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Canonical metrics on the moduli space of Riemann surfaces. I

Kefeng Liu Xiaofeng Sun Shing-Tung Yau

Differential Geometry mathscidoc:1912.43051

J. Differential Geom, 68, (3), 571-637, 2004.1
One of the main purposes of this paper is to understand the geometry of the moduli and the Teichmller spaces of Riemann surfaces. The most interesting results we have in this paper are the detailed understanding of two new complete Khler metrics with nice properties and the KhlerEinstein metric on the Teichmller and the moduli spaces of Riemann surfaces. The two new metrics, the Ricci metric and the perturbed Ricci metric, are naturally defined as the negative Ricci curvature of the WeilPetersson metric and a combination of it with the WeilPetersson metric. We prove that these new metrics and the KhlerEinstein metric on the Teichmller and moduli spaces all have Poincar type boundary behavior, and further, in [7], we prove that they all have bounded geometry. Note that the KhlerEinstein metric is the key link between the differential geometric and algebraic geometric aspects of these spaces
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@inproceedings{kefeng2004canonical,
  title={Canonical metrics on the moduli space of Riemann surfaces. I},
  author={Kefeng Liu, Xiaofeng Sun, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221111247496249611},
  booktitle={J. Differential Geom},
  volume={68},
  number={3},
  pages={571-637},
  year={2004},
}
Kefeng Liu, Xiaofeng Sun, and Shing-Tung Yau. Canonical metrics on the moduli space of Riemann surfaces. I. 2004. Vol. 68. In J. Differential Geom. pp.571-637. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221111247496249611.
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