Canonical metrics on the moduli space of Riemann Surfaces I

Abstract

In this paper, we continue our study of the canonical metrics on the moduli space of curves. We first prove the bounded geometry of the Ricci and perturbed Ricci metrics. By carefully choosing the pertubation constant and by studying the asymptotics, we show that the Ricci and holomorphic sectional curvatures of the perturbed Ricci metric are bounded from above and below by negative constants. Based on our understanding of the Kahler–Einstein metric, we show that the logarithmic cotangent bundle over the Deligne–Mumford moduli space is stable with respect to the canonical polarization. Finally, in the last section, we prove the strongly bounded geometry of the Kahler–Einstein metric by using the Kahler–Ricci flow and a priori estimates of the complex Monge-Ampere equation.