Dans cette thèse, nous étudions principalement les applications semi-conformes et leur influence sur la résolution de certaines équations géométriques importantes comme celle d’un soliton de Ricci et celle d’une application biharmonique. Dans la première partie, nous appliquons un ansatz qui permet de construire des applications semi-conformes à partir d’une équation différentielle en une fonction de deux variables. Nous caractérisons les solutions réelles-analytiques. Parmi les solutions explicites obtenues, nous trouvons le premier exemple d’une application semi-conforme non-harmonique définie entièrement sur R3 à valeurs dans le plan complexe. Dans la deuxième partie, nous étudions les solitons de Ricci. Nous nous intéressons aux solitons de dimension 3, où ils peuvent être décrits, au moins localement, en terme d’une application semi-conforme. Nous développons une nouvelle méthode de construction de ces solitons à partir des transformations biconformes, particulièrement adaptées à l’étude de l’unicité de la structure. Finalement, nous introduisons une nouvelle notion de morphisme harmonique généralisé qui, comme son nom l’indique, contient les morphismes harmoniques comme un cas particulier. Cette classe d’applications a une importance dans la théorie d’applications biharmoniques. Les morphismes harmoniques généralisés ont une caractérisation nette qui permet de donner plusieurs exemples et méthodes de construction d’applications biharmoniques non-harmonique. / In this work, we primarily study semiconformal mappings and their influence in the resolution of important geometric equations, such as those for a Ricci soliton and those for a biharmonic maps. In the first part of this thesis, we exploit an ansatz for the construction of semi-conformal mappings from a differential equation in a function of two variables. We characterize real-analytic solutions.Among the resulting explicit solutions, we find the first known example of an entire semi-conformal mapping into the plane which is not harmonic. In the second part, we study Ricci solitons.We are particularly interested in 3-dimensional Ricci solitons, as they can be described at least locally, in terms of a semi-conformal map. We develop a construction method of solitons from biconformal deformations, particularly adapted to the study of the structure unicity. Finally, we introduce a new notion of generalized harmonic morphism, which, as the name suggests, contain the harmonic morphisms as a special case. These mappings have an elegant characterization which enables the construction of explicit examples, as well as impacting on the theory of biharmonic mappings.

Le but de cette thèse est d’obtenir des informations sur le spectre discret d’opérateurs non auto-adjoints définis par des perturbations relativement compactes d’opérateurs auto-adjoints. Ces opérateurs auto-adjoints sont choisis parmi les opérateurs classiques de mécanique quantique. Il s’agit des opérateurs de Dirac, de Klein-Gordon et le laplacien fractionnaire qui généralise l’opérateur de Schrödinger habituellement considéré pour de tels problèmes. La principale méthode utilisée ici relève d’un théorème d’analyse complexe donnant une condition de type Blaschke sur les zéros d’une fonction holomorphe du disque unité. Cette condition traduit lecomportement des valeurs propres de l’opérateur perturbé sous forme d’inégalités de type Lieb-Thirring. Une autre méthode venant d’analyse fonctionnelle a été employée pour obtenir de telles inégalités et les deux méthodes sont comparées entre elles. / The topic of this thesis concerns the discrete spectrum of non-selfadjoint operators defined by relatively compact perturbation of selfadjoint operators. These selfadjoint operators are choosen among classical operators of quantum mechanics. These areDirac operator, Klein-Gordon operator, and the fractional Laplacian who generalize the Schrödinger operator. The main method is based on a theorem of complex analysis which gives Blaschke-type condition on the zeros of a holomorphic function on the unit disc. This Blaschke condition gives the information on the behaviour of eigenvalues of the perturbed operator by mean of Lieb-Thirring-type inequalities. Another method using functional analysis is also used to obtain these kind of inequalities and both methods are compared to each other.

Submitted by Fabio Sobreira Campos da Costa (fabio.sobreira@ufpe.br) on 2016-06-29T13:52:59Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) tese_final.pdf: 5635071 bytes, checksum: b300efb627e9ece412ad5936ab67e8e2 (MD5) / Made available in DSpace on 2016-06-29T13:52:59Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) tese_final.pdf: 5635071 bytes, checksum: b300efb627e9ece412ad5936ab67e8e2 (MD5) Previous issue date: 2013-08-26 / CNPq / No presente trabalho duas classes de problemas são abordadas. Primeiramente, são apresentados estudos computacionais sobre o modelo O(n) de spins na rede quadrada, e em seguida apresentamos novas soluções exatas para a dinâmica de bolhas na célula de Hele-Shaw. O estudo do modelo O(n) é feito utilizando sua representação em laços (cadeias fechadas), a qual é obtida a partir de uma expansão para altas temperaturas. Nesse representação, a função de partição do modelo possui uma expansão diagramática em que cada termo depende do número e comprimento total de laços e do número de (auto)interseções entre esses laços. Propriedades críticas do modelo de laços O(n) são obtidas através de conceitos oriundos da teoria de percolação. Para executar as simulações Monte Carlo, usamos o eficiente algoritmo WORM, o qual realiza atualizações locais através do movimento da extremidade de uma cadeia aberta denominada de verme e não sofre com o problema de "critical slowing down". Para implementar esse algoritmo de forma eficiente para o modelo O(n) na rede quadrada, fazemos uso de um nova estrutura de dados conhecida como listas satélites. Apresentamos estimativas para o ponto crítico do modelo para vários valores de n no intervalo de 0 < n ≤ 2. Usamos as estatísticas de laços e vermes para extrair, respectivamente, os expoentes críticos térmicos e magnéticos do modelo. No estudo de dinâmica de interfaces, apresentamos uma solução exata bastante geral para um arranjo periódico de bolhas movendo-se com velocidade constante ao longo de uma célula de Hele-Shaw. Usando a periodicidade da solução, o domínio relevante do problema pode ser reduzido a uma célula unitária que contém uma única bolha. Nenhuma imposição de simetria sobre forma da bolha é feita, de modo que a solução é capaz de produzir bolhas completamente assimétricas. Nossa solução é obtida por métodos de transformações conformes entre domínios duplamente conexos, onde utilizamos a transformação de Schwarz-Christoffel generalizada para essa classe de domínios. / In this thesis two classes of problems are discussed. First, we present computational studies of the O(n) spin model on the square lattice and determine its critical properties, whereas in the second part of the thesis we present new exact solutions for bubble dynamics in a Hele-Shaw cell. The O(n) model is investigated by using its loop representation which is obtained from a high-temperature expansion of the original model. In this representation, the partition function admits an diagrammatic expansion in which each term depends on the number and total length of loops (closed graphs) as well as on the number of intersections between these loops. Critical properties of the O(n) model are obtained by employing concepts from percolation theory. To perform Monte Carlo simulations of the model, we use the WORM algorithm, which is an efficient algorithm that performs local updates through the motion of one of the ends (called head) of an open chain (called worm) and hence does not suffer from “critical slowing down”. To implement this algorithm efficiently for the O(n) model on the square lattice, we make use of a new data structure known as a satellite list. We present estimates for the critical point of the model for various values of n in the range 0 < n ≤ 2. We use the statistics about the loops and the worm to extract the thermal and magnetic critical exponents of the model, respectively. In our study about interface dynamics, we present a rather general exact solution for a periodic array of bubbles moving with constant velocity in a Hele-Shaw cell. Using the periodicity of the solution, the relevant domain of the problem can be reduced to a unit cell containing a single bubble. No symmetry requirement is imposed on the bubble shape, so that the solution is capable of generating completely asymmetrical bubbles. Our solution is obtained by using conformal mappings between doubly-connected domains and employing the generalized Schwarz-Christoffel formula for this class of domains.