Canonical Metrics on the Moduli Space of Riemann Surfaces I

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.