Thesis (Ph. D.) - University of Washington. / Bibliography: ℓ.[184]-185.

Monopoles are solutions of an SU(2) gauge theory in R3 satisfying a lower bound for energy and certain asymptotic conditions, which translate as topological properties encoded in their charge. Using methods from integrable systems, monopoles can be described in algebraic-geometric terms via their spectral curve, i.e. an algebraic curve, given as a polynomial P in two complex variables, satisfying certain constraints. In this thesis we focus on the Ercolani-Sinha formulation, where the coefficients of P have to satisfy the Ercolani-Sinha constraints, given as relations amongst periods. In this thesis a particular class of such monopoles is studied, namely charge 3 monopoles with a symmetry by C3, the cyclic group of order 3. This class of cyclic 3-monopoles is described by the genus 4 spectral curve X , subject to the Ercolani-Sinha constraints: the aim of the present work is to establish the existence of such monopoles, which translates into solving the Ercolani-Sinha constraints for X . Exploiting the symmetry of the system,we manage to recast the problem entirely in terms of a genus 2 hyperelliptic curve X, the (unbranched) quotient of X by C3 . A crucial step to this aim involves finding a basis forH1( X; Z), with particular symmetry properties according to a theorem of Fay. This gives a simple formfor the period matrix of X ; moreover, results by Fay and Accola are used to reduce the Ercolani-Sinha constraints to hyperelliptic ones on X. We solve these constraints onX numerically, by iteration using the tetrahedral monopole solution as starting point in the moduli space. We use the Arithmetic-GeometricMean method to find the periods onX: this method iswell understood for a genus 2 curve with real branchpoints; in this work we propose an extension to the situation where the branchpoints appear in complex conjugate pairs, which is the case for X. We are hence able to establish the existence of a curve of solutions corresponding to cyclic 3-monopoles.

Wen, Chengfeng. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 77-83). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

Tse, Wang Hung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 75-77). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- On bounded interval with n jumps in inhibitor diffusivity --- p.3 / Chapter 2.1 --- Introduction --- p.3 / Chapter 2.2 --- Preliminaries --- p.5 / Chapter 2.3 --- Review of previous results in the two segment case: interior spike and spike near the jump discontinuity of the diffusion coefficient --- p.7 / Chapter 2.4 --- The construction and analysis of spiky steady-state solutions --- p.9 / Chapter 2.5 --- Stability Analysis --- p.10 / Chapter 2.6 --- Spikes near the jump discontinuity xb of the inhibitor diffusivity --- p.11 / Chapter 2.7 --- Stability Analysis II: Small Eigenvalues of the Spike near the Jump --- p.16 / Chapter 2.8 --- Existence of interior spikes for N segments --- p.20 / Chapter 2.9 --- Existence of a spike near a jump for N segments --- p.24 / Chapter 2.10 --- Appendix: The Green´ةs function for three segments --- p.25 / Chapter 3 --- On a compact Riemann surface without boundary --- p.30 / Chapter 3.1 --- Introduction --- p.30 / Chapter 3.2 --- Some Preliminaries --- p.35 / Chapter 3.3 --- Existence --- p.43 / Chapter 3.4 --- Refinement of Approximate Solution --- p.50 / Chapter 3.5 --- Stability --- p.52 / Chapter 3.6 --- Appendix I: Expansion of the Laplace-Beltrami Operator --- p.67 / Chapter 3.7 --- Appendix II: Some Technical Calculations --- p.73

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

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Univ., Diss.--Sendai.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, August 1999. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Texas at Austin, 1998. / Vita. Includes bibliographical references (leaves 77-80). Available also in a digital version from Dissertation Abstracts.