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The Weil-Petersson geometry on the thick part of the moduli space of Riemann surfaces


Author: Zheng Huang
Journal: Proc. Amer. Math. Soc. 135 (2007), 3309-3316
MSC (2000): Primary 30F60, 32G15
Published electronically: June 21, 2007
MathSciNet review: 2322763
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Abstract: In the thick part of the moduli space of Riemann surfaces, we show that the sectional curvature of the Weil-Petersson metric is bounded independently of the genus of the surface.


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  • 1. Lars V. Ahlfors, Some remarks on Teichmüller’s space of Riemann surfaces, Ann. of Math. (2) 74 (1961), 171–191. MR 0204641
  • 2. Lars V. Ahlfors, Curvature properties of Teichmüller’s space, J. Analyse Math. 9 (1961/1962), 161–176. MR 0136730
  • 3. Lipman Bers, Spaces of degenerating Riemann surfaces, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973) Princeton Univ. Press, Princeton, N.J., 1974, pp. 43–55. Ann. of Math. Studies, No. 79. MR 0361051
  • 4. Clifford J. Earle and James Eells, A fibre bundle description of Teichmüller theory, J. Differential Geometry 3 (1969), 19–43. MR 0276999
  • 5. James Eells Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. MR 0164306
  • 6. Philip Hartman, On homotopic harmonic maps, Canad. J. Math. 19 (1967), 673–687. MR 0214004
  • 7. Zheng Huang, Asymptotic flatness of the Weil-Petersson metric on Teichmüller space, Geom. Dedicata 110 (2005), 81–102. MR 2136021, 10.1007/s10711-003-0816-x
  • 8. Z. Huang, On asymptotic Weil-Petersson geometry of Teichmüller space of Riemann surfaces, Asian J. Math., to appear
  • 9. Jürgen Jost, Two-dimensional geometric variational problems, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1991. A Wiley-Interscience Publication. MR 1100926
  • 10. Howard Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J. 43 (1976), no. 3, 623–635. MR 0417456
  • 11. Yair N. Minsky, Harmonic maps, length, and energy in Teichmüller space, J. Differential Geom. 35 (1992), no. 1, 151–217. MR 1152229
  • 12. M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Ann. of Math.(2), to appear
  • 13. M. Mirzakhani, Weil-Petersson volumes and intersection theory on the moduli space of curves, Preprint, 2004
  • 14. Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511
  • 15. H. L. Royden, Intrinsic metrics on Teichmüller space, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 217–221. MR 0447636
  • 16. J. H. Sampson, Some properties and applications of harmonic mappings, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 2, 211–228. MR 510549
  • 17. Richard Schoen and Shing Tung Yau, On univalent harmonic maps between surfaces, Invent. Math. 44 (1978), no. 3, 265–278. MR 0478219
  • 18. Stefano Trapani, 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), no. 4, 681–705. MR 1176026, 10.1007/BF01444740
  • 19. A. J. Tromba, On a natural algebraic affine connection on the space of almost complex structures and the curvature of Teichmüller space with respect to its Weil-Petersson metric, Manuscripta Math. 56 (1986), no. 4, 475–497. MR 860734, 10.1007/BF01168506
  • 20. Michael Wolf, The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989), no. 2, 449–479. MR 982185
  • 21. Michael Wolf, Infinite energy harmonic maps and degeneration of hyperbolic surfaces in moduli space, J. Differential Geom. 33 (1991), no. 2, 487–539. MR 1094467
  • 22. Scott Wolpert, Noncompleteness of the Weil-Petersson metric for Teichmüller space, Pacific J. Math. 61 (1975), no. 2, 573–577. MR 0422692
  • 23. Scott Wolpert, On the homology of the moduli space of stable curves, Ann. of Math. (2) 118 (1983), no. 3, 491–523. MR 727702, 10.2307/2006980
  • 24. Scott A. Wolpert, Chern forms and the Riemann tensor for the moduli space of curves, Invent. Math. 85 (1986), no. 1, 119–145. MR 842050, 10.1007/BF01388794
  • 25. Scott A. Wolpert, Geometry of the Weil-Petersson completion of Teichmüller space, Surveys in differential geometry, Vol. VIII (Boston, MA, 2002) Surv. Differ. Geom., VIII, Int. Press, Somerville, MA, 2003, pp. 357–393. MR 2039996, 10.4310/SDG.2003.v8.n1.a13
  • 26. S. Wolpert, Weil-Petersson perspectives, preprint, 2005

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Additional Information

Zheng Huang
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: zhengh@umich.edu

DOI: https://doi.org/10.1090/S0002-9939-07-08868-5
Received by editor(s): March 16, 2006
Received by editor(s) in revised form: July 26, 2006
Published electronically: June 21, 2007
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.