On positive scalar curvature and moduli of curves

TL;DR

This paper proves that covers of the moduli space of closed Riemann surfaces cannot support certain types of Riemannian metrics with nonnegative or positive scalar curvature, confirming a conjecture of Farb-Weinberger.

Contribution

It establishes new curvature obstructions on moduli spaces, extending understanding of their geometric structure and confirming a conjecture about scalar curvature.

Findings

No nonnegative scalar curvature metrics dominate the Teichmüller metric on covers.

No complete metrics of positive scalar curvature exist in the quasi-isometry class of the Teichmüller metric.

Confirms Farb-Weinberger conjecture on scalar curvature obstructions.

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.