On positive scalar curvature and moduli of curves

Kefeng Liu UCLA Yunhui Wu Tsinghua University

Geometric Analysis and Geometric Topology mathscidoc:1904.15002

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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