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Mathematische Annalen

, Volume 373, Issue 1–2, pp 1–36 | Cite as

The Weil–Petersson curvature operator on the universal Teichmüller space

  • Zheng HuangEmail author
  • Yunhui Wu
Article

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.

Mathematics Subject Classification

Primary 30F60 Secondary 32G15 

Notes

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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Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Department of MathematicsThe City University of New YorkNew YorkUSA
  2. 2.The Graduate CenterThe City University of New YorkNew YorkUSA
  3. 3.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina

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