Survey on the canonical metrics on the Teichmüller spaces and the moduli spaces of Riemann surfaces.

• Other Authors: Chan, Kin Wai.
• Format: Others
• Language: English; Chinese
• Published: 2010
• Subjects: Riemann surfaces; Teichmüller spaces; Moduli theory
• Online Access: http://library.cuhk.edu.hk/record=b5896649; http://repository.lib.cuhk.edu.hk/en/item/cuhk-327174

Chan, Kin Wai. === "September 2010." === Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. === Includes bibliographical references (leaves 103-106). === Abstracts in English and Chinese. === Chapter 1 --- Introduction --- p.8 === Chapter 2 --- Background Knowledge --- p.13 === Chapter 2.1 --- Results from Riemann Surface Theory and Quasicon- formal Mappings --- p.13 === Chapter 2.1.1 --- Riemann Surfaces and the Uniformization The- orem --- p.13 === Chapter 2.1.2 --- Fuchsian Groups --- p.15 === Chapter 2.1.3 --- Quasiconformal Mappings and the Beltrami Equation --- p.17 === Chapter 2.1.4 --- Holomorphic Quadratic Differentials --- p.20 === Chapter 2.1.5 --- Nodal Riemann Surfaces --- p.21 === Chapter 2.2 --- Teichmuller Theory --- p.24 === Chapter 2.2.1 --- Teichmiiller Spaces --- p.24 === Chapter 2.2.2 --- Teichmuller's Distance --- p.26 === Chapter 2.2.3 --- The Bers Embedding --- p.26 === Chapter 2.2.4 --- Teichmuller Modular Groups and Moduli Spaces of Riemann Surfaces --- p.27 === Chapter 2.2.5 --- Infinitesimal Theory of Teichmiiller Spaces --- p.28 === Chapter 2.2.6 --- Boundary of Moduli Spaces of Riemann Sur- faces --- p.29 === Chapter 2.3 --- Schwarz-Yau Lemma --- p.30 === Chapter 3 --- Classical Canonical Metrics on the Teichnmuller Spaces and the Moduli Spaces of Riemann Surfaces --- p.31 === Chapter 3.1 --- Finsler Metrics and Bergman Metric --- p.31 === Chapter 3.1.1 --- Definitions and Properties of the Metrics --- p.32 === Chapter 3.1.2 --- Equivalences of the Metrics --- p.33 === Chapter 3.2 --- Weil-Petersson Metric --- p.36 === Chapter 3.2.1 --- Definition and Properties of the Weil-Petersson Metric --- p.36 === Chapter 3.2.2 --- Results about Harmonic Lifts --- p.37 === Chapter 3.2.3 --- Curvature Formula for the Weil-Petersson Met- ric --- p.41 === Chapter 4 --- Kahler Metrics on the Teichmiiller Spaces and the Moduli Spaces of Riemann Surfaces --- p.42 === Chapter 4.1 --- McMullen Metric --- p.42 === Chapter 4.1.1 --- Definition of the McMullen Metric --- p.42 === Chapter 4.1.2 --- Properties of the McMullen Metric --- p.43 === Chapter 4.1.3 --- Equivalence of the McMullen Metric and the Teichmuller Metric --- p.45 === Chapter 4.2 --- Kahler-Einstein Metric --- p.50 === Chapter 4.2.1 --- Existence of the Kahler-Einstein Metric --- p.50 === Chapter 4.2.2 --- A Conjecture of Yau --- p.50 === Chapter 4.3 --- Ricci Metric --- p.51 === Chapter 4.3.1 --- Definition of the Ricci Metric --- p.51 === Chapter 4.3.2 --- Curvature Formula of the Ricci Metric --- p.53 === Chapter 4.4 --- The Asymptotic Behavior of the Ricci Metric --- p.61 === Chapter 4.4.1 --- Estimates on the Asymptotics of the Ricci Metric --- p.61 === Chapter 4.4.2 --- Estimates on the Curvature of the Ricci Metric --- p.83 === Chapter 4.5 --- Perturbed Ricci Metric --- p.92 === Chapter 4.5.1 --- Definition and the Curvature Formula of the Perturbed Ricci Metric --- p.92 === Chapter 4.5.2 --- Estimates on the Curvature of the Perturbed Ricci Metric --- p.93 === Chapter 4.5.3 --- Equivalence of the Perturbed Ricci Metric and the Ricci Metric --- p.96 === Chapter 5 --- Equivalence of the Kahler Metrics on the Teichmuller Spaces and the Moduli Spaces of Riemann Surfaces --- p.98 === Chapter 5.1 --- Equivalence of the Ricci Metric and the Kahler-Einstein Metric --- p.98 === Chapter 5.2 --- Equivalence of the Ricci Metric and the McMullen Metric --- p.99 === Bibliography --- p.103