## Journal of Differential Geometry

### On positive scalar curvature and moduli of curves

#### Abstract

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

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

Dates
First available in Project Euclid: 6 February 2019

https://projecteuclid.org/euclid.jdg/1549422104

Digital Object Identifier
doi:10.4310/jdg/1549422104

Mathematical Reviews number (MathSciNet)
MR3909910

Zentralblatt MATH identifier
07015572

#### Citation

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. https://projecteuclid.org/euclid.jdg/1549422104