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

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 and , 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 . 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…
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · advanced mathematical theories · Advanced Algebra and Geometry
