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The Weil-Petersson curvature operator on the universal Teichmüller space

Zheng Huang The City University of New York Yunhui Wu Tsinghua University

Differential Geometry mathscidoc:1909.43006

Mathematische Annalen, 373, 1-36, 2019
The universal Teichmu¨ller space is an infinitely dimensional generalization of the classical Teichmu¨ller 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 ˜ Q of the universal Teichmu¨ller space with the Hilbert structure, and prove the following: (i) ˜ Q is non-positive definite. (ii) ˜ Q is a bounded operator. (iii) ˜ Q is not compact; the set of the spectra of ˜ 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 Teichmu¨ller space endowed with the Weil-Petersson metric.
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@inproceedings{zheng2019the,
  title={The Weil-Petersson curvature operator on the universal Teichmüller space},
  author={Zheng Huang, and Yunhui Wu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190904112252359712483},
  booktitle={Mathematische Annalen},
  volume={373},
  pages={1-36},
  year={2019},
}
Zheng Huang, and Yunhui Wu. The Weil-Petersson curvature operator on the universal Teichmüller space. 2019. Vol. 373. In Mathematische Annalen. pp.1-36. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190904112252359712483.
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