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

摘要

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