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The Weil–Petersson curvature operator on the universal Teichmüller space

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Abstract

The universal Teichmüller space is an infinitely dimensional generalization of the classical Teichmüller space of Riemann surfaces. It carries a natural Hilbert structure, on which one can define a natural Riemannian metric, the Weil–Petersson metric. In this paper we investigate the Weil–Petersson Riemannian curvature operator \(\tilde{Q}\) of the universal Teichmüller space with the Hilbert structure, and prove the following:

  1. (i)

    \(\tilde{Q}\) is non-positive definite.

  2. (ii)

    \(\tilde{Q}\) is a bounded operator.

  3. (iii)

    \(\tilde{Q}\) is not compact; the set of the spectra of \(\tilde{Q}\) is not discrete.

As an application, we show that neither the Quaternionic hyperbolic space nor the Cayley plane can be totally geodesically immersed in the universal Teichmüller space endowed with the Weil–Petersson metric.

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References

  1. Ahlfors, L.V.: Some remarks on Teichmüller’s space of Riemann surfaces. Ann. Math. 2(74), 171–191 (1961)

    Article  MATH  Google Scholar 

  2. Bers, L.: Automorphic forms and general Teichmüller spaces. In: Proc. Conf. Complex Analysis (Minneapolis, 1964), pp. 109–113. Springer, Berlin (1965)

  3. Brock, J., Farb, B.: Curvature and rank of Teichmüller space. Am. J. Math. 128(1), 1–22 (2006)

    Article  MATH  Google Scholar 

  4. Chavel, I.: Riemannian symmetric spaces of rank one. In: Lecture Notes in Pure and Applied Mathematics, vol. 5. Marcel Dekker, Inc., New York (1972)

  5. Chu, T.: The Weil–Petersson metric in the moduli space. Chin. J. Math. 4(2), 29–51 (1976)

    MathSciNet  MATH  Google Scholar 

  6. Corlette, K.: Archimedean superrigidity and hyperbolic geometry. Ann. Math. (2) 135(1), 165–182 (1992)

  7. Cui, G.: Integrably asymptotic affine homeomorphisms of the circle and Teichmüller spaces. Sci. China Ser. A. Math. 43(3), 267–279 (2000)

    Article  MATH  Google Scholar 

  8. Daskalopoulos, G., Mese, C.: Rigidity of Teichmüller space, ArXiv e-prints. arXiv:1502.03367 (2015)

  9. Duchesne, B.: Infinite-dimensional nonpositively curved symmetric spaces of finite rank. Int. Math. Res. Not. IMRN 7, 1578–1627 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Duchesne, B.: Superrigidity in infinite dimension and finite rank via harmonic maps. Groups Geom. Dyn. 9(1), 133–148 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gay-Balmaz, F., Ratiu, T.S.: The geometry of the universal Teichmüller space and the Euler–Weil–Petersson equation. Adv. Math. 279, 717–778 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gromov, M., Schoen, R.: Harmonic maps into singular spaces and \(p\)-adic superrigidity for lattices in groups of rank one. Inst. Hautes Études Sci. Publ. Math. 76, 165–246 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hejhal, D.A.: The Selberg trace formula for \({\rm PSL}(2,R)\). Vol. I, Lecture Notes in Mathematics, vol. 548. Springer, Berlin (1976)

  14. Huang, Z.: Asymptotic flatness of the Weil–Petersson metric on Teichmüller space. Geom. Dedic. 110, 81–102 (2005)

    Article  MATH  Google Scholar 

  15. Huang, Z.: On asymptotic Weil–Petersson geometry of Teichmüller space of Riemann surfaces. Asian J. Math. 11(3), 459–484 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Huang, Z.: The Weil-Petersson geometry on the thick part of the moduli space of Riemann surfaces. Proc. Am. Math. Soc. 135(10), 3309–3316 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jost, J., Yau, S.-T.: Harmonic Maps and Superrigidity, Tsing Hua Lectures on Geometry and Analysis (Hsinchu, 1990–1991), pp. 213–246. Int. Press, Cambridge (1997)

  18. Lang, S.: Fundamentals of differential geometry. In: Graduate Texts in Mathematics, vol. 191. Springer, New York (1999)

  19. Liu, K., Sun, X., Yau, S.-T.: Canonical metrics on the moduli space of Riemann surfaces. I. J. Differ. Geom. 68(3), 571–637 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu, K., Sun, X., Yau, S.-T.: Good geometry on the curve moduli. Publ. Res. Inst. Math. Sci. 44(2), 699–724 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, K., Sun, X., Yang, X., Yau, S.-T.: Curvatures of moduli space of curves and applications, ArXiv e-prints. arXiv:1312.6932 (2013)

  22. Mok, N., Siu, Y.T., Yeung, S.-K.: Geometric superrigidity. Invent. Math. 113(1), 57–83 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  23. Schumacher, G.: Harmonic maps of the moduli space of compact Riemann surfaces. Math. Ann. 275(3), 455–466 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  24. Schmüdgen, K.: Unbounded self-adjoint operators on Hilbert space. In: Graduate Texts in Mathematics, vol. 265. Springer, Dordrecht (2012)

  25. Teo, L.-P.: The Weil–Petersson geometry of the moduli space of Riemann surfaces. Proc. Am. Math. Soc. 137(2), 541–552 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tromba, A.J.: 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. Manuscr. Math. 56(4), 475–497 (1986)

    Article  MATH  Google Scholar 

  27. Takhtajan, L.A., Teo, L.-P: Weil–Petersson metric on the universal Teichmüller space. Mem. Amer. Math. Soc. 183(861) (2006)

  28. Wolpert, S.: Noncompleteness of the Weil–Petersson metric for Teichmüller space. Pac. J. Math. 61(2), 573–577 (1975)

    Article  MATH  Google Scholar 

  29. Wolpert, S.A.: Chern forms and the Riemann tensor for the moduli space of curves. Invent. Math. 85(1), 119–145 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wolpert, S.A.: Geodesic length functions and the Nielsen problem. J. Differ. Geom. 25(2), 275–296 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wolpert, S.A.: Families of Riemann surfaces and Weil–Petersson geometry. In: CBMS Regional Conference Series in Mathematics, vol. 113, the American Mathematical Society, Providence (2010)

  32. Wolpert, S.A.: Understanding Weil–Petersson curvature. In: Geometry and Analysis. No. 1, Adv. Lect. Math. (ALM), vol. 17, Int. Press, Somerville, pp. 495–515 (2011)

  33. Wolf, M.: The Weil–Petersson Hessian of length on Teichmüller space. J. Differ. Geom. 91(1), 129–169 (2012)

    Article  MATH  Google Scholar 

  34. Wolpert, S.A.: Geodesic-length functions and the Weil–Petersson curvature tensor. J. Differ. Geom. 91(2), 321–359 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Yunhui, W.: The Riemannian sectional curvature operator of the Weil–Petersson metric and its application. J. Differ. Geom. 96(3), 507–530 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Yunhui, W.: On the Weil–Petersson curvature of the moduli space of Riemann surfaces of large genus. Int. Math. Res. Not. 2017(4), 1066–1102 (2017)

    MathSciNet  MATH  Google Scholar 

  37. Wolf, M., Wu, Y.: Uniform bounds for Weil–Petersson curvatures. ArXiv e-prints: arXiv:1501.03245 (2015)

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Acknowledgements

We acknowledge supports from U.S. National Science Foundation Grants DMS 1107452, 1107263, 1107367 “RNMS: Geometric Structures and Representation varieties” (the GEAR Network). This work was supported by a grant from the Simons Foundation (#359635, Zheng Huang) and a research award from the PSC-CUNY. Part of the work is completed when the second named author was a G. C. Evans Instructor at Rice University, he would like to thanks to the mathematics department for their support. He would also like to acknowledge a start-up research fund from Tsinghua University to finish this work. The authors would like to thank an anonymous referee whose comments are very helpful to improve the paper.

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Authors and Affiliations

  1. Department of Mathematics, The City University of New York, Staten Island, New York, NY, 10314, USA

    Zheng Huang

  2. The Graduate Center, The City University of New York, 365 Fifth Ave., New York, NY, 10016, USA

    Zheng Huang

  3. Yau Mathematical Sciences Center, Tsinghua University, Haidian District, Beijing, 100084, China

    Yunhui Wu

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  1. Zheng Huang
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  2. Yunhui Wu
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Correspondence to Zheng Huang.

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Communicated by Ngaiming Mok.

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Huang, Z., Wu, Y. The Weil–Petersson curvature operator on the universal Teichmüller space. Math. Ann. 373, 1–36 (2019). https://doi.org/10.1007/s00208-017-1616-1

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  • DOI: https://doi.org/10.1007/s00208-017-1616-1

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