Journal of Differential Geometry

On positive scalar curvature and moduli of curves

Kefeng Liu and Yunhui Wu

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In this article we first show that any finite cover of the moduli space of closed Riemann surfaces of genus $g$ with $g \geqslant 2$ does not admit any Riemannian metric $ds^2$ of nonnegative scalar curvature such that ${\lVert \, \cdotp \rVert}_{ds^2} \succ {\lVert \, \cdotp \rVert}_T$ where ${\lVert \, \cdotp \rVert}_T$ is the Teichmüller 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 Teichmüller metric, which implies a conjecture of Farb–Weinberger in [9].

Article information

J. Differential Geom., Volume 111, Number 2 (2019), 315-338.

Received: 28 January 2016
First available in Project Euclid: 6 February 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30F60: Teichmüller theory [See also 32G15] 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx] 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]

moduli space scalar curvature Teichmüller metric Riemannian metric


Liu, Kefeng; Wu, Yunhui. On positive scalar curvature and moduli of curves. J. Differential Geom. 111 (2019), no. 2, 315--338. doi:10.4310/jdg/1549422104.

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