by Ho Chor Yin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 100-103). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.4 / Chapter 2 --- CR Manifolds and ab Complex --- p.8 / Chapter 2.1 --- Almost Complex Structures --- p.8 / Chapter 2.2 --- CR Structures --- p.10 / Chapter 2.3 --- The Tangential Cauchy-Riemann Complex (ab Com- Plex) --- p.12 / Chapter 3 --- Subelliptic Estimates for □b --- p.18 / Chapter 3.1 --- Preliminaries --- p.18 / Chapter 3.2 --- Subelliptic Estimates for the Tangential Caucliy-R.iemann Complex --- p.34 / Chapter 3.3 --- Local Regularity and the Hodge Theorem for □b --- p.44 / Chapter 4 --- Embeddability of CR manifolds --- p.60 / Chapter 4.1 --- CR Embedding and Embeddability of Real Analytic CR Manifold --- p.60 / Chapter 4.2 --- Boutet de Monvel's Global CR Embedding Theorem --- p.62 / Chapter 4.3 --- Rossi's Globally Nonembeddable CR Manifold --- p.69 / Chapter 4.4 --- Nirenberg's Locally Nonembeddable CR Manifold --- p.72 / Chapter 5 --- Geometry of Strongly Pseudoconvex CR Manifolds --- p.79 / Chapter 5.1 --- Equivalence Problem and Pseudoconformal Geometry --- p.79 / Chapter 5.2 --- Pseudo-hermitian Geometry --- p.82 / Chapter 5.3 --- A Geometric Approach to the Hodge Theorem for □b --- p.85 / Bibliography --- p.100

We study mirror symmetry via Fourier-Mukai-type transformations, which we call SYZ mirror transformations, in view of the ground-breaking Strominger-Yau-Zaslow Mirror Conjecture which asserted that the mirror symmetry for Calabi-Yau manifolds could be understood geometrically as a T-duality modified by suitable quantum corrections. We apply these transformations to investigate a case of mirror symmetry with quantum corrections, namely the mirror symmetry between the A-model of a toric Fano manifold X¯ and the B-model of a Landau-Ginzburg model (Y, W). Here Y is a noncompact Kahler manifold and W : Y → C is a holomorphic function. We construct an explicit SYZ mirror transformation which realizes canonically the isomorphism QH*X&d1; ≅Ja cW between the quantum cohomology ring of X¯ and the Jacobian ring of the function W. We also show that the symplectic structure oX¯ of X¯ is transformed to the holomorphic volume form eWOY of ( Y, W). Concerning the Homological Mirror Symmetry Conjecture, we exhibit certain correspondences between A-branes on X¯ and B-branes on (Y, W) by applying the SYZ philosophy. / Chan, Kwok Wai. / Adviser: Nai Chung Conan Leung. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3536. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 52-56). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Ip Tsz Ho. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 86-89). / Chapter 0 --- Introduction --- p.5 / Chapter 1 --- Preliminary --- p.11 / Chapter 1.1 --- Background in analysis --- p.11 / Chapter 1.1.1 --- Holder Continuity --- p.11 / Chapter 1.1.2 --- Hausdorff Measure --- p.12 / Chapter 1.1.3 --- Weak Derivatives --- p.13 / Chapter 1.2 --- Basic Facts of Harmonic Functions --- p.14 / Chapter 1.2.1 --- Harmonic Approximation --- p.14 / Chapter 1.2.2 --- Elliptic Regularity --- p.15 / Chapter 1.3 --- Background in geometry --- p.16 / Chapter 1.3.1 --- Notations and Symbols --- p.16 / Chapter 1.3.2 --- Nearest Point Projection --- p.16 / Chapter 2 --- Monotonicity formula and Regularity of Harmonic maps --- p.17 / Chapter 2.1 --- Energy Minimizing Maps --- p.17 / Chapter 2.2 --- Variational Equations --- p.18 / Chapter 2.3 --- Monotonicity Formula --- p.21 / Chapter 2.4 --- A Technical Lemma --- p.22 / Chapter 2.5 --- Luckhau's Lemma --- p.28 / Chapter 2.6 --- Reverse Poincare Inequality --- p.40 / Chapter 2.7 --- ε-Regularity of Energy Minimizing Maps --- p.45 / Chapter 3 --- Monotonicity Formulae and Vanishing Theorems --- p.52 / Chapter 3.1 --- Stress energy tensor and basic formulae for harmonic p´ؤforms --- p.52 / Chapter 3.2 --- Monotonicity formula --- p.59 / Chapter 3.2.1 --- Monotonicity Formula for Harmonic Maps --- p.64 / Chapter 3.2.2 --- Bochner-Weitzenbock Formula --- p.65 / Chapter 3.3 --- Conservation Law and Vanishing Theorem --- p.68 / Chapter 4 --- On conformally compact Einstein Manifolds --- p.71 / Chapter 4.1 --- Energy Decay of Harmonic Maps with Finite Total Energy --- p.73 / Chapter 4.2 --- Vanishing Theorem of Harmonic Maps --- p.81 / Bibliography --- p.86

Ma, Chit. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 61-63). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Sasaki Geometry --- p.8 / Chapter 2.1 --- Sasakian manifolds --- p.8 / Chapter 2.2 --- Transversal Kahler geometry --- p.11 / Chapter 2.3 --- Sasaki-Futaki invariant --- p.16 / Chapter 2.3.1 --- Space of Kahler cone metric --- p.17 / Chapter 2.3.2 --- Futaki invariant --- p.20 / Chapter 2.3.3 --- A formula for volume variation --- p.27 / Chapter 3 --- Toric Geometry --- p.30 / Chapter 3.1 --- Toric Kahler geometry --- p.30 / Chapter 3.2 --- Toric Sasakian manifold --- p.42 / Chapter 3.3 --- Ricci flat metric in toric Kahler-Sasaki cone --- p.47 / Chapter 4 --- Chern numbers inequality --- p.55

Thesis (M. Phil.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

Thesis (Ph. D.)--University of Washington, 2001. / Vita. Includes bibliographical references (p. 120-124).

Thesis (M. Phil.)--University of Hong Kong, 2010. / Includes bibliographical references (leaves 68-70). Also available in print.

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1's) representing the points of the primary space fails to be complete; but it can be completed to give a 4- dimensional family, effecting a unique embedding of the original 3-fold in a 4-fold with conformal structure, of which the conformal curvature is selfdual, in such a way that the induced conformal structure is the original one and such that the conformal torsion is related to the second conformal fundamental form of the hypersurface in a canonical linear fashion. In any case, the small deformations of the complex structure of the space of null geodesies correspond precisely to the small deformations of the conformal connexion. It is shown that a space of torsion-free null geodesies admits a holomorphic contact structure, and that conversely, for n ≥ 4, the admission of a contact structure forces the conformal torsion to vanish; for n=3, the contact form constructs automatically a unique metric on the ambient 4-fold in the previously constructed self-dual conformal class which solves Einstein's equations with cosmological constant 1 and blov/s up on the 3-fold, which is a general umbilic hypersurface. These results are in turn used to show that a real-analytic 3-fold with real-analytic positive definite conformal structure and a real-analytic symmetric form of conformal weight 1 can be embedded (in a locally unique fashion) in a real-analytic 4-fold with positive-definite conformal structure for which the conformal curvature is self-dual in such a way as to realize the given structures as the first and second conformal fundamental forms of the hypersurface; and it is shown that a real analytic 3-fold with positivedefinite conformal bounds a locally unique positive-definite solution of Einstein's equations with cosmological constant -1 as its umbilic conformal infinity. By contrast, these results fail when "real-analytic" is replaced by "smooth".

Thesis (Ph. D.)--University of Washington, 1997. / Vita. Includes bibliographical references (leaves [114]-116).