  In this paper we continue our study on the canonical metrics on the
Teichm\"uller and the moduli space of Riemman surfaces. We first prove the
equivalence of the Bergman metric and the Carath\'eodory metric to the
K\"ahler-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 K\"ahler-Einstein metric and prove that it has
bounded geometry, and all of the covariant derivatives of its curvature are
uniformly bounded on the Teichm\"uller 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.
