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Canonical metrics on the moduli space of Riemann surfaces II

Kefeng Liu Xiaofeng Sun ShingTung Yau

Differential Geometry mathscidoc:1912.43052

Journal of Differential Geometry, 69, (1), 163-216, 2005
In this paper, we continue our study of the canonical metrics on the moduli space of curves. We first prove the bounded geometry of the Ricci and perturbed Ricci metrics. By carefully choosing the pertubation constant and by studying the asymptotics, we show that the Ricci and holomorphic sectional curvatures of the perturbed Ricci metric are bounded from above and below by negative constants. Based on our understanding of the KhlerEinstein metric, we show that the logarithmic cotangent bundle over the DeligneMumford moduli space is stable with respect to the canonical polarization. Finally, in the last section, we prove the strongly bounded geometry of the KhlerEinstein metric by using the KhlerRicci flow and a priori estimates of the complex Monge-Ampere equation.
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@inproceedings{kefeng2005canonical,
  title={Canonical metrics on the moduli space of Riemann surfaces II},
  author={Kefeng Liu, Xiaofeng Sun, and ShingTung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221111250858065612},
  booktitle={Journal of Differential Geometry},
  volume={69},
  number={1},
  pages={163-216},
  year={2005},
}
Kefeng Liu, Xiaofeng Sun, and ShingTung Yau. Canonical metrics on the moduli space of Riemann surfaces II. 2005. Vol. 69. In Journal of Differential Geometry. pp.163-216. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221111250858065612.
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