Curvatures of moduli space of curves and applications

In this paper, we investigate the geometry of the moduli space of curves by using the curvature properties of direct image sheaves of vector bundles. We show that the moduli space $(M_g,{\omega }_{WP})$ of curves with genus $g>1$ has dual-Nakano negative and semi-Nakano-negative curvature, and in particular, it has non-positive Riemannain curvature operator and also non-positive complex sectional curvature. As applications, we prove that any submanifold in $M_g$ which is totally geodesic in $A_g$ with finite volume must be a ball quotient.