On positive scalar curvature and moduli of curves

Kefeng Liu UCLA Yunhui Wu Tsinghua University

Geometric Analysis and Geometric Topology mathscidoc:1904.15002

Distinguished Paper Award in 2019

Journal of Differential Geometry, 111, (2), 315–338, 2019
In this article we first show that any finite cover of the moduli space of closed Riemann surfaces of genus $g$ with $g\geq 2$ does not admit any Riemannian metric $ds^2$ of nonnegative scalar curvature such that $ds^2 \succ ||\cdot||_{T}$ where $||\cdot||_T$ is the Teichm\"uller metric. Our second result is the proof that any cover $M$ of the moduli space $\mathbb{M}_{g}$ of a closed Riemann surface $S_{g}$ does not admit any complete Riemannian metric of uniformly positive scalar curvature in the quasi-isometry class of the Teichm\"uller metric, which implies a conjecture of Farb-Weinberger in \cite{Farb-prob}.
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@inproceedings{kefeng2019on,
  title={On positive scalar curvature and moduli of curves},
  author={Kefeng Liu, and Yunhui Wu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190430210100479286304},
  booktitle={Journal of Differential Geometry},
  volume={111},
  number={2},
  pages={315–338},
  year={2019},
}
Kefeng Liu, and Yunhui Wu. On positive scalar curvature and moduli of curves. 2019. Vol. 111. In Journal of Differential Geometry. pp.315–338. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190430210100479286304.
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