  We prove the equivalences of several classical complete metrics on the
Teichm\"uller and the moduli spaces of Riemann surfaces. We use as bridge two
new K\"ahler metrics, the Ricci metric and the perturbed Ricci metric and prove
that the perturbed Ricci metric is a complete K\"ahler metric with bounded
negative holomorphic sectional curvature and bounded bisectional and Ricci
curvature. As consequences we prove that these two new metrics are equivalent
to several famous classical metrics, which inlcude the Teichm\"uller metric,
therefore the Kabayashi metric, the K\"ahler-Einstein metric and the McMullen
metric. This also solves a conjecture of Yau in the early 80s.
