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 $g\geq 2$ does not admit any Riemannian metric $ds^2$ of nonnegative scalar curvature such that $ds^2 \succ ds_{T}^2$ where $ds_{T}^2$ 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.