Uniform bounds on harmonic Beltrami differentials and Weil-Petersson curvatures

Martin Bridgeman Boston College, Chestnut Hill (Boston) Yunhui Wu Qinghua (Tsing Hua) University, Beijing, PEOPLES REPUBLIC OF CHINA

Differential Geometry Geometric Analysis and Geometric Topology mathscidoc:2203.10005

Journal für die Reine und Angewandte Mathematik, 770, 159-181, 2021.12
In this article we show that for every finite area hyperbolic surface X of type (g,n) and any harmonic Beltrami differential μ on X , then the magnitude of μ at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil–Petersson norm of μ over the square root of the systole of X up to a uniform positive constant multiplication. We apply the uniform bound above to show that the Weil–Petersson Ricci curvature, restricted at any hyperbolic surface of short systole in the moduli space, is uniformly bounded from below by the negative reciprocal of the systole up to a uniform positive constant multiplication. As an application, we show that the average total Weil–Petersson scalar curvature over the moduli space is uniformly comparable to -g as the genus g goes to infinity.
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@inproceedings{martin2021uniform,
  title={Uniform bounds on harmonic Beltrami differentials and Weil-Petersson curvatures},
  author={Martin Bridgeman, and Yunhui Wu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220317145252245199986},
  booktitle={Journal für die Reine und Angewandte Mathematik},
  volume={770},
  pages={159-181},
  year={2021},
}
Martin Bridgeman, and Yunhui Wu. Uniform bounds on harmonic Beltrami differentials and Weil-Petersson curvatures. 2021. Vol. 770. In Journal für die Reine und Angewandte Mathematik. pp.159-181. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220317145252245199986.
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