Canonical Metrics on the Moduli Space of Riemann Surfaces II

Abstract

In this paper we continue our study on the canonical metrics on the Teichmüller and the moduli space of Riemman surfaces. We first prove the equivalence of the Bergman metric and the Carathéodory metric to the Kähler-Einstein metric, solving another old conjecture of Yau. We then prove that the Ricci curvature of the perturbed Ricci metric has negative upper and lower bounds, and it also has bounded geometry. Then we study in detail the boundary behaviors of the Kähler-Einstein metric and prove that it has bounded geometry, and all of the covariant derivatives of its curvature are uniformly bounded on the Teichmüller space. As an application of our detailed understanding of these metrics, we prove that the logarithmic cotangent bundle of the moduli space is stable in the sense of Mumford.