Survey on the Canonical Metrics on the Teichm\"uller Spaces and the Moduli Spaces of Riemann Surfaces

This thesis results from an intensive study on the canonical metrics on the Teichm\"uller spaces and the moduli spaces of Riemann surfaces. There are several renowned classical metrics on Tg and Mg, including the Weil-Petersson metric, the Teichm\"uller metric, the Kobayashi metric, the Bergman metric, the Carath\'eodory metric and the K\"ahler-Einstein metric. The Teichm\"uller metric, the Kobayashi metric and the Carath\'eodory metric are only (complete) Finsler metrics, but they are effective tools in the study of hyperbolic property of Mg. The Weil-Petersson metric is an incomplete K\"ahler metric, while the Bergman metric and the K\"ahler-Einstein metric are complete K\"ahler metrics. However, McMullen introduced a new complete K\"ahler metric, called the McMullen metric, by perturbing the Weil-Petersson metric. This metric is indeed equivalent to the Teichm\"uller metric. Recently, Liu-Sun-Yau proved that the equivalence of the K\"ahler-Einstein metric to the Teichm\"uller metric, and hence gave a positive answer to a conjecture proposed by Yau. Their approach in the proof is to introduce two new complete K\"ahler metrics, namely, the Ricci metric and the perturbed Ricci metric, and then establish the equivalence of the Ricci metric to the K\"ahler-Einstein metric and the equivalence of the Ricci metric to the McMullen metric. The main purpose of this thesis is to survey the properties of these various metrics and the geometry of Tg and Mg induced by these metrics.