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.

Dans cette thèse nous étudions deux problèmes issus de la relativité générale : la construction de données initiales pour le problème de Cauchy des équations d’Einstein et le théorème de la masse positive. Nous construisons tout d’abord des données initiales en utilisant la méthode dite conforme introduite par Lichnerowicz [Lichnerowicz, 1944], Y. Choquet-Bruhat–J. York [Choquet-Bruhat et York, 1980] et Y. Choquet-Bruhat–J. Isenberg– D. Pollack [Choquet-Bruhat et al., 2007a]. Plus particulièrement, nous étudions les équations –de contrainte conforme– qui apparaissent dans cette méthode sur des variétés riemanniennes compactes de dimension n > 3. Dans cette thèse, nous donnons une preuve simplifiée du résultat de [Dahl et al., 2012], puis nous étendons et nous généralisons les théorèmes de M. Holst–G. Nagy–G. Tsogtgerel [Holst et al., 2009] et de D. Maxwell [Maxwell, 2009] dans le cas de données initiales à courbure moyenne fortement nonconstante. Nous donnons au passage un point de vue unifié sur ces résultats. En parallèle, nous donnons des résultats de non-existence et de non-unicité pour les équations de la méthode conforme sous certaines hypothèses. / The aim of this thesis is the study of two topical issues arising from general relativity: finding initial data for the Cauchy problem with respect to the Einstein equations and the positive mass theorem. For the first issue, in the context of the conformal method introduced by Lichnerowicz [Lichnerowicz, 1944], Y. Choquet-Bruhat–J. York [Choquet-Bruhat et York, 1980] and Y. Choquet-Bruhat–J. Isenberg–D. Pollack [Choquet-Bruhat et al., 2007a], we consider the conformal constraint equations on compact Riemannian manifolds of dimension n > 3. In this thesis, we simplify the proof of [Dahl et al., 2012, Theorem 1.1], extend and sharpen the far-from CMC result proven by Holst– Nagy–Tsogtgerel [Holst et al., 2009], Maxwell [Maxwell, 2009] and give an unifying viewpoint of these results. Besides discussing the solvability of the conformal constraint equations, we will also show nonexistence and nonuniqueness results for solutions to the conformal constraint equations under certain assumptions.

In this project a recent model of inflation in which the Standard Model Higgs field with a nonminimal coupling to gravity takes on the role of the inflaton field is investigated. The tensor to scalar ratio, spectral index and the running of the spectral index  is calculated for a tree level analysis and compared with the Planck experiment. The value of the nonminimal coupling constant $\xi$ is estimated by obtaining a relation between the amplitude of scalar perturbations and the Higgs mass, it is found that $\xi \sim 10^4$. The basic aspects of how the results are modified through quantum corrections and what the consequences of the nonminimal coupling are for the effective field theory description is discussed. It is found that a tree level analysis yields predictions which are inside the allowed regions of the cosmological parameters given by the Planck experiment. The large value of the nonminimal coupling leads to unitarity problems for this model of inflation.  However quantum effects will have a significant effect and how they modify the results of the tree level analysis is what decides if Higgs inflation is a viable theory. / I detta projekt undersöker vi en modell av kosmisk inflation där Higgsfältet med en ickeminimal koppling till tyngdkraften är mekanismen bakom inflation. Vi utför en klassisk analys och beräknar modellens föresägelser för ett antal kosmologiska parametrar som jämförs med Planck experimentet. Vi uppskattar värdet på den ickeminimala kopplingen $\xi$ och finner att $\xi \sim 10^4$. De grundläggande aspekterna bakom kvantanalysen samt vad effekten av den ickeminimala kopplingen har på beskrivningen i termer av en effektiv fältteori diskuteras. Vi finner att en klassisk analys ger förutsägelser som passar väl med Planckexperimentet men att den ickeminimala kopplingen leder till unitaritetsproblem för denna modell av inflation. Kvanteffekter kan dock ha en avsevärd effekt på resultat och en utförlig analys som tar dem till hänsyn krävs för att avgöra om Higgsinflation är en möjlig modell för inflation.

abstract: There is a pervasive need in the defense industry for conformal, low-profile, efficient and broadband (HF-UHF) antennas. Broadband capabilities enable shared aperture multi-function radiators, while conformal antenna profiles minimize physical damage in army applications, reduce drag and weight penalties in airborne applications and reduce the visual and RF signatures of the communication node. This dissertation is concerned with a new class of antennas called Magneto-Dielectric wire antennas (MDWA) that provide an ideal solution to this ever-present and growing need. Magneto-dielectric structures (μr>1;εr>1) can partially guide electromagnetic waves and radiate them by leaking off the structure or by scattering from any discontinuities, much like a metal antenna of the same shape. They are attractive alternatives to conventional whip and blade antennas because they can be placed conformal to a metallic ground plane without any performance penalty. A two pronged approach is taken to analyze MDWAs. In the first, antenna circuit models are derived for the prototypical dipole and loop elements that include the effects of realistic dispersive magneto-dielectric materials of construction. A material selection law results, showing that: (a) The maximum attainable efficiency is determined by a single magnetic material parameter that we term the hesitivity: Closely related to Snoek's product, it measures the maximum magnetic conductivity of the material. (b) The maximum bandwidth is obtained by placing the highest amount of μ" loss in the frequency range of operation. As a result, high radiation efficiency antennas can be obtained not only from the conventional low loss (low μ") materials but also with highly lossy materials (tan(δm)>>1). The second approach used to analyze MDWAs is through solving the Green function problem of the infinite magneto-dielectric cylinder fed by a current loop. This solution sheds light on the leaky and guided waves supported by the magneto-dielectric structure and leads to useful design rules connecting the permeability of the material to the cross sectional area of the antenna in relation to the desired frequency of operation. The Green function problem of the permeable prolate spheroidal antenna is also solved as a good approximation to a finite cylinder. / Dissertation/Thesis / Ph.D. Electrical Engineering 2013

Neste trabalho, discutem-se aspectos das transformações conformes na eletrodinâmica clássica com ênfase na invariância e nas leis de conservação. Inicialmente, abordaram-se aspectos gerais das transformações conformes e fez-se um resumo histórico da evolução dessas transformações. Procurou-se fazer uma apresentação didática, revisando-se a formulação Lagrangiana e o Teorema de Noether para campos aplicado à eletrodinâmica. Estudaram-se as transformações conformes no espaço plano, onde se mostrou que para dimensões maiores ou iguais a três o número de transformações é finito. A partir das equações de Maxwell em coordenadas curvilíneas, chegou-se à condição para que essas equações mantivessem sua forma cartesiana. Com essa condição, mostrou-se que a eletrodinâmica clássica é invariante para o grupo de transformações conformes. Foram discutidas as leis de conservação associadas à invariância conforme da eletrodinâmica clássica a partir do teorema de Noether. Das simetrias por translações no espaço-tempo, obtiveram-se as leis de conservação do momento linear e da energia. Das simetrias associadas às rotações, obtiveram-se seis quantidades conservadas: três delas ligadas à conservação do momento angular e, com relação às três restantes, observou-se, a partir de analogias com a mecânica, que estavam associadas ao movimento do centro de energia do campo. Para a interpretação da grandeza conservada por simetria de escala, verificou-se, também a partir de uma analogia mecânica, que essa simetria somente é verificada para partículas não massivas ou para partículas massivas a altas energias. Finalmente, para as transformações conformes especiais, verificou-se que as leis de conservação resultantes são consequências das leis anteriores de conservação para o campo eletromagnético, e neste caso, essa simetria também somente se manifesta para partículas de massa nula ou para altas energias. / In this work, aspects of conformal transformations in classical electrodynamics are discussed with emphasis on the invariance and conservation laws. Initially, a general view of conformal transformations was shown and a summary of the historical evolution of those transformations was presented. The work was approached didactically, and Noethers theorem based on the electrodynamics Lagrangian formulation was revised. The conformal transformations were studied in plane spaces and it was shown that, for dimensions greater than or equal to three, the number of transformations is finite. Starting from Maxwells equations in curvilinear coordinates, a condition for maintaining those equations in Cartesian form was established. With that condition, it was shown that the classical electrodynamics laws are invariant for the group of conformal transformations. The conservation laws associated with the conformal invariance of classical electrodynamics were discussed, based on Noethers theorem. From the space-time translation symmetry, the laws of conservation of linear momentum and of energy were obtained. From rotational symmetry, six conserved quantities were obtained: three of them associated with angular momentum and the remaining three, observed, starting from analogies with mechanics, were associated with the movement of the center of energy of the field. For the interpretation of the quantity conserved by scale symmetry, it was verified, also from a mechanical analogy, that that symmetry is only valid for null mass particles or for high energies. Finally, for the special conformal transformations, it was verified that the resultant laws of conservation are consequences of the previous laws, and in that case, symmetry is also valid only for particles of null mass or for high energies.

Nesta tese estudamos as propriedades críticas dos modelos anisotrópicos (isotrópicos) de Heisenberg com spin s arbitrário. O espectro das Hamiltonianas, com condições periódicas de contorno, foi calculado para redes finitas, resolvendo-se as equações do Bethe ansatz associadas. Nossos resultados indicam que a anomalia conforme destes modelos tem o valor c=3s/(1+s), independente da anisotropia, e os expoentes críticos variam continuamente com a anisotropia assim como no modelo de 8-vértices. O conteúdo de operadores destes modelos indica que a teoria de campos que governa a criticalidade destes modelos de spin é descrita por operadores formados pelo produto de um operador Gaussiano por outro com simetria Z(2s). Estudando estes modelos, com certas condições especiais de contorno, mostramos que eles são relacionados com uma nova classe de teorias unitárias recentemente propostas / This thesis is concerned with the critical properties of anisotropic (isotropic) Heisenberg chain,with arbitrary spin-s. The eigenspectrum of these Hamiltoniana, with periodic boundaries, are calculated for finite chains by solving numerically their associated Bethe ansatz equations. The results indicate that the conformal anomaly hás the value c=3s/1+s, independently of the anisotropy, and the exponentes vary continuously with the anisotropy like in the 8-vertex model. The operator content of these models indicate that the underlying field theory governing these critical spin-s models are described by composite fields formed by the product of Gaussian and Z(2s) fields. Studying these models, with some special boundary conditions, we show that they are related with a large class of unitary conformal field theories recntly introduced

Submitted by Sandra Maria Neri Santiago (sandra.neri@ufpe.br) on 2016-03-07T19:42:45Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) DISSERTAÇÃO versão finalt.pdf: 2479188 bytes, checksum: 40682d874a9a13182595ec8c40992750 (MD5) / Made available in DSpace on 2016-03-07T19:42:45Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) DISSERTAÇÃO versão finalt.pdf: 2479188 bytes, checksum: 40682d874a9a13182595ec8c40992750 (MD5) Previous issue date: 2013 / Neste trabalho resumimos o estudo do empacotamento de fios em uma região bidimensional planar. Abordamos o problema de um ponto de vista teórico, usando técnicas de campo conforme, e propriedades de escala do modelo, no regime de empacotamento-rígido, são derivadas, de sorte que os expoentes críticos para a energia elástica e para o número de laços da conformação são obtidos. Os resultados apresentam razoável concordância com dados advindos de experimentos e simulações. Também esboçamos uma analogia entre esse sistema e gravitação em duas dimensões, via gravitação de Liouville. / In this work we summarize the study of the packaging of wire in a planar two-dimensional region. We approach the problem from a theoretical point of view, using techniques of conformal field, and scaling properties of the model, in the tight-packing configuration, are derived, so that the critical exponents for the elastic energy and the number of loops of the conformation are obtained. The results show reasonable agreement with data coming from experiments and simulations. We also outline an analogy between this system and gravitation in two dimensions, via Liouville gravity.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Nesta tese estudamos vÃrios aspectos da geometria de variedades de Lorentz conformemente estacionÃrias e, particularmente, de espaÃos generalizados de Robertson-Walker, sob a presenÃa de um campo vetorial conforme fechado. Inicialmente, nÃs desenvolvemos um estudo sobre a r-estabilidade e a r-estabilidade forte de hipersuperfÃcies tipo-espaÃo fechadas em ambientes conformemente estacionÃrios de curvatura seccional constante; mais precisamente,nÃs obtemos uma caracterizaÃÃo das hipersuperfÃcies r-estÃveis pelo primeiro autovalor de um certo operador elÃptico naturalmente associado à sua r-Ãsima curvatura, bem como classificamos as hipersuperfÃcies fortemente r-estÃveis por meio de uma condiÃÃo adequada sobre o fator conforme do campo conforme do ambiente. Em seguida, estabelecemos teoremas gerais tipo-Bernstein para hipersuperfÃcies tipo-espaÃo em variedades de Lorentz conformemente estacionÃrias, um dos quais nÃo exige que a hipersuperfÃcie possua curvatura mÃdia constante. Finalmente, estendemos para variedades de Lorentz conformemente estacionÃrias um resultado de J. Simons sobre a minimalidade de certos cones em espaÃos Euclidianos, e aplicamos este resultado para construir subvariedades mÃnimas completas e nÃo-compactas no espaÃo de de Sitter e no espaÃo anti-de Sitter. / In this thesis we study several aspects of the geometry of conformally stationary Lorentz manifolds and, more particularly, of generalized Robertson-Walker spaces, under the presence of a closed conformal vector field. We initiate by focusing our study on the r-stability and on the strong r-stability of closed spacelike hypersurfaces of conformally stationary ambient spaces of constant sectional curvature; more precisely, we obtain a characterization of the r-stable ones by means of the first eigenvalue of a suitable elliptic operator naturally associated to its r-th mean curvature, as well classify the strongly r-stable ones by means of an appropriate condition on the conformal factor of the conformal vector field on the ambient space. Following,we establish general Bernstein-type theorems for spacelike hypersurfaces of conformally stationary Lorentz manifolds, one of which does not require the hypersurface to be of constant mean curvature. We end by extending, to conformally stationary Lorentz manifolds, a result of J. Simons on the minimality of certain cones in Euclidean spaces, and apply this result to build complete, non-compact minimal submanifolds in the de Sitter space and in the anti-de Sitter space.

O objetivo deste trabalho é calcular a ação de teoria de campos de supercordas para os dois primeiros níveis de massa da supercorda, incluindo os dois setores de projeção GSO. Considerando uma corda tipo II-A na presença de uma D9-brana instável, calcula-se a ação para o táquion, o campo de gauge e os férmions GSO(+) e GSO(-). O trabalho é realizado usando o formalismo híbrido e usando-se a ação de campos de supercordas de Berkovits, que inclui o setor Ramond. Para tanto, inclui-se amplo material de revisão sobre teorias e teorias de campos de supercordas. A construção de operadores de vértice GSO(-) no formalismo híbrido é feita em detalhes. Considerações sobre a ação obtida e perspectivas futuras do trabalho são discutidas no final. / The goal of this work is to compute the superstring field theory action contribution for the two first mass level of the superstring, including both GSO sectors. A type IIA superstring in the presence of an unstable non-BPS D9 brane is considered and the computation of the action for the Tachyon, Gauge Field and Massless fermions from GSO(+) and GSO(-) sectors is done. The main work is accomplished using the hybrid formalism and the superstring field theory action of Berkovits, including the Ramond Sector. This task is accomplished by including revision material thoroughly, for conformal and super conformal field theory. Construction of physical GSO(-) vertex operators is considered in detail. At the end, theres a discussion about the action for these fields and some future perspectives are considered.

No presente trabalho consideramos o produto de uma variedade Riemanniana compacta sem bordo de curvatura escalar zero e uma variedade Riemanniana compacta com bordo, curvatura escalar zero e curvatura media constante no bordo, e fazemos uso da teoria de bifurcação para provar a existência de um numero infinito de classes conforme com, pelo menos, duas métricas Riemannianas não homotéticas de curvatura escalar zero e curvatura média constante no bordo, sobre a variedade produto. / In this work, we consider the product of a compact Riemannian manifold without boundary, null scalar curvature and a compact Riemannian manifold with boundary, null scalar curvature and constant mean curvature on the boundary and we use the bifurcation theory to prove the existence of a infinite number of conformal classes with at least two non homothetic Riemannian metrics of null scalar curvature and constant mean curvature of the boundary on the product manifold.