Communications in Analysis and Geometry

Volume 25 (2017)

Number 2

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

Pages: 465 – 484



Yunhui Wu (Department of Mathematics, Rice University, Houston, Texas, U.S.A.; and Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)


In this article we show that any finite cover of the moduli space of closed Riemann surfaces of $g$ genus with $g \geqslant 2$ 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 Teichmüller metric, on any finite cover of the moduli space is negative provided that the scalar curvature is bounded from below.


moduli space, scalar curvature, Teichmüller metric

2010 Mathematics Subject Classification

30F60, 32G15

Received 30 April 2015

Published 4 August 2017