnau R., "A quasiconformal dynamic property of the disk," J. Anal. Math., 72, 93–103 (1997).Google Scholar
  55. 55.
    Krushkal S. L., "Exact coefficient estimates for univalent functions with quasiconformal extension," Ann. Acad. Sci. Fenn. Ser. AI. Math., 20, 349–357 (1995).Google Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • S. L. Krushkal
    • 1
    • 2
  1. 1.Bar-Ilan UniversityRamat Gan
  2. 2.Sobolev Institute of MathematicsNovosibirsk

Personalised recommendations