Ultracold atoms provide a powerful tool for studying quantum control of interacting many-body systems with well-characterized and controllable Hamiltonians. In this thesis, we demonstrate quantum control of a many-body system consisting of a ferromagnetic spin-1 Bose-Einstein condensate (BEC). By tuning the Hamiltonian of the system, we can generate either a phase space with an unstable hyperbolic fixed point or a phase space with an elliptical fixed point. A classical pendulum with a stable oscillation about the "down" position and an inverted pendulum with unstable non-equilibrium dynamics about the "up" position are classical analogs of the quantum spin dynamics we investigate in this thesis. In one experiment, we dynamically stabilize the system about an unstable hyperbolic fixed point, which is similar to stabilizing an inverted pendulum. In a second experiment, we parametrically excite the system by modulating the quadratic Zeeman energy. In addition, we demonstrate rectifier phase control as a new method to manipulate the quantum states of the many-body system. This is similar to parametric excitation and manipulation of the oscillation angle of a classical pendulum. These experiments demonstrate the ability to control a quantum system realized in a spinor BEC, and they also can be applied to other quantum systems. In addition, we extend our studies to atoms above the Bose-Einstein transition temperature, and we present results on thermal spin relaxation processes and equilibrium spin populations.

My dissertation treats the physics of ultracold gases, in particular of Bose-Einstein condensates with long-ranged interactions induced by admixing a small fraction of a Rydberg state to the atomic ground state. The resulting interaction leads to the emergence of supersolid states and to the self-trapping of a Bose-Einstein condensate.

Nous investiguons dans ce travail la création d'échantillons permettant l'étude du comportement des polaritons excitoniques dans les matériaux semi-conducteurs organiques. Le couplage fort entre les états excités d'électrons et des photons impose la création de nouveaux états propres dans le milieu. Ces nouveaux états, les polaritons, ont un comportement bosonique et sont donc capables de se condenser dans un état fortement dégénéré. Une occupation massive de l'état fondamental permet l'étude de comportements explicables uniquement par la mécanique quantique. La démonstration, au niveau macroscopique, d'effets quantiques promet d'éclairer notre compréhension de la matière condensée. De plus, la forte localisation des excitons dans les milieux organiques permet la condensation des polaritons excitoniques organiques à des températures beaucoup plus hautes que dans les semi-conducteurs inorganiques. À terme, les échantillons proposés dans ce travail pourraient donc servir à observer une phase cohérente macroscopique à des températures facilement atteignables en laboratoire. Les cavités proposées sont des résonateurs Fabry-Perot ultraminces dans lesquels est inséré un cristal unique d'anthracène. Des miroirs diélectriques sont fabriqués par une compagnie externe. Une couche d'or de 60 nanomètres est ensuite déposée sur leur surface. Les miroirs sont ensuite mis en contact, or contre or, et compressés par 2,6 tonnes de pression. Cette pression soude la cavité et laisse des espaces vides entre les lignes d'or. Une molécule organique, l'anthracène, est ensuite insérée par capillarité dans la cavité et y est cristallisée par la suite. Dans leur état actuel, les cavités présentent des défauts majeurs quant à la planarité des miroirs et à l'uniformité des cristaux. Un protocole détaillé est présenté et commenté dans ce travail. Nous y proposons aussi quelques pistes pour régler les problèmes courants de l'appareil. / In this work we investigate the creation of samples for the study of the behavior of excitonic polaritons in organic semiconductor materials. The strong coupling between the excited states of electrons and photons implies the creation new eigenstates in the medium. These new states, called polaritons, are composite bosons and are therefore capable of condensing in a strongly degenerated state. A massive occupation of the ground state allows the study of behaviors that are only explainable by quantum mechanics. A macroscopic demonstration of quantum effects offers a rare opportunity for scientific research and discoveries. The strong localization of excitons in organic materials allows condensation of exciton polaritons at temperatures much higher than in inorganic semiconductors. Therefore the samples proposed in this work could ultimately be used to observe a macroscopic coherent phase at temperatures easily attainable in a laboratory. The cavities proposed in this work are Fabry-Perot resonators in which anthracene is inserted and crystalized. The mirrors used in the resonator are dielectric reflectors made by a external company according to our specifications. A gold layer of 60 nm is deposited on their surface. The mirrors are then brought into contact, gold against gold, and compressed by 2.6 tons of pressure. This pressure seals the cavity and leaves voids between the gold lines. An organic molecule, anthracene, is then inserted in by capillary inside the cavity voids and subsequently crystallized by controlled cooling. In their current state cavities have defects regarding the planarity of the mirrors and the uniformity of the crystals. A detailed protocol is presented and discussed in this work.

L'aimant organique NIT-2Py a été caractérisé expérimentalement et ses propriétés ont été simulées numériquement à partir de la théorie de la fonctionnelle de la densité. Le magnétisme dans ce matériau provient de la présence d'un électron non apparié sur chaque molécule qui a ainsi un moment magnétique non nul. Ceci a été confirmé par des simulations sur une molécule isolée. Les molécules de NIT-2Py cristallisent dans le groupe d'espace P21/c avec huit molécules par maille élémentaire pour former la structure cristalline Alpha étudiée dans ce document. Le moment effectif de la susceptibilité et l'entropie magnétique totale montre que ce matériau est un système de spins 1/2 avec un spin par molécule. Les mesures de chaleur spécifique ont mis en évidence la présence de deux phases magnétiques ordonnées à basse température qui sont séparées par un plateau en aimantation. Une première phase est observée à des champs magnétiques inférieurs à 2.2 T et a une température de transition de 1.32 K en champ nul. Les mesures de susceptibilité magnétique et d'aimantation ont permis d'établir que cette phase ordonnée est antiferromagnétique. Ceci est confirmé par les simulations numériques. La deuxième phase est induite par le champ magnétique avec une température de transition de 0.53 K à 6 T. L'information disponible sur cette phase est limitée et l'étude du système à l'extérieur des phases ordonnées en donne une meilleure compréhension. Un modèle de spins S=1/2 isolés et de dimères S=0 isolés reproduit bien les mesures d'aimantation et de chaleur spécifique au-dessus de 3 K. L'application d'un champ magnétique réduit l'écart d'énergie entre le singulet et le triplet du dimère jusqu'au croisement qui se produit à 6 T. La phase induite émerge précisément à ce croisement et on spécule l'existence d'un condensat de Bose-Einstein des états triplets. / The organic magnet built from NIT-2Py molecules has been characterized experimentally and its properties have been simulated using density functional theory. In this material, an unpaired electron carrying a magnetic moment on each molecule is responsible for the magnetism. This has been confirmed by numeric simulations on an isolated molecule. NIT-2Py molecules crystallize in space group P21/c with eight molecules per unit cell to form crystalline phase Alpha studied in this document. The effective moment obtained from magnetic susceptibility and the total magnetic entropy show that this material is a spin 1/2 system with one spin per molecule. Specific heat measurements have highlighted the presence of two magnetically ordered phases at low temperature, which are separated by a plateau in magnetization. A first phase is observed at magnetic field lower than 2.2 T and has a transition temperature of 1.32 K in zero field. Magnetic susceptibility and magnetization measurements have established that this ordered phase is antiferromagnetic. This is confirmed by numeric simulations. The second phase is induced by a magnetic field and has a transition temperature of 0.53 K at 6 T. Information concerning the field induced phase is limited and a study of the system above the transition temperatures helps to gain a better understanding. A model of isolated spins S=1/2 and isolated dimers S=0 reproduces nicely the specific heat and magnetization data above 3 K. The application of a magnetic field reduces the energy gap between the singlet and the triplet of the dimer and the crossover between these levels is observed at 6 T. The field induced phase emerges precisely at this crossover suggesting the occurrence of a Bose-Einstein condensation of triplets states.

Dans ce manuscrit, nous présentons une étude expérimentale d’un gaz de Bose de spin-1 avec des interactions antiferromagnétiques, réalisée pour des atomes de sodium ultra-froids dans l’état hyperfin F=1. Gr au refroidissement évaporatif, nous obtenons un condensat de Bose-Einstein (CBE) spineur, soit dans un piège très confinant (« piège 0D »), soit sous la forme d’un quasi-condensat quasi-unidimensionnel dans un piège très allongé. Les deux systèmes présentent un ordre magnétique a très basse température, qui résulte de la compétition entre les interactions d’échange et l’énergie Zeeman quadratique q dans un champ magnétique externe. Nous étudions dans un premier temps l’ordre magnétique se forme dans le piège 0D. À très bassetempérature deux phases magnétiques sont possible : une phase dite « antiferromagnétique » pour q < Us, ou une phase dite « à aimantation transverse » dans le cas inverse. Dans ce travail, nous nous plaçons près de la température critique. Nous mesurons plusieurs scénarios de condensation séquentielles en changeant la magnétisation et le champ magnétique externe, ou une composante Zeeman condense toujours en premier et ou l’ordre magnétique n’apparait qu’à une seconde température de condensation. Les résultats expérimentaux pour les températures critiques sont bien décrits par une théorie d’Hartree-Fock simplifiée dans les cas ou une seule composante Zeeman est condensée. Dans un second temps, nous étudions l’ordre magnétique du système quasi-unidimensionnel a basse température. On observe la formation de domaines de spin ou les composantes Zeeman se sépare spontanément en domaines disjoints en l’absence de force extérieure (par exemple, un gradient de champ magnétique). On étudie l’état d’équilibre du système en fonction de la magnétisation et du champ magnétique. On observe une transition de phase entre une phase miscible et une phase immiscible ou la composante Zeeman mF = 0 forme un domaine séparé de mF = ±1 dans le centre du piège. L’équation d’état d’un nuage polarisé (atomes dans l’état mF = +1) est utilisée pourmesurer la température du système. Enfin, nous mesurons la réponse mécanique a une force magnétique appliquée pour un système binaire mF = 0, +1. Nous mesures une exaltation de la réponse par rapport a l’attente na basée sur l’effet Zeeman habituel, d’un facteur qui peut varier de plusieurs dizaines a environ cent. La configuration spatiale des domaines est ainsi sensible a de très faibles gradients de champ magnétique inférieurs au mG/cm. / In this thesis we present the experimental study of a spin-1 Bose gas of ultra-cold Na atoms with antiferromagnetic interactions in the F=1 manifold. Thanks to evaporative cooling in optical traps we obtain, depending on the trap geometry, quasi-pure spinor Bose-Einstein condensates (BEC) in 0D traps and quasi-condensates in quasi-1D traps. The quantum-statistical Bose enhancement, typical of BEC, allows inter-component interactions (between the different Zeeman components) to order the system just below the Bose-Einstein condensation temperature. The magnetic ordering of the system is set: by contact interactions, that do not change the Zeeman populations, by spin-exchange interactions (U_s spin-exchange energy), that do, and by the quadratic Zeeman energy q. In particular, for q < U_s the system is in the antiferromagnetic phase while, for q > U_s, is in the transverse magnetised phase. We study first in which order the magnetic ordering appears, in the 0D trap, near to the critical temperature for BEC. We experimentally study different condensations scenarii varying q and magnetisation. The condensation of the different components is sequential and strongly influenced by interactions. We find a good agreement between the experimental data and a simplified Hartree-Fock model.Then we study the magnetic ordering, at T=0, in a quasi-1D trap. The system presents the formation of spin domains. We study the ground state of the system varying magnetisation and q. We observe a transition from the miscible to the immiscible phase, associated with the transition from the antiferromagnetic to the transverse magnetised phase. This is due to the relative strengths of inter-species contact interaction. To measure the temperature of the system, we measure the equation of state for a polarised cloud (all atoms in m_F=+1). Finally, we prepare the system in the immiscible phase m_F=0,+1 and we measure the spin-dipole polarisability of the system.

Le résultat principal de cette thèse est l'obtention d'une condition nécessaire et suffisante pour l'existence d'une métrique de Kähler-Einstein sur une compactification bi-équivariante lisse et Fano d'un groupe complexe réductif connexe. Ces variétés comprennent les variétés toriques et les compactifications magnifiques de groupes semisimples adjoints.Dans la première partie de ce travail sont développés les outils nécessaires à l'étude de l'existence de métriques de Kähler-Einstein sur ces variétés. Nous calculons en particulier la Hessienne complexe d'une fonction $Ktimes K$-invariante sur la complexification d'un groupe compact $K$. Nous associonségalement, à toute métrique invariante à courbure positive sur un fibré linéarisé ample sur une compactification de groupe, une fonction convexe dont le comportement asymptotique est prescrit. Ceci est utilisé une première fois pour obtenir une formule pour l'invariant alpha d'un fibré en droite ample sur une compactification de groupe Fano. Cette formule est obtenue par le calcul des seuils log canoniques des métriques hermitiennes invariantes à courbure positive, et induit, dans le cas particulier des variétés toriques, un résultat obtenu auparavant, figurant dans l'article par ailleurs inclus en appendice de la thèse.Nous prouvons ensuite le résultat principal en obtenant des estimées $C^0$ le long de la méthode de continuité, en se ramenant à une équation de Monge-Ampèreréelle sur un cône. La condition obtenue est que le barycentre du polytope associé à la compactification de groupe, par rapport à la mesure de Duistermaat-Heckman, doit être dans une zone particulière du polytope. Cette condition peut être vérifiée sur les exemples, donne de nouveaux exemples de variétés deKähler-Einstein Fano, et donne aussi un exemple qui n'admet aucun soliton de Kähler-Ricci. Nous calculons de plus la plus grande borne inférieure de Ricci lorsqu'il n'y a pas de métrique de Kähler-Einstein. / The main result of this work is a necessary and sufficient condition for the existence of a Kähler-Einstein metric on a smooth and Fano bi-equivariant compactification of a complex connected reductive group. Examples of such varieties include wonderful compactifications of adjoint semisimple groups.The tools needed to study the existence of Kähler-Einstein metrics on these varieties are developed in the first part of the work, including a computation of the complex Hessian of a $Ktimes K$-invariant function on the complexification of a compact group $K$. Another step is to associate to any non-negatively curved invariant hermitian metric on an ample linearized line bundle on a group compactification a convex function with prescribed asymptotic behavior. This is used a first time to derive a formula for the alpha invariantof an ample line bundle on a Fano group compactification. This formula is obtained through the computation of the log canonical thresholds of any non-negatively curved invariant hermitian metric, and gives the sameresult, for toric manifolds, as the one we obtained before, in an article that is included in this thesis as an appendix.Then we prove the main result by obtaining $C^0$ estimates along the continuity method, using the tools developed to reduce to a real Monge-Ampère equation on a cone. The condition obtained is that the barycenter of the polytope associated to the group compactification, with respect to the Duistermaat-Heckman measure, lies in a certain zone in the polytope. This condition can be checked on examples, gives new examples of Fano Kähler-Einstein manifolds, and also gives an example that admits no Kähler-Ricci solitons. We also compute the greatest Ricci lower bound when there are no Kähler-Einstein metrics.

Submitted by Emanuel Varela Cardoso (emanuel.varela@ufcg.edu.br) on 2018-10-17T17:33:09Z No. of bitstreams: 1 Dissertação-Ildemar Barreto Veloso -(OK).pdf: 627444 bytes, checksum: ba9ac192a954812365d05a8c75c86647 (MD5) / Made available in DSpace on 2018-10-17T17:33:09Z (GMT). No. of bitstreams: 1 Dissertação-Ildemar Barreto Veloso -(OK).pdf: 627444 bytes, checksum: ba9ac192a954812365d05a8c75c86647 (MD5) Previous issue date: 2016-09 / Capes / A Relatividade Geral, que foi construída baseada nos princípios da equivalência e da covariância,é a teoria da gravitação que admite um espaço-tempo não euclidiano. Ela permite que o espaço-tempo se torne curvo próximo à distribuições de matéria. Os buracos negros são objetos com imensas concentrações de matéria, que provocam grandes perturbações no espaço a sua volta. A métrica correspondente ao buraco negro que analisamos foi a de Schwarzschild. Tais objetos também podem ser analisados utilizando-se a termodinâmica, com a obtenção de variáveis como temperatura e entropia. Este último conceito, é de fundamental importância para entendermos a questão do emaranhamento entre dois sistemas físicos correlacionados e a teoria de informação. Assim, nosso objetivo foi analisar alguns fatores sobre o paradoxo da conservação da informação em buracos negros. Além do mais, também analisamos como o emaranhamento entre duas regiões desconectadas pode gerar espaços-tempos conectados, e qual sua relação com as pontes deEinstein-Rosen. / The General Relativity, that was built on the equivalence and covariance principles, is the gravitation the or which admits an on-euclide an space time. It permits that the space time be comes curve near matter distributions. The black holes are objects with huge concentrations of matter, which cause major disruption in the space around it.The corresponding metric to the black hole that were view edit is the Schwarzschildmetric. Such objects canal sobe analyzed using the thermo dynamic concepts, to obtain variables such as temperature and entropy. This one, will be of fundamental importance to understanding the question of the entanglement between two related physical systems and the concept of information. So, ourgo alitis to examine some factors aboutt he information conservation paradox in black holes. Moreover, also we analyzedas a entan- glement between two disconnected regions can generate space time connected,and what is its relationship with Einstein-Rosenbridges.

Among the wide range of remarkable experimentson dilute Bose-Einstein condensates has been the observed dynamics of attractive condensatesexhibiting collapse and subsequent explosion. For attractive condensates the collapse occurs when the number of atoms N becomes higher than a critical value Nc. After a collapse, the number of atoms N in the condensate is reduced so that for N below Nc A stable configuration is attained. By increasing the number of atoms in the condensate up to the point where N>Nc a further collapse is induced and so on, this process may be repeated and a series of collapses may be observed.In this work we investigate analytically the behavior of the collapsing condensate within the framework of a nonlinear Gross-Pitaevskii equation, suitable to describe the dynamics of the order parameter Ψ(r, t ) of a Bose-Einstein condensatemagnetically trapped in a harmonic three-dimensional potential.Two and three-body inelastic collisions which remove atoms from the condensate are included.By using a variational approach based on d’Alembert ́s principle and suitable for non-conservative systems wefindananalyticalsolutionforacollapsingBose-Einsteincondensate.We demonstrate that a Gaussianansatzcapturesremarkablywellthesequenceofimplosionand explosionobservedinattractivecondensates. / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Entre o vasto leque de experiências notáveis em condensados de Bose-Einstein diluídos, foi observada a dinâmica de condensados atrativos exibindo colapso e subseqüente explosão. Para condensados atrativos, o colapso ocorre quando o número de átomos N torna-se maior que um valor crítico Nc'N>Nc. Após um colapso, o número de átomos no condensado é reduzido tal que, para N abaixo de Nc uma configuração estável é atingida. Aumentando o número de átomos no condensado até o ponto onde N>Nc outro colapso é induzido e, assim por diante, esse processo será repetido e uma série de colapsos pode ser observada. Neste trabalho, nós investigamos analiticamente o comportamento do condensado colapsante no âmbito de uma equação de Gross-Pitaevskii não-linear, apropriada para descrever a dinâmica do parâmetro de ordem Ψ(r, t ) de um condensado de Bose-Einstein magneticamente aprisionado em um potencial harmônico tridimensional. Colisões inelásticas de dois e três corpos que removem átomos do condensado são incluídas. Usando uma abordagem variacional baseada no princípio de D’Alembert e apropriada para sistemas não-conservativos nós encontramos uma solução analítica para o condensado de Bose-Einstein colapsante. Nós demonstramos que um ansatz Gaussiano captura notavelmente bem a seqüência de implosões e explosões observada em condensados atrativos.

SANTOS, A. S. L. Problema de Yamabe modificado em variedades compactas de dimensão quatro e métricas críticas do funcional curvatura escalar. 2017. 58 f. Tese (Doutorado em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017. / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-05-25T19:34:47Z No. of bitstreams: 1 2017_tese_aslsantos.pdf: 535461 bytes, checksum: 8c3ddbdd33d74c4eb7b265354b3bafb3 (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Boa tarde, Eu revisei a Tese de ALEX SANDRO LOPES SANTOS, e encontrei um pequeno erro na capa, ele colocou os seguintes elementos: UNIVERSIDADE FEDERAL DO CEARÁ CENTRO DE CIÊNCIAS PROGRAMA DE PÓS-GRADUAÇÃO EM MATEMÁTICA DOUTORADO EM MATEMÁTICA Mas deve ser alterado para: UNIVERSIDADE FEDERAL DO CEARÁ CENTRO DE CIÊNCIAS DEPARTAMENTO DE MATEMÁTICA PROGRAMA DE PÓS-GRADUAÇÃO EM MATEMÁTICA Com os demais elementos da Tese, não há nenhum problema de formatação. Atenciosamente, on 2017-05-26T15:06:03Z (GMT) / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-05-29T13:47:44Z No. of bitstreams: 1 2017_tese_aslsantos.pdf: 536279 bytes, checksum: f37ece7d8035a2d9c788c45d2e7807ae (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-05-29T14:08:17Z (GMT) No. of bitstreams: 1 2017_tese_aslsantos.pdf: 536279 bytes, checksum: f37ece7d8035a2d9c788c45d2e7807ae (MD5) / Made available in DSpace on 2017-05-29T14:08:17Z (GMT). No. of bitstreams: 1 2017_tese_aslsantos.pdf: 536279 bytes, checksum: f37ece7d8035a2d9c788c45d2e7807ae (MD5) Previous issue date: 2017-05-19 / In the fisrt part of this work we investigate the modified Yamabe problem on four-dimensional manifolds whose the modifiers invariants depending on the eigenvalues of the Weyl curvature tensor and they are described in terms of maximum and minimum of the biorthogonal (sectional) curvature. We provide some geometrical and topological properties on four-dimensional manifolds in terms of these invariants. In the second part we investigate the critical points of the total scalar curvature functional restricted to space of metrics with constant scalar curvature of unitary volume, for simplicity CPE metrics. It was conjectured in the 1980’s that every CPE metric must be Einstein. We prove that such a conjecture is true under a second-order vanishing condition on the Weyl tensor. / Na primeira parte deste trabalho investigamos o problema de Yamabe modificado em variedades de dimensão quatro cujos invariantes modificadores dependem dos autovalores do tensor de Weyl e são descritos em termos do máximo e mínimo da curvatura biortogonal (seccional). Fornecemos algumas propriedades geométricas e topológicas para tais variedades em termos destes invariantes. Na segunda parte investigamos os pontos críticos do funcional curvatura escalar total restrito ao espaço de métricas com curvatura escalar constante e volume unitário, abreviadamente chamamos de métricas CPE. Conjecturou-se na década de 1980 que toda métrica CPE deve ser Einstein. Provamos que tal conjectura é verdadeira sob uma condição de nulidade sobre o divergente de segunda ordem do tensor de Weyl.

SILVA, Adam Oliveira da. Rigidez de métricas críticas para funcionais riemannianos. 2017. 78 f. Tese (Doutorado em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017. / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-09-19T19:08:04Z No. of bitstreams: 1 2017_tese_aosilva.pdf: 481005 bytes, checksum: 2bdfc6ab68b042a5cfd4f67caf1e21e4 (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Bom dia, Estou devolvendo a Tese de ADAM OLIVEIRA DA SILVA, para que o arquivo seja substituído, pois o aluno já veio na BCM e orientei quais eram as correções a serem feitas. Atenciosamente, on 2017-09-20T14:03:26Z (GMT) / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-09-20T16:47:21Z No. of bitstreams: 1 2017_tese_aosilva.pdf: 480774 bytes, checksum: a1267dd82f8a82a19f79902004e1afb5 (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-09-21T12:26:34Z (GMT) No. of bitstreams: 1 2017_tese_aosilva.pdf: 480774 bytes, checksum: a1267dd82f8a82a19f79902004e1afb5 (MD5) / Made available in DSpace on 2017-09-21T12:26:35Z (GMT). No. of bitstreams: 1 2017_tese_aosilva.pdf: 480774 bytes, checksum: a1267dd82f8a82a19f79902004e1afb5 (MD5) Previous issue date: 2017-09-15 / The aim of this work is to study metrics that are critical points for some Riemannian functionals. In the first part, we investigate critical metrics for functionals which are quadratic in the curvature on closed Riemannian manifolds. It is known that space form metrics are critical points for these functionals, denoted by F t,s (g). Moreover, when s = 0, always Einstein metrics are critical to F t (g). We proved that under some conditions the converse is true. For instance, among others results, we prove that if n ≥ 5 and g is a Bach-flat critical metric to F −n/4(n−1) , with second elementary symmetric function of the Schouten tensor σ 2 (A) > 0, then g should be Einstein. Furthermore, we show that a locally conformally flat critical metric with some additional conditions are space form metrics. In the second part, we study the critical metrics to volume functional on compact Riemannian manifolds with connected smooth boundary. We call such critical points of Miao-Tam critical metrics due to the variational study making by Miao and Tam (2009). In this work, we show that the geodesics balls in space forms Rn , Sn and Hn have the maximum possible boundary volume among Miao-Tam critical metrics with connected boundary provided that the boundary be an Einstein manifold. In the same spirit, we also extend a rigidity theorem due to Boucher et al. (1984) and Shen (1997) to n-dimensional static metrics with positive constant scalar curvature, which give us another way to get a partial answer to the Cosmic no-hair conjecture already obtained by Chrusciel (2003). / Este trabalho tem como principal objetivo estudar métricas que são pontos críticos de alguns funcionais Riemannianos. Na primeira parte, investigaremos métricas críticas de funcionais que são quadráticos na curvatura sobre variedades Riemannianas fechadas. É de conhecimento que métricas tipo formas espaciais são pontos críticos para tais funcionais, denotados aqui por F t,s (g). Além disso, no caso s = 0, métricas de Einstein são sempre críticas para F t (g). Provamos que sob algumas condições, a recíproca destes fatos são verdadeiras. Por exemplo, dentre outros resultados, provamos que se n ≥ 5 e g é uma métrica Bach-flat crìtica para F−n/4(n−1) com segunda função simétrica elementar do tensor de Schouten σ 2 (A) > 0, então g tem que ser métrica de Einstein. Ademais, mostramos que uma métrica crítica localmente conformemente plana, com algumas hipóteses adicionais, tem que ser tipo forma espacial. Na segunda parte, estudamos as métricas críticas do funcional volume sobre variedades Riemannianas compactas com bordo suave conexo. Chamamos tais pontos críticos de métricas críticas de Miao-Tam, devido ao estudo variacional feito por Miao e Tam (2009). Neste trabalho provamos que as bolas geodésicas das formas espaciais Rn , S n e H n possuem o valor máximo para o volume do bordo dentre todas as métricas críticas de Miao-Tam com bordo conexo, desde que o bordo seja uma variedade de Einstein. No mesmo sentido, também estendemos um teorema de rigidez devido à Boucher et al. (1984) e Shen (1997) para métricas estáticas de dimensão n e com curvatura escalar constante positiva, o qual nos fornece outra maneira para obter uma resposta parcial para a Cosmic no-hair conjecture já obtida por Chrusciel (2003).