Provamos resultados sobre a geometria de hipersuperfícies em diferentes espaços ambiente. Primeiro, definimos uma aplicação de Gauss generalizada para uma hipersuperfície Mn-1 c/ Nn, onde N é um espaço simétrico de dimensão n ≥ 3. Em particular, generalizamos um resultado de Ruh-Vilms e apresentamos aplicações. Em seguida, estudamos superfícies em espaços de dimensão 3: estudamos a equação da curvatura média em um produto semidireto R2oAR e obtemos estimativas da altura e a existência de gráficos mínimos do tipo Scherk. Finalmente, no espaço ambiente de uma variedade hiperbólica de dimensão 3: nós apresentamos condições suficientes para que um mergulho completo de uma superfície ∑ de topologia finita em N com curvatura média |H∑| ≤ 1 seja próprio. / We prove results concerning the geometry of hypersurfaces on di erent ambient spaces. First, we de ne a generalized Gauss map for a hypersurface Mn-1 c/ Nn, where N is a symmetric space of dimension n ≥ 3. In particular, we generalize a result due to Ruh-Vilms and make some applications. Then, we focus on surfaces on spaces of dimension 3: we study the mean curvature equation of a semidirect product R2 oA R to obtain height estimates and the existence of a Scherk-like minimal graph. Finally, on the ambient space of a hyperbolic manifold N of dimension 3 we give su cient conditions for a complete embedding of a nite topology surface ∑ on N with mean curvature |H∑| ≤ 1 to be proper.

Provamos resultados sobre a geometria de hipersuperfícies em diferentes espaços ambiente. Primeiro, definimos uma aplicação de Gauss generalizada para uma hipersuperfície Mn-1 c/ Nn, onde N é um espaço simétrico de dimensão n ≥ 3. Em particular, generalizamos um resultado de Ruh-Vilms e apresentamos aplicações. Em seguida, estudamos superfícies em espaços de dimensão 3: estudamos a equação da curvatura média em um produto semidireto R2oAR e obtemos estimativas da altura e a existência de gráficos mínimos do tipo Scherk. Finalmente, no espaço ambiente de uma variedade hiperbólica de dimensão 3: nós apresentamos condições suficientes para que um mergulho completo de uma superfície ∑ de topologia finita em N com curvatura média |H∑| ≤ 1 seja próprio. / We prove results concerning the geometry of hypersurfaces on di erent ambient spaces. First, we de ne a generalized Gauss map for a hypersurface Mn-1 c/ Nn, where N is a symmetric space of dimension n ≥ 3. In particular, we generalize a result due to Ruh-Vilms and make some applications. Then, we focus on surfaces on spaces of dimension 3: we study the mean curvature equation of a semidirect product R2 oA R to obtain height estimates and the existence of a Scherk-like minimal graph. Finally, on the ambient space of a hyperbolic manifold N of dimension 3 we give su cient conditions for a complete embedding of a nite topology surface ∑ on N with mean curvature |H∑| ≤ 1 to be proper.

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Cette thèse est dédiée à l'analyse mathématique de quelques problèmes variationnels motivés par le modèle de Ginzburg-Landau en théorie de la supraconductivité. Dans la première partie on étudie l'existence de solutions pour les équations de Ginzburg-Landau sans champ magnétique et avec données au bord de type semi-rigides. Ces données consistent à prescrire le module de la fonction sur le bord du domaine ainsi que son degré topologique. C'est un cas particulier de problèmes à bord libre, ou la donnée complète de la fonction sur le bord est une inconnue du problème. L'existence de solutions à ce problème n'est pas assurée. En effet la méthode directe du calcul des variations ne peut pas s'appliquer car le degré sur le bord n'est pas continu pour la convergence faible dans l'espace de Sobolev adapté. On dit que c'est un problème sans compacité. En étudiant le phénomène de "bubbling" qui apparaît dans l'étude de tels problèmes on donne des résultats d'existence et de non existence de solutions. Dans le Chapitre 1 on étudie des conditions qui permettent d'affirmer que la différence entre deux niveaux d'énergie est strictement optimale. Pour cela on adapte une technique due à Brezis-Coron. Ceci nous permet de redémontrer un résultat (précédemment obtenu par Berlaynd Rybalko et Dos Santos) d'existence de solutions stables pour les équations de Ginzburg-Landau dans des domaines multiplement connexes. Dans le Chapitre 2 on considère les applications harmoniques a valeurs dans $R^2$ avec des conditions au bord de type degrés prescrits sur un anneau. On fait un lien entre ce problème et la théorie des surfaces minimales dans $R^3$ grâce à la différentielle quadratique de Hopf. Ceci nous conduit à l'étude des surfaces minimales bordées par deux cercles dans des plans parallèles. On prouve l'existence de telles surfaces qui ne sont pas des catenoides grâce a un résultat de bifurcation. On utilise alors les résultats obtenus pour déduire des théorèmes d'existence et de non existence de minimiseurs de l'énergie de Ginzburg-Landau à degrés prescrits dans un anneau. Dans ce troisième Chapitre on obtient des résultats pour une valeur du paramètre " grand. Le Chapitre 4 a pour objet l'étude des problèmes a degrés prescrits en dimension n3. On y montre la non existence des minimiseurs de la n-énergie de Ginzburg-Landau a degrés prescrits dans un domaine simplement connexe. On étudie ensuite des points critiques de type min-max pour une énergie perturbée. La deuxième partie est consacrée a l'analyse asymptotique des solutions des équations deGinzburg-Landau lorsque " tend vers zero. Sandier et Serfaty ont étudié le comportement asymptotique des mesures de vorticité associées aux équations. Ils ont notamment trouvé des conditions critiques sur les mesures limites dans le cas des équations avec et sans champ magnétique. Nous nous intéressons alors à ces conditions critiques dans le cas sans champ magnétique. Le problème de la régularité locale des mesures limites se ramène ainsi a l'étude de la régularité des fonctions stationnaires harmoniques dont le Laplacien est une mesure. Nous montrons que localement de telles mesures sont supportées par une union de lignes appartenant à l'ensemble des zéros d'une fonction harmonique / This thesis is devoted to the mathematical analysis of some variational problems. These problem sare motivated by the Ginzburg-Landau model related to the super conductivity. In the first part we study existence of solutions of the Ginzburg-Landau equations without magnetic eld but with semi-sti boundary conditions. These conditions are obtained by prescribing the modulus of the function on the boundary of the domain along with its topological degree. This is a particular case of free boundary problems, where the function on the boundary is an unknown of the problem. Existence of solutions of that problem does not necessary hold. Indeed we can not apply the direct method of the calculus of variations since the degree on the boundaryis not continuous with respect to the weak convergence in an appropriated Sobolev space. This is problem with loss of compactness. By studying the bublling" phenomenon which come upin such problems we obtain some existence and non existence results .In Chapter 1 we study conditions under which the dierence between two energy levels is strictly optimal. In order to do that we adapt a technique due to Brezis-Coron. This allow us to recover known existence results (previously obtained by Berlyand and Rybalko and DosSantos) for stable solutions of the Ginzburg-Landau equations in multiply connected domains. In Chapter 2 we are interested in harmonic maps with values in $R^2$ with prescribed degree boundary condition in an annulus. We make a link between this problem and the minimal surface theory in $R^3$ thanks to the so-called Hopf quadratic differential. This leads us to study immersed minimal surfaces bounded by two circles in parallel planes. We prove the existence of such surfaces die rent from catenoids by using a bifurcation argument. We then apply the results obtained to deduce existence and non existence results for minimizers of the Ginzburg-Landau energy with prescribed degrees. This is done in Chapter 3 where the results are obtained for large ".Chapter 4 is devoted to prescribed degree problems in dimension n3 . We prove the non existence of minimizers of the Ginzburg-Landau energy in simply connected domains. We then study min-max critical points of a perturbed energy. The second part is devoted to the asymptotic analysis of solutions of the Ginzburg-Landau equations when "goes to zero. Sandier and Serfaty studied the asymptotic behavior of the vorticity measures associated to these equations. They derived critical conditions on the limiting measures both with and without magnetic Field. We are interested by these conditions when there is no magnetic Field. The problem of the local regularity of the limiting measures is then equivalent to the study of regularity of stationary harmonic functions whose Laplacianis a measure. We show that locally such measures are concentrated on a union of lines which belong to the zero set of an harmonic function

[pt] Neste trabalho, estudamos superfícies com curvatura média afim zero. Elas são chamadas de superfícies mínimas afins e para superfícies convexas, também são chamadas de superfícies máximas afins. Provamos que uma superfície mínima euclidiana também é uma superfície mínima afim se, e somente se, as linhas de curvatura da superfície mínima euclidiana conjugada são planas. Para uma superfície máxima afim, descrevemos como recuperá-la do campo de vetor conormal ao longo de uma determinada curva. Para algumas escolhas do vector conormal, a superfície máxima é singular e descrevemos as condições sob as quais as singularidades são arestas cuspidais ou swallowtails. / [en] In this work we study surfaces with zero affine mean curvature. They are called affine minimal surfaces and for convex surfaces, they are also called affine maximal surfaces. We prove that an euclidean minimal surface is also an affine minimal surface if and only if the curvature lines of the conjugate euclidean minimal surface are planar. For an affine maximal surface, we describe how to recover it from the conormal vector field along a given curve. For some choices of the conormal vector, the maximal surface is singular and we describe conditions under which the singularities are cuspidal edges or swallowtails.

Diese Arbeit wird durch die AdS/CFT Korrespondenz, sowie durch die Dualität zwischen lichtartigen, polygonalen Wilsonschleifen und Gluonenstreuamplituden in N=4 Super-Yang-Mills-Theorie motiviert. Bei starker Kopplung haben lichtartige, polygonale Wilsonschleifen und Gluonenstreuamplituden eine Beschreibung über raumartige Minimalflächen in AdS5. Wir benutzen eine Pohlmeyerreduktion, um eine Klassifikation aller raumartigen Minimalflächen in AdS3xS3 mit flachen Projektionen herzuleiten. Diese Klassifikation enthält neun verschiedene Klassen von Flächen. Dabei treten raumartige, zeitartige und degenerierte AdS3-Projektionen auf. Bei denjenigen Lösungen, die einen geschlossenen, polygonalen und lichtartigen Rand besitzen, berechnen wir den regularisierten Flächeninhalt. Bei schwacher Kopplung erfüllen lichtartige, polygonale Wilsonschleifen und Gluonenstreuamplituden den um eine Remainderfunktion korrigierten BDS-Ansatz. Wir präsentieren eine Technik, die auf einer Renormierungsgruppengleichung für selbstschneidende Wilsonschleifen beruht, mit der wir die Divergenzen der Remainderfunktion in diesem Limes berechnen können. Mittels dieser Technik analysieren wir zwei Arten des Selbstschnittes. Im Falle des Selbstschnittes zwischen zwei Ecken berechnen wir die führenden Divergenzen bis zur vierten Schleifenordnung. Beim Selbstschnitt zwischen zwei Kanten berechnen wir die führenden und nächstfolgenden Divergenzen bis zur vierten Schleifenordnung und präsentieren eine analytische Fortsetzung in die Region der Euklidischen Wilsonschleifen und sagen bestimmte Terme vorher, die in dem unbekannten analytischen Ausdruck für die Remainderfunktion enthalten sein müssen. / This thesis is motivated by the AdS/CFT correspondence and the duality between gluon scattering amplitudes and light-like polygonal Wilson loops in N=4 super Yang-Mills theory. At strong coupling light-like polygonal Wilson loops and gluon scattering amplitudes have a description in terms of space-like minimal surfaces in AdS5. We use a Pohlmeyer reduction to derive a classification of all space-like minimal surfaces in AdS3xS3 that have flat projections. The classification consists of nine different classes and contains space-like, time-like and degenerated AdS3 projections. For solutions that admit a closed light-like polygonal boundary we calculate the regularized area. At weak coupling light-like polygonal Wilson loops and gluon scattering amplitudes obey the BDS Ansatz corrected by a remainder function. We present a renormalisation group equation technique using self-crossing Wilson loops to extract the divergences of the remainder function in this limit. Using this technique we analyse two different types of self-crossing. We present the leading and sub-leading divergences up to four loops for a crossing between two edges and the leading divergences for a crossing between two vertices. For a crossing between two edges we present an analytic continuation to the euclidean regime to predict certain terms that have to occur in the unknown analytic expression of the remainder function.

Les problèmes inverses consistent à retrouver une donnée qui a été transformée ou perturbée. Ils nécessitent une régularisation puisque mal posés. En traitement d'images, la variation totale en tant qu'outil de régularisation a l'avantage de préserver les discontinuités tout en créant des zones lisses, résultats établis dans cette thèse dans un cadre continu et pour des énergies générales. En outre, nous proposons et étudions une variante de la variation totale. Nous établissons une formulation duale qui nous permet de démontrer que cette variante coïncide avec la variation totale sur des ensembles de périmètre fini. Ces dernières années les méthodes non-locales exploitant les auto-similarités dans les images ont connu un succès particulier. Nous adaptons cette approche au problème de complétion de spectre pour des problèmes inverses généraux. La dernière partie est consacrée aux aspects algorithmiques inhérents à l'optimisation des énergies convexes considérées. Nous étudions la convergence et la complexité d'une famille récente d'algorithmes dits Primal-Dual.

Made available in DSpace on 2014-07-29T16:02:18Z (GMT). No. of bitstreams: 1 Dissertacao Lidiane Mayumi Hieda.pdf: 465165 bytes, checksum: a5ce3ff47770899f6a4edcca3e40ed69 (MD5) Previous issue date: 2011-05-13 / This work was based on papers On Compact Minimal Surfaces with non-negative Gaussian Curvature in a Space of Constant Curvature: I and Minimal Surfaces with Constant Curvature in 4-dimensional Space Forms, by Katsuei Kenmotsu, consisting in the classification of minimal surfaces with constant Gaussian curvature K in a 4-dimensional space form without any global assumption. We will show that an isometric minimal immersion x: M2(K) → M4(c), where c is sectional curvature, is either totally geodesic, or locally Clifford Torus, or locally a Veronese surface. As a corollary, we have that there is not isometric minimal immersions with constant negative Gaussian curvature into unit sphere S4(1) even locally. / Este trabalho foi baseado nos artigos On CompactMinimal Surfaces with non-negative Gaussian Curvature in a Space of Constant Curvature: I e Minimal Surfaces with Constant Curvature in 4-dimensional Space Forms de Katsuei Kenmotsu que consistem em classificar superfícies mínimas com curvatura Gaussiana constante K nas formas espaciais 4-dimensionais, sem alguma hipótese global. Mostraremos que uma imersão isométrica mínima x : M2(K) → M4(c), onde c é a curvatura seccional, ou é totalmente geodésica, ou localmente um Toro de Clifford, ou localmente uma superfície de Veronese. Como corolário, temos que não existe uma imersão isométrica mínima com curvatura Gaussiana constante negativa numa esfera unitária S4(1) mesmo que localmente.

[pt] A Geometria Diferencial Discreta tem por objetivo desenvolver uma teoria discreta que respeite os aspectos fundamentais da teoria suave. Com isto em mente, são apresentados incialmente resultados da teoria suave da Geometria Afim que terão suas versões discretas tratadas a posteriori. O primeiro objetivo deste trabalho é construir uma estrutura afim discreta para as redes assintóticas definidas no espaço tridimensional, com métrica de Blaschke indefinida e parâmetros assintóticos. Com este intuito, são definidos um campo conormal, que satisfaz as equações de Lelieuvre e está associado a um parâmetro real, e um normal afim que define a forma cúbica da rede e torna a estrutura bem definida. Esta estrutura permite, por exemplo, o estudo das superfícies regradas, com ênfase nas esferas afins impróprias. Além disso, propõe-se uma definição para as singularidades no caso das esferas afins impróprias discretas a partir da construção centrocorda. Outro objetivo deste trabalho é propor uma definição para as superfícies afins discretas com curvatura afim média constante (CAMC), de forma que englobe as superfícies afins mínimas e as esferas afins. As superfícies afins mínimas discretas recebem uma caracterização geométrica bastante interessane e ligada diretamente às quádricas de Lie discretas. O trabalho se completa com o principal resultado, referente à versão discreta das superfícies de Cayley, esferas afins impróprias regradas caracterizadas a partir da conexão afim induzida: uma rede assintótica com CAMC é congruente equiafim à uma superfície de Cayley se, e somente se, a forma cúbica é não nula e a conexão afim induzida é paralela. / [en] Discrete Differential Geometry aims to develop a discrete theory which respects fundamental aspects of smooth theory. With this in mind, some results of smooth theory of Affine Geometry are firstly introduced since their discrete counterparts shall be treated a posteriori. The first goal of this work is construct a discrete affine structure for nets in a three-dimensional space with indefinite Blaschke metric and asymptotic parameters. For this purpose, one defines a conormal vector field, which satisfies Lelieuvre s equations and it is associated to a real parameter; and an affine normal vector field, which defines the cubic form of the net and makes the structure well defined. This structure allows to study, e.g., ruled surfaces with emphasis on improper affine spheres. Moreover, a definition for singularities is proposed in the case of discrete improper affine spheres from the center-chord construction. Another goal here is to propose a definition for an asymptotic net with constant affine mean curvature (CAMC), in a way that encompasses discrete affine minimal surfaces and discrete affine spheres. Discrete affine minimal surfaces receive a beautiful geometrical characterization directly linked to discrete Lie quadrics. This work is completed with the main result about a discrete version of Cayley surfaces, which are ruled improper affine spheres that can be characterized by the induced connection as: an asymptotic net with CAMC is equiaffinely congruent to a Cayley surface if and only if the cubic form does not vanish and the affine induced connection is parallel.