Skip to main content
Log in

The Weil–Petersson Isometry Group

  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

We prove that, for 3g−3+n>1 and (g,n)≠(1,2), the group of Weil–Petersson isometries of the Teichmüller space T g,n coincides with the extended mapping class group.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Abikoff, W.: Augmented Teichmuller spaces, Bull. Amer. Math. Soc. 82 (1971), 333–334.

    Google Scholar 

  2. Abikoff, W.: Degenerating families of Riemann surfaces, Ann. of Math. 105 (1977), 29–44.

    Google Scholar 

  3. Ahlfors, L.: Some remarks on Teichmüller's space of Riemann surfaces, Ann. Math. 74 (1961), 171–191.

    Google Scholar 

  4. Ballman, W.: Lectures on Spaces of Nonpositive Curvature, Birkhäuser, Basel, 1995.

  5. Bers, L.: Spaces of Degenerating Riemann Surfaces in Discontinous Groups and Riemann Surfaces, Ann. of Math Stud. 79, Princeton Univ. Press, Princeton, New Jersey, 1974.

    Google Scholar 

  6. Bridson, M. and Haefliger, A.: Metric Spaces of Non-Positive Curvature, Springer, Berlin, 1999.

    Google Scholar 

  7. Chu, T.: The Weil—Petersson metric in moduli space, Chinese J. Math. 4 (1976), 29–51.

    Google Scholar 

  8. Earle, C. and Kra, I.: On isometries between Teichmuller spaces, Duke J. Math 41 (1974), 583–591.

    Google Scholar 

  9. Ivanov, N.: Automorphism of complexes of curves and of Teichmuller spaces, Internat. Math. Res. Notices 14 (1997), 651–666.

    Google Scholar 

  10. Korkmaz, M.: Automorphisms of complexes of curves in punctured spheres and on punctured tori, Topology Appl. 95 (1999), 85–111.

    Google Scholar 

  11. Luo, F.: Automorphisms of the complex of curves, Topology 39 (2000), 283–298.

    Google Scholar 

  12. Masur, H.: The extension of the Weil—Petersson metric to the boundary of Teichmuller space, Duke Math J. 43 (1976), 623–635.

    Google Scholar 

  13. Mumford, D.: A remark on Mahler's compactness theorem, Proc. Amer. Math. Soc. 28 (1971), 289–294.

    Google Scholar 

  14. Nag, S.: Complex Analytic Theory of Teichmuller Space, Wiley, New York, 1988.

    Google Scholar 

  15. Royden, H.: Automorphisms and isometries of Teichmuller spaces. Advances in the theory of Riemann surfaces, Ann. of Math. Stud. 66, Princeton, Univ. Press, Princeton, New Jersey, 1970, pp. 369–383.

    Google Scholar 

  16. Royden, H.: Oral communication.

  17. Schoen, R. and Yau, S.-T.: Compact group actions and the topology of manifolds with nonpositive curvature, Topology 18 (1979), 361–380.

    Google Scholar 

  18. Strebel, K.: Quadratic Differentials, Springer, Berlin, 1984.

    Google Scholar 

  19. Trapani, S.: On the determinant of the bundle of meromorphic quadratic differentials on the Deligne-Mumford compactification of the moduli space of Riemann surfaces, Math. Ann. 293 (1992), 681–705.

    Google Scholar 

  20. Tromba, A.: On a natural algebraic affine connection on the space of almost complex structures and the curvature of Teichmuller space with respect to its Weil—Petersson metric, Manuscripta Math. 56 (1986), 475–497.

    Google Scholar 

  21. Tromba, A.: Teichmuller Theory in Riemannian Geometry, Birkhäuser, Basel, 1992.

  22. Wolpert, S.: Noncompleteness of the Weil—Petersson metric for Teichmuller space, Pacific J. Math. 61 (1975), 573–577.

    Google Scholar 

  23. Wolpert, S.: The finite Weil—Petersson diameter of Riemann space, Pacific J. Math. 70 (1977), 281–288.

    Google Scholar 

  24. Wolpert, S.: Chern forms and the Riemann tensor for the moduli space of curves, Invent. Math. 85 (1986), 119–145.

    Google Scholar 

  25. Wolpert, S.: Geodesic length functions and the Nielsen problem, J.Differential Geom. 25 (1987), 275–295.

    Google Scholar 

  26. Yamada, S.: Weil—Petersson completion of Teichmuller spaces and mapping class group actions, Manuscript.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Masur, H., Wolf, M. The Weil–Petersson Isometry Group. Geometriae Dedicata 93, 177–190 (2002). https://doi.org/10.1023/A:1020300413472

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1020300413472

Navigation