by Li Cheung Shing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1991. / Bibliography: leaves 111-115. / Introduction --- p.1 / Chapter Chapter 1 --- "Real hypersurfaces,CR manifolds and the imbedding problem" --- p.5 / Chapter § 1.1 --- Non-equivalence of real analytic hypersurfaces in C2 --- p.5 / Chapter § 1.2 --- The Lewy operator --- p.8 / Chapter § 1.3 --- CR manifolds --- p.19 / Chapter § 1.4 --- Imbedding of CR manifolds --- p.24 / Chapter Chapter 2 --- Geometry of the real hyperquadric --- p.30 / Chapter § 2.1 --- The real hyperquadric --- p.30 / Chapter § 2.2 --- Q-frames --- p.31 / Chapter § 2.3 --- Maurer Cartan forms --- p.33 / Chapter § 2.4 --- Structural equations and chains --- p.36 / Chapter Chapter 3 --- Moser normal form --- p.40 / Chapter § 3.1 --- Formal theory of the normal form --- p.40 / Chapter § 3.2 --- Geometric theory of the normal form --- p.48 / Chapter Chapter 4 --- Cartan-Chern invariants and pseudohermitian geometry --- p.67 / Chapter §4.1 --- Cartan's solution of the equivalence problem --- p.67 / Chapter § 4.2 --- Chern's construction in higher dimensions --- p.69 / Chapter §4.3 --- Webster's invariants for pseudohermitian manifolds --- p.72 / Chapter § 4.4 --- Geometric interpretation of Webster's invariants --- p.76 / Chapter § 4.5 --- Applications --- p.80 / Chapter Chapter 5 --- Fefferman metric --- p.86 / Chapter § 5.1 --- Differential geometry on the boundary --- p.86 / Chapter § 5.2 --- Computations --- p.93 / Chapter §5.3 --- An example of spiral chains --- p.103 / References --- p.111

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.

In this thesis we unconditionally show that certain K3 surfaces satisfy the Hasse principle. Our method involves the 2-Selmer groups of simultaneous quadratic twists of two elliptic curves, only with places of good or additive reduction. More generally we prove that, given finitely many such elliptic curves defined over a number field (with rational 2-torsion and satisfying some mild conditions) there exists an explicit quadratic extension such that the quadratic twist of each elliptic curve has essential 2-Selmer rank one. Furthermore, given a 2-covering in each of the 2-Selmer groups, the quadratic extension above can be chosen so that the 2-Selmer group of the quadratic twist of each elliptic curve is generated by the given 2-covering and the image of the 2-torsion. Our approach to the Hasse Principle is outlined below and was introduced by Skorobogatov and Swinnerton-Dyer. We also generalize the result proved in their paper. If each elliptic curve has a distinct multiplicative place of bad reduction, then we find a quadratic extension such that the quadratic twist of each elliptic curve has essential 2-Selmer rank one. Furthermore, given a 2-covering in each of the 2-Selmer groups, the quadratic extension above can be chosen so that the 2-Selmer group of the quadratic twist of each elliptic curve is generated by the given 2-covering and the image of the 2-torsion. If we further assume the finiteness of the Shafarevich-Tate groups (of the twisted elliptic curves) then each elliptic curve has Mordell-Weil rank one. If K = Q, then under the above assumptions the analytic rank of each elliptic curves is one. Furthermore, with the assumption on the Shafarevich-Tate group (and K = Q), we describe a single quadratic twist such that each elliptic curve has analytic rank zero and Mordell-Weil rank zero, again under some mild assumptions.

This thesis examines orbifold versions of three results concerning the existence of canonical metrics in the Kahler setting. The first of these is Yau's solution to Calabi's conjecture, which demonstrates the existence of a Kahler metric with prescribed Ricci form on a compact Kahler manifold. The second is a variant of Yau's solution in a certain non-compact setting, namely, the setting in which the Kahler manifold is assumed to be asymptotic to a cone. The final result is one due to Uhlenbeck and Yau which asserts the existence of Kahler-Einstein metrics on stable vector bundles over compact Kahler manifolds.

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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

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

Cheung, Yat Ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 75-78). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Symplectic Geometry and Special Lagrangian Graphs in Cn --- p.10 / Chapter 2.1 --- Symplectic and Lagrangian Geometry of Cn --- p.10 / Chapter 2.2 --- Calibrated and Spccial Lagrangian Geometries in Cn --- p.13 / Chapter 2.3 --- Special Lagrangian Differential Equation --- p.16 / Chapter 3 --- Contact Geometry in S2n-1 --- p.20 / Chapter 3.1 --- Contact and Legendrian Geometries in S2n-1 --- p.20 / Chapter 3.2 --- Special Lagrangian Cone in R2n --- p.24 / Chapter 3.3 --- The Second Fundamental Form of Lagrangian Cone in E2n --- p.26 / Chapter 4 --- Geometry of Grassmannians --- p.29 / Chapter 4.1 --- Locally Symmetric Space --- p.29 / Chapter 4.2 --- "The Grassmann manifold G(n, m)" --- p.33 / Chapter 4.3 --- "Leichtweiss' Formula for Curvature Tensor in G(n, m)" --- p.36 / Chapter 4.4 --- "Normal Neighbourhoods of a Point in G(n, m)" --- p.39 / Chapter 4.5 --- Some Remarks on Lagrangian Grassmannians --- p.49 / Chapter 5 --- Harmonic Maps between Riemannian Manifolds --- p.51 / Chapter 5.1 --- Energy Functional and Tension Field --- p.52 / Chapter 5.2 --- Harmonic Map and Euler-Lagrange Equation --- p.56 / Chapter 5.3 --- The Gauss Map and its Tension Field --- p.59 / Chapter 5.4 --- Simple Riemannian Manifolds and A Liouville-Type Result of Har- monic Maps --- p.63 / Chapter 6 --- Bernstein-Type Results for Special Lagrangian Graphs --- p.65 / Chapter 6.1 --- Convexity and Bounded Slope Assumption --- p.65 / Chapter 6.2 --- Spherical Bernstein-Type Result --- p.68 / Chapter 6.3 --- Bernstein-Type Result with only Bounded Slope --- p.72 / Bibliography --- p.75

Ng, Kwong Wing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (p. 123-126). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgments --- p.iii / Contents --- p.v / Introduction --- p.vii / Chapter 1 --- Preliminaries --- p.1 / Chapter 1.1 --- Basic Notations and Conventions --- p.1 / Chapter 1.2 --- Fundamental Results in Banach Space Theory and Variational Analysis --- p.4 / Chapter 1.3 --- Set-Valued Mappings --- p.6 / Chapter 1.4 --- Enlargements and Projections --- p.8 / Chapter 1.5 --- Subdifferentials --- p.11 / Chapter 1.6 --- Sets of Normals --- p.18 / Chapter 1.7 --- Coderivatives --- p.24 / Chapter 2 --- The Generalized Distance Function - Basic Estimates --- p.27 / Chapter 2.1 --- Elementary Properties of the Generalized Distance Function --- p.27 / Chapter 2.2 --- Frechet-Like Subdifferentials of the Generalized Distance Function --- p.32 / Chapter 2.3 --- Limiting and Singular Subdifferentials of the Generalized Distance - Function --- p.44 / Chapter 3 --- The Generalized Distance Function - Estimates via Intermediate Points --- p.73 / Chapter 3.1 --- Frechet-Like and Limiting Subdifferentials of the Generalized Dis- tance Function via Intermediate Points --- p.74 / Chapter 3.2 --- Frechet and Proximal Subdifferentials of the Generalized Distance Function via Intermediate Points --- p.90 / Chapter 4 --- The Marginal Function --- p.95 / Chapter 4.1 --- Singular Subdifferentials of the Marginal Function --- p.95 / Chapter 4.2 --- Singular Subdifferentials of the Generalized Marginal Function . . --- p.102 / Chapter 5 --- The Perturbed Distance Function --- p.107 / Chapter 5.1 --- Elementary Properties of the Perturbed Distance Function --- p.107 / Chapter 5.2 --- The Convex Case - Subdifferentials of the Perturbed Distance Function --- p.111 / Chapter 5.3 --- The Nonconvex Case - Frechet-Like and Proximal Subdifferentials of the Perturbed Distance Function --- p.113 / Bibliography --- p.123