O presente trabalho tem por objeto fazer uma introdução ao estudo das subvariedades isoparamétricas do espaço Euclidiano. Começamos com uma introdução ao desenvolvimento histórico desses objetos. A seguir apresentamos os conceitos básicos da teoria de subvariedades de formas espaciais. Deduzimos as equações fundamentais de primeira e segunda ordem e demonstramos o teorema fundamental da teoria de subvariedades. Em seguida damos a definição de subvariedade isoparamétrica e desenvolvemos conceitos elementares para o caso do espaço Euclidiano como são normais de curvatura, grupo de Coxeter, câmera de Weyl e variedades paralelas e focais. Provamos dois teoremas referentes à decomposição de subvariedades isoparamétricas do espaço Euclidiano adaptando ferramentas usadas em [HL97] para ocaso de subvariedades isoparamétricas de espaços de Hilbert. Demonstramos o teorema da fatia e discutimos sobre subvariedades isoparamétricas desde o ponto de vista clássico, a saber, aplicações isoparamétricas. Concluímos com alguns exemplos: hipersuperfécies isoparamétricas da esfera e órbitas principais da ação adjunta de um grupo de Lie sobre a respectiva álgebra de Lie. / The goal of this dissertation is to present an introduction to the study of isoparametric submanifolds of Euclidean space. We begin with an introduction to the history of the subject. Then we present the basic results of submanifold theory of space forms. We compute the fundamental equations of first and second order, and we prove the fundamental theorem of submanifold theory. Next, we define isoparametric submanifolds and discuss some basic constructions, as curvature normals, Coxeter groups, Weyl chambers and parallel and focal submanifolds. We prove two decomposition theorems about isoprametric submanifolds using techniques that we learnt from [HL97], paper in which the case of submanifolds of Hilbert spaces is studied. Then we prove slice theorem. We also discuss those submanifold from the classical point of view, namely, isoparametric maps. We finish by explaining some examples: isoparametric hipersurfaces of spheres and principal orbits of the adjoint action of a Lie group on its Lie algebra.

We study a natural Dirac operator on a Lagrangian submanifold of a Kähler manifold. We first show that its square coincides with the Hodge - de Rham Laplacian provided the complex structure identifies the Spin structures of the tangent and normal bundles of the submanifold. We then give extrinsic estimates for the eigenvalues of that operator and discuss some examples.

This thesis consists of a summary of two papers dealing with questions related to Legendrian submanifolds of contact manifolds together with exact Lagrangian cobordisms between Legendrian submanifolds. The focus is on studying Legendrian submanifolds from the perspective of their handle decompositions. The techniques used are mainly from Symplectic Field Theory. In Paper I, a series of examples of Legendrian surfaces in standard contact 5-space are studied. For every g > 0, we produce g+1 Legendrian surfaces of genus g, all with g+1 transverse Reeb chords, which lie in distinct Legendrian isotopy classes. For each g, exactly one of the constructed surfaces has a Legendrian contact homology algebra admitting an augmentation. Moreover, it is shown that the same surface is the only one admitting a generating family. Legendrian contact homology with Novikov coefficients is used to classify the different Legendrian surfaces. In particular, we study their augmentation varieties. In Paper II, the effect of a Legendrian ambient surgery on a Legendrian submanifold is studied. Given a Legendrian submanifold together which certain extra data, a Legendrian ambient surgery produces a Legendrian embedding of the manifold obtained by surgery on the original submanifold. The construction also provides an exact Lagrangian handle-attachment cobordism between the two submanifolds. The Legendrian contact homology of the submanifold produced by the Legendrian ambient surgery is then computed in terms of pseudo-holomorphic disks determined by data on the original submanifold. Also, the cobordism map induced by the exact Lagrangian handle attachment is computed. As a consequence, it is shown that a sub-critical standard Lagrangian handle attachment cobordism induces a one-to-one correspondence between the augmentations of the Legendrian contact homology algebras of its two ends.

Let M be smooth symplectic closed manifold and L a closed Lagrangian submanifold of M. It was shown by Ozan that Ham(M,L): the relative Hamiltonian diffeomorphisms on M fixing the Lagrangian submanifold L setwise is a subgroup which is equal to the kernel of the restriction of the flux homomorphism to the universal cover of the identity component of the relative symplectomorphisms. In this thesis we show that Ham(M,L) is a non-simple perfect group, by adopting a technique due to Thurston, Herman, and Banyaga. This technique requires the diffeomorphism group be transitive where this property fails to exist in our case.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / O problema de Bernstein clÃssico, resolvido por S. Bernstein em 1915-1917 em seu artigo [12], pergunta se existe um grÃfico mÃnimo completo em R3 alÃm do plano. Bernstein mostrou que a resposta para este problema à nÃo, utilizando mÃtodos analÃticos para o estudo de equaÃÃes de curvatura prescrita. Veremos aqui como este problema està relacionado com a aplicaÃÃo de Gauss deste grÃfico, e como conseqÃÃncia desta relaÃÃo iremos generalizar este teorema para uma classe de superfÃcies maior (nÃo necessariamente grÃficos), seguindo a prova dada por R. Osserman em [51]. Veremos a seguir generalizaÃÃes deste teorema em dimensÃes maiores, seguindo essencialmente os mÃtodos introduzidos Por W. Fleming em [31], e refinados posteriormente por E. De Giorgi, em [20], F. Almgren, em [6], e J. Simons, em [62], que resolvem o problema para grÃficos em Rn, n < 9 mostrando que o Ãnico grÃfico mÃnimo completo nesses espaÃos à o hiperplano. Mostraremos tambÃm que em dimensÃo n &#8805; 9, à possÃvel construir grÃficos mÃnimos completos em Rn, seguindo a prova apresentada por E. Bombieri, E. Di Giorgi e E. Giusti em [14]. Por fim, concluÃmos com uma extensÃo do teorema de Bernstein para a classe das subvariedades estÃveis com respeito à segunda variaÃÃo de volume, sob certas condiÃÃes de crescimento de curvatura ou volume, e investigaremos ainda o caso que a variedade ambiente nÃo à o espaÃo euclidiano. / The classical Bernstein problem, solved by S. Bernstein in 1915-1917 in his article [12], asks if there is a complete minimal graph in R3 besides the plane. Bernstein showed that the answer to this question is no using analytical methods for study of equations of prescribed curvature. We will see here how this problem is related to the Gauss map of the graph, and as consequence of this relationship we generalize this theorem to a larger class of surfaces (not necessarily graphs), following the proof given by R. Osserman in [51]. We will see next generalizations of this theorem in higher dimensions, following essentially the methods introduced by W. Fleming in [31], and later refined by E. De Giorgi in [20], F. Almgren in [6] and J. Simons in [62]. In fact, they solve the problem for graphs in Rn, n < 9, namely they prove that the only complete minimal graph in these espaces is the hyperplane. Following the proof given by E. Bombieri, E. De Giorgi and E. Giusti in [14], we also show that, in dimension n &#8805; 9, it is possible to construct complete minimal graphs in Rn. At last, we conclude with an extension of Bernsteinâs theorem to the class of submanifolds stable with respect to the second variation of volume, under certain conditions of curvature and volume growth, and yet we investigate the case in which the ambient manifold is not the Euclidean space.

Neste trabalho estudamos alguns resultados importantes sobre a teoria das subvariedades bi-harmônicas de espaços homogêneos tridimensionais. Existem três classes de espaços homogêneos tridimensionais simplesmente conexos dependendo da dimensão do grupo de isometrias, que pode ser: 3, 4 ou 6. No caso da dimensão ser 6, M é uma forma espacial; se a dimensão do grupo de isometrias for 4, M é isométrica a: \'H IND. 3\' (grupo de Heisenberg), SU(2) (grupo unitário especial), ~SL(2,R) (revestimento universal do grupo linear especial), ou aos espaços produtos \'S POT. 2\' × R e \'H POT. 2\' × R. Feita exceção para \'H POT. 3\', no caso da dimensão ser 4 ou 6 o espaço homogêneo é localmente isométrico a (uma parte de) \'R POT. 3\', munido de uma métrica que depende de dois parâmetros reais. Tal família de métricas aparece primeiramente no trabalho [3] de L. Bianchi e, mais tarde, nos artigos [14, 35] de É. Cartan e G. Vranceanu, respectivamente. Nesse projeto de mestrado, queremos estudar (essencialmente) resultados de existência e classificação de subvariedades bi-harmônicas nesses espaços, também conhecidos como variedades de Bianchi-Cartan-Vranceanu / In this work we study some important results about the theory of the biharmonic submanifolds of tridimensional homogeneous spaces. There exist three classes of simply connected tridimensional homogeneous spaces depending on the dimension of the group of isometries, which can be: 3, 4 or 6. In the case of dimension 6, M will be a space form; if the dimension of the group of isometries is 4, M will be isometric to: either \'H IND. 3\' (Heisenbergs group), or SU(2) (special unitary group), or ~SL(2,R) (universal recovering of the special linear group), or the product spaces \'S POT. 2\' × R and \'H POT. 2\' × R. Except for \'H POT. 3\', in the case of dimension 4 or 6 the homogeneous space is locally isometric to (a part of) \'R POT. 3\', endowed with a metric that depends on two real parameters. Such family of metrics first appears in the work [3] of L. Bianchi and later in the articles [14, 35] of ´E. Cartan and G. Vranceanu, respectively. In this master thesis, we want to study (essentially) results of existence and classification of bi-harmonic submanifolds in these spaces, also known as Bianchi-Cartan-Vranceanus manifolds

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18444号 / 理博第4004号 / 新制||理||1577(附属図書館) / 31322 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 加藤 毅, 教授 堤 誉志雄, 教授 小野 薫 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Die hier dargelegte Dissertation ist motiviert durch die Vorschläge von Joyce, Doan und Walpuski zur Definitionen enumerativer Invarianten für G2-Mannigfaltigkeit, durch das Zählen gewisser kalibrierter Untermannigfaltigkeiten, sogenannter assoziativen Untermannigfaltigkeiten. In Kapitel 1, werde ich Definitionen und grundlegende Fakten über G2-Mannigfaltigkeit und deren assoziative Untermannigfaltigkeit wiederholen. Darüber hinaus erläutere ich die Konstruktion von G2-Mannigfaltigkeit als verdrehte verbundener Summe. Kapitel 2 schafft die nötige Grundlage für das darauf folgende dritte Kapitel. Hier definiere ich den Modul-Raum der asymptotisch zylindrischen assoziativen Untermannigfaltigkeiten zusammen mit seiner natürlichen Topologie und zeige, dass der Modul-Raum lokal homeomorph zur Urbild-Menge der Null einer glatten Abbildung zwischen zwei endlich-dimensionalen Räu- men ist. In besonderen Fällen ist dieser Modul-Raum eine Lagrangesche Untermannigfaltigkeit des Modul Raums der holomorphen Kurven einer asymptotisch zylindrischen Calabi-Yau Man- nigfaltigkeit. In Kapitel 3 beweise ich ein Klebe-Theorem für ein Paar von asymptotisch zylindrischen as- soziativen Untermannigfaltigkeiten in einem zusammenpassenden Paar von asymptotisch zylin- drischen G2-Mannigfaltigkeiten. Hiermit konstruiere ich neue geschlossene und starre (rigid) assoziative Untermannigfaltigkeiten in verdrehten verbundenen Summe G2-Mannigfaltigkeiten. In Kapitel 4 untersuche ich den Modul-Raum der konisch singulären assoziativen Un- termannigfaltigkeiten in G2-Mannigfaltigkeiten. Durch das Umformulieren des Indexes des Operators, der die Deformationstheorie kontrolliert, in bestimmte Stabilität-Indizes des zu- grundeliegenden assoziativen Kegels begründe ich, dass in einem generischen Pfad in dem Raum der ko-geschlossenen G2-Strukturen keine asymptotisch konischen assoziative Unter- mannigfaltigkeiten existieren, die mindestens eine Singularität besitzen, die auf einem Kegel mit Stabiltätsindex größer als eins modeliert werden. Dieses Resultat lässt sich auf alle speziellen Lagrangesche-Kegel außer den Harvey-Lawson-T2-Kegel und die Vereinigung zweier speziellen Lagrangesche-Flächen anwenden. Zusätzlich lässt sich das Ergebnis auch auf alle konischen assoziativen Untermannigfaltigkeiten anwenden, deren zugrundeliegende Verschlingung (link) holomorphe Kurven mit Null-Torsion in S6 sind. Des Weiteren dienen Teile des vierten Kapitels als Grundlage für das darauf folgende Kapitel 5. Aufgrund einiger Übergangsphänomene entlang eines generischen Pfades von G2-Strukturen, führt das naive Zählen von assoziativen Untermannigfaltigkeiten zu keiner Invariante. Tat- sächlich wurde vermutet, dass a) eine assoziative Untermannigfaltigkeit aus einer assoziativen Untermannigfaltigkeit mit Selbstsschnitt (self-intersection) geboren werden kann, und, dass b) drei assoziative Untermannigfaltigkeiten aus einer konisch singulären assoziativen Un- termannigfaltigkeit, deren Singularität durch den Harvey-Lawson-T2-Kegel modelliert wird, entspringen. In Kapitel 5, beweise ich ein Desingularitätstheorem für konisch singulären assoziative Untermannigfaltigkeit entlang eines Pfades von ko-geschlossenen G2-Strukturen. Somit verifiziere ich Vermutung b) bewiesen und teilweise auch Vermutung a). / The dissertation presented here is motivated from the proposals made by Joyce, Doan and Walpuski to define enumerative invariants of G2-manifolds by counting certain calibrated submanifolds, called associative submanifolds. In Chapter 1, I review the definitions and basic facts of G2-manifolds and associative submanifolds. Moreover, I explain the construction of G2-manifolds as twisted connected sums. Chapter 2 serves as a necessary groundwork for Chapter 3. Here, I define the moduli space of asymptotically cylindrical associative submanifolds with its natural topology and prove that the moduli space is locally homeomorphic to the zero set of a smooth map between two finite-dimensional spaces. In the best scenario, this moduli space is a Lagrangian submanifold of the moduli space of holomorphic curves in the asymptotic Calabi-Yau 3-fold. In Chapter 3, I prove a gluing theorem for a pair of asymptotically cylindrical associative submanifolds in a matching pair of asymptotically cylindrical G2-manifolds. Using this I construct new closed and rigid associative submanifolds of twisted connected sum G2-manifolds. In Chapter 4, I study the moduli space of conically singular associative submanifolds in G2-manifolds. By reformulating the index of the operator that controls the deformation theory in terms of certain stability-index of the associative cones, I establish that in a generic path of co-closed G2-structures there are no conically singular associative submanifolds that have at least one singularity modeled on a cone of stability-index greater than one. This result applies to all special Lagrangian cones, except the Harvey-Lawson T2-cone and a union of two special Lagrangian planes. Additionally, it applies to all associative cones whose links are null-torsion holomorphic curves in S6. Furthermore, parts of Chapter 4 also serve as a necessary groundwork for Chapter 5. The naive counting of associative submanifolds does not lead to an invariant due to several transitions that may occur along a generic path of G2-structures. In fact it was conjectured that a) an associative submanifold born out of an associative submanifold with self intersection, and b) three associative submanifolds arise from a conically singular associative submanifold whose singularity is modeled on Harvey-Lawson T2-cone. In Chapter 5, I prove a desingularization theorem for conically singular associative submanifolds along a path of co-closed G2-structures. Consequently, I verify conjecture b) and partially confirm conjecture a).

This thesis is devoted to the study of the effect of Legendrian surgery on contact manifolds. In particular, we study the effect of this surgery on the Reeb dynamics of the contact manifold on which we perform such a surgery along Legendrian links. We obtain an exact sequence of cyclic Legendrian homology for the Legendrian links. Then we present the applications in 3-dimension and higher dimensions. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

This thesis consists of three scientific papers dealing with invariants of Legendrian and Lagrangian submanifolds. Besides the scientific papers, the thesis contains an introduction to contact and symplectic geometry, and a brief outline of Symplectic field theory with focus on Legendrian contact homology. In Paper I we give an orientation scheme for moduli spaces of rigid flow trees in Legendrian contact homology. The flow trees can be seen as the adiabatic limit of sequences of punctured pseudo-holomorphic disks with boundary on the Lagrangian projection of the Legendrian. So to equip the trees with orientations corresponds to orienting the determinant line bundle of the dbar-operator over the space of Lagrangian boundary conditions on the punctured disk. We define an  orientation of this line bundle and prove that it is well-defined in the limit. We also prove that the chosen orientation scheme gives rise to a combinatorial algorithm for computing the orientation of the trees, and we give an explicit description of this algorithm. In Paper II we study exact Lagrangian cobordisms with cylindrical Legendrian ends, induced by Legendrian isotopies. We prove that the combinatorially defined DGA-morphisms used to prove invariance of Legendrian contact homology for Legendrian knots over the integers can be derived analytically.  This is proved using the orientation scheme from Paper I together with a count of abstractly perturbed flow trees  of the Lagrangian cobordisms. In Paper III we prove a flexibility result for closed, immersed Lagrangian submanifolds in the standard symplectic plane.