Survey on the Canonical Metrics on the Teichm\"{u}ller Spaces and the Moduli Spaces of Riemann Surfaces

TL;DR

This survey explores various canonical metrics on Teichmüller and moduli spaces of Riemann surfaces, analyzing their properties, equivalences, and implications for the geometry of these spaces.

Contribution

It provides a comprehensive overview of classical and recent canonical metrics, including new results on metric equivalences and their geometric significance.

Findings

The Teichmüller, Kobayashi, and Carathéodory metrics are effective in studying hyperbolic properties.

The McMullen metric is equivalent to the Teichmüller metric.

The Kähler-Einstein metric is proven to be equivalent to the Teichmüller metric.

Abstract

This thesis results from an intensive study on the canonical metrics on the Teichm\"{u}ller 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 Teichm\"{u}ller metric, the Kobayashi metric, the Bergman metric, the Carath\'{e}odory metric and the K\"{a}hler-Einstein metric. The Teichm\"{u}ller metric, the Kobayashi metric and the Carath\'{e}odory 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 K\"{a}hler metric, while the Bergman metric and the K\"{a}hler-Einstein metric are complete K\"{a}hler metrics. However, McMullen introduced a new complete K\"{a}hler metric, called the McMullen metric, by perturbing the Weil-Petersson metric. This metric is indeed equivalent to the Teichm\"{u}ller metric. Recently, Liu-Sun-Yau proved that the equivalence of the K\"{a}hler-Einstein metric to the Teichm\"{u}ller 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\"{a}hler metrics, namely, the Ricci metric and the perturbed Ricci metric, and then establish the equivalence of the Ricci metric to the K\"{a}hler-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.