Chan, Kin Wai (2024) Survey on the canonical metrics on the Teichmüller spaces and the moduli spaces of Riemann surfaces. The Chinese University of Hong Kong, Masters Thesis, 108pp.
Abstract
This thesis results from an intensive study on the canonical metrics on the Teichmüller spaces and the moduli spaces of Riemann surfaces. There are several renowned classical metrics on $T_g$ and $\mathcal{M}_g$, including the Weil-Petersson metric, the Teichmüller metric, the Kobayashi metric, the Bergman metric, the Carathéodory metric and the Kähler-Einstein metric. The Teichmüller metric, the Kobayashi metric and the Carathéodory metric are only (complete) Finsler metrics, but they are effective tools in the study of hyperbolic property of $\mathcal{M}_g$. The Weil-Petersson metric is an incomplete Kähler metric, while the Bergman metric and the Kähler-Einstein metric are complete Kähler metrics. However, McMullen introduced a new complete Kähler metric, called the McMullen metric, by perturbing the Weil-Petersson metric. This metric is indeed equivalent to the Teichmüller metric. Recently, Liu-Sun-Yau proved that the equivalence of the Kähler-Einstein metric to the Teichmüller metric, and hence gave a positive answer to a conjecture proposed by Yau. Their approach in the proof is to introduce two new complete Kähler metrics, namely, the Ricci metric and the perturbed Ricci metric, and then establish the equivalence of the Ricci metric to the Kähler-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 $T_g$ and $\mathcal{M}_g$ induced by these metrics.
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