Scalar curvatures of Hermitian metrics on the moduli space of Riemann surfaces

TL;DR

This paper proves that finite covers of the moduli space of Riemann surfaces cannot have complete finite-volume Hermitian metrics with non-negative scalar curvature, and shows the total scalar curvature is negative for metrics comparable to the Teichmüller metric.

Contribution

It establishes new curvature restrictions on Hermitian metrics on moduli spaces of Riemann surfaces, extending understanding of their geometric structure.

Findings

Finite covers do not admit complete finite-volume Hermitian metrics with non-negative scalar curvature.

The total scalar curvature of metrics comparable to the Teichmüller metric is negative.

Scalar curvature bounds impose strong geometric constraints on moduli space metrics.

Abstract

In this article we show that any finite cover of the moduli space of closed Riemann surfaces of $g$ genus with $g\geq 2$ does not admit any complete finite-volume Hermitian metric of non-negative scalar curvature. Moreover, we also show that the total mass of the scalar curvature of any almost Hermitian metric, which is equivalent to the Teichm\"uller metric, on any finite cover of the moduli space is negative provided that the scalar curvature is bounded from below.