Η παρούσα διπλωματική εργασία εντάσεται στο πεδίο μελέτης της Διαφορικής Γεωμετρίας και ειδικότερα την θεωρία υποπολλαπλοτήτων Ευκλείδειων και ψευδο-Ευκλείδειων χώρων. Η μελέτη μας εστιάζεται γύρω από μια εικασία που διατύπωσε ο B-Y. Chen. Η εικασία αναφέρει πως: <<Κάθε διαρμονική υποπολλαπλότητα στους Ευκλείδειους χώρους είναι ελάχιστης έκτασης.>> Η εικασία αν και φαίνεται απλή εν'τουτις μέχρι τις μέρες μας δεν έχει αποδειχθεί. Με την εργασία αυτή θα επιχειρήσουμε μια αναδρομή για το πως προέκυψε αυτή η εικασία και πια αποτελέσματατα έχουν παρουσιαστεί σε σχέση με αυτή μέχρι και σήμερα. / -

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

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

Or Ming-keung Ben. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 75-78). / Abstracts in English and Chinese. / Chapter 0 --- Introduction --- p.3 / Chapter 1 --- CR manifolds --- p.6 / Chapter 1.1 --- Introduction to CR manifolds --- p.6 / Chapter 1.2 --- CR functions --- p.11 / Chapter 1.3 --- CR maps and imbedding of CR manifolds --- p.15 / Chapter 1.4 --- Non-degenerate CR structures --- p.19 / Chapter 1.5 --- CR structures by means of differential forms --- p.21 / Chapter 2 --- Segre Family --- p.25 / Chapter 2.1 --- The Segre family associated to a real analytic hyper- surface --- p.25 / Chapter 2.2 --- G-structures on Segre family --- p.30 / Chapter 2.3 --- Local Computations --- p.37 / Chapter 3 --- Projective Structure --- p.41 / Chapter 3.1 --- Construction of the frame bundle over Segre family 。 --- p.41 / Chapter 3.2 --- The associated Cartan Connection --- p.45 / Chapter 3.3 --- Formulation in terms of Projective Connection --- p.54 / Chapter 4 --- Riemann Mapping Theorem --- p.57 / Chapter 4.1 --- Preliminary --- p.57 / Chapter 4.2 --- Generalizations of Poincare's theorem --- p.59 / Chapter 4.3 --- Local G-stucture on the space of hyperplane elements --- p.62 / Chapter 4.4 --- Extension of induced G-structure --- p.66 / Chapter 4.5 --- Proof of Theorem B --- p.70 / Chapter 4.6 --- Domains with continuous boundary --- p.72 / Bibliography --- p.75

Chao, Khek Lun Harold. / On t.p. "n" is superscript. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 67-68). / Abstracts in English and Chinese. / Chapter 1 --- Overview --- p.6 / Chapter 1.1 --- Introduction --- p.6 / Chapter 1.2 --- Domain of holomorphy --- p.7 / Chapter 1.3 --- Strongly pseudoconvex domains --- p.7 / Chapter 1.4 --- Geometry on the boundary --- p.10 / Chapter 1.5 --- Geometry in the interior --- p.12 / Chapter 1.6 --- Outline of the thesis --- p.13 / Chapter 2 --- Kahler-Einstein metric --- p.14 / Chapter 2.1 --- Problem --- p.14 / Chapter 2.2 --- Analysis of the domain --- p.15 / Chapter 2.3 --- Proof of openness --- p.23 / Chapter 2.4 --- Proof of closedness --- p.25 / Chapter 2.5 --- Uniqueness of solution --- p.33 / Chapter 2.6 --- Boundary behavior of the metric --- p.36 / Chapter 3 --- Boundary geometry of pseudo convex domains --- p.45 / Chapter 3.1 --- Background --- p.45 / Chapter 3.2 --- Monge-Ampere equation --- p.46 / Chapter 3.3 --- Differential geometry on the boundary --- p.51 / Chapter 3.4 --- Explicit calculation of the metric --- p.54 / Chapter 3.5 --- An example of spiralling chains --- p.63 / Bibliography --- p.67

Nous décrivons un nouvelle famille d'exemples d'hypersurfaces de la sphère satisfaisant le cas d'égalité de la majoration extrinsèque de C. Bär de la plus petite valeur propre de l'opérateur de Dirac. / We describe a new family of examples of hypersurfaces in the sphere satisfying the limitingcase in C. Bär's extrinsic upper bound for the smallest eigenvalue of the Dirac operator.

Neste trabalho estudamos as subvariedades lagrangeanas mínimas e autossimilares do espaço paracomplexo Dn. Começamos definindo o conceito de variedade para-Kähler e, como exemplo, descrevemos o espaço projetivo paracomplexo. Em seguida, estudamos as subvariedades paracomplexas e lagrangeanas. Após mostrarmos que toda subvariedade paracomplexa não-degenerada é mínima, dedicamos a atenção ao estudo das subvariedades lagrangeanas, restringindo-nos ao ambiente Dn. Em particular, estudamos as lagrangeanas que são invariantes sob a ação canônica do grupo SO(n), e as superfícies de Castro-Chen. Em ambos os casos, analisamos a minimalidade e a autossimilaridade das mesmas. / In this work, we study minimal and self-similar Lagrangian submanifolds in the para-complex space Dn. Firstly, we define the concept of para-Kähler manifold and, to exemplify, we describe the para-complex projective space.Then, we study para-complex submanifolds and Lagrangian submanifolds. After proving that every non-degenerate para-complex submanifold is minimal, we pay attention to Lagrangian submanifolds, restricting us to the case of Dn. In particular, we study Lagrangian submanifolds which are invariant by the canonical SO(n)-action of Dn, and Castro-Chen\'s surfaces. In both cases, we analyse the minimality and self-similarity.

Neste trabalho estudamos as subvariedades lagrangeanas mínimas e autossimilares do espaço paracomplexo Dn. Começamos definindo o conceito de variedade para-Kähler e, como exemplo, descrevemos o espaço projetivo paracomplexo. Em seguida, estudamos as subvariedades paracomplexas e lagrangeanas. Após mostrarmos que toda subvariedade paracomplexa não-degenerada é mínima, dedicamos a atenção ao estudo das subvariedades lagrangeanas, restringindo-nos ao ambiente Dn. Em particular, estudamos as lagrangeanas que são invariantes sob a ação canônica do grupo SO(n), e as superfícies de Castro-Chen. Em ambos os casos, analisamos a minimalidade e a autossimilaridade das mesmas. / In this work, we study minimal and self-similar Lagrangian submanifolds in the para-complex space Dn. Firstly, we define the concept of para-Kähler manifold and, to exemplify, we describe the para-complex projective space.Then, we study para-complex submanifolds and Lagrangian submanifolds. After proving that every non-degenerate para-complex submanifold is minimal, we pay attention to Lagrangian submanifolds, restricting us to the case of Dn. In particular, we study Lagrangian submanifolds which are invariant by the canonical SO(n)-action of Dn, and Castro-Chen\'s surfaces. In both cases, we analyse the minimality and self-similarity.

In this thesis we present a slight generalisation of the Pearl complex or relative quantum homology to some non monotone Lagrangian submanifolds. First we develop the theory for the so called almost monotone Lagrangian submanifolds, We apply it to uniruling problems as well as estimates for the relative Gromov width. In the second part we develop the theory for toric fiber in toric Fano manifolds, recovering previous computaional results of Floer homology .

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