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

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.