Doutorado em Matemática (IME)
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Item Inexact variants of the alternating direction method of multipliers and their iteration-complexity analyses(Universidade Federal de Goiás, 2019-03-27) Adona, Vando Antônio; Gonçalves, Max Leandro Nobre; http://lattes.cnpq.br/7841103869154032; Melo, Jefferson Divino Gonçalves de; http://lattes.cnpq.br/8296171010616435; Melo, Jefferson Divino Gonçalves de; Gonçalves, Max Leandro Nobre; Prudente, Leandro da Fonseca; Pérez, Luis Roman Lucambio; Andreani, RobertoThis thesis proposes and analyzes some variants of the alternating direction method of multipliers (ADMM) for solving separable linearly constrained convex optimization problems. This thesis is divided into three parts. First, we establish the iteration-complexity of a proximal generalized ADMM. This ADMM variant, proposed by Bertsekas and Eckstein, introduces a relaxation parameter into the second ADMM subproblem in order to improve its computational performance. We show that, for a given tolerance ρ>0, the proximal generalized ADMM with α in (0, 2) provides, in at most O(1/ρ^2) iterations, an approximate solution of the Lagrangian system associated to the optimization problem under consideration. It is further demonstrated that, in at most O(1/ρ) iterations, an approximate solution of the Lagrangian system can be obtained by means of an ergodic sequence associated to a sequence generated by the proximal generalized ADMM with α in (0, 2]. Second, we propose and analyze an inexact variant of the aforementioned proximal generalized ADMM. In this variant, the rst subproblem is approximately solved using a relative error condition whereas the second one is assumed to be easy to solve. It is important to mention that in many ADMM applications one of the subproblems has a closed-form solution; for instance, l_1-regularized convex composite optimization problems. We show that the proposed method possesses iteration-complexity bounds similar to its exact version. Third, we develop an inexact proximal ADMM whose rst subproblem is inexactly solved using an approximate relative error criterion similar to the aforementioned inexact proximal generalized ADMM. Pointwise and ergodic iteration-complexity bounds for the proposed method are established. Our approach consists of interpreting these ADMM variants as an instance of a hybrid proximal extragradient framework with some special properties. Finally, in order to show the applicability and advantage of the inexact ADMM variants proposed here, we present some numerical experiments performed on a setting of problems derived from real-life applications.Item Subgradient and gradient methods with feasible inexact projections for constrained convex optimization problems(Universidade Federal de Goiás, 2021-04-30) Aguiar, Ademir Alves; Prudente, Leandro da Fonseca; http://lattes.cnpq.br/4573611419840935; Ferreira, Orizon Pereira; http://lattes.cnpq.br/0201145506453251; Ferreira, Orizon Pereira; Prudente, Leandro da Fonseca; Melo, Jefferson Divino Gonçalves de; Bento, Glaydston de Carvalho; Andreani, RobertoO método subgradiente com uma projeção inexata viável é proposto na presente tese para resolver problemas de otimização convexa com restrições não diferenciáveis. Para realizar a projeção inexata proposta em um conjunto restrito, uma tolerância de erro relativa é introduzida. Além do mais, em cada iteração, o algoritmo requer o cálculo de um subgradiente aproximado da função. Limitantes para a complexidade na iteração e resultados de convergência assintótica para a sequência gerada pelo método empregando os conhecidos tamanhos de passo exógeno, Polyak e dinâmico são estabelecidos. Finalmente, relatamos alguns resultados numéricos a fim de ilustrar o comportamento prático do algoritmo quando o conjunto de restrição é convexo e compacto. Aqui também consideramos um novo método gradiente com projeção inexata usando tolerância de erro relativa. Convergência assintótica e limitantes para a complexidade na iteração do método empregando tamanho de passo constante e de Armijo são apresentados. Resultados numéricos são relatados ilustrando as vantagens potenciais de considerar projeções inexatas em vez de exatas em alguns exemplos de média escala em problemas de mı́ nimos quadrados sobre o espectraedro.Item Conditional gradient methods for multiobjective optimization(Universidade Federal de Goiás, 2021-08-06) Assunção Filho, Pedro Bonfim de; Prudente, Leandro da Fonseca; http://lattes.cnpq.br/4573611419840935; Ferreira, Orizon Pereira; http://lattes.cnpq.br/0201145506453251; Ferreira, Orizon Pereira; Prudente, Leandro da Fonseca; Melo, Jefferson Divino Gonçalves de; Bento, Glaydston de Carvalho; Souza, João Carlos de OliveiraNeste trabalho, analisamos o método do gradiente condicional, também conhecido como método de Frank-Wolfe, para resolver problemas de otimização multiobjetivo restrita. Também propomos e analisamos uma versão generalizada deste método para resolver problemas de otimização composta multiobjetivo que consistem em minimizar simultaneamente várias funções objetivo. Cada função objetiva é a soma de duas funções, uma é considerada continuamente diferenciável e a outra não é necessariamente diferenciável. Ambos os métodos são analisados com três estratégias de obtenção dos tamanhos dos passos, a saber: tipo Armijo, adaptativos e tamanhos decrescentes dos passos. Propriedades de convergência assintótica e limites de complexidade de iteração com e sem suposições de convexidade na função objetivo são estabelecidas. Experimentos numéricos para o método do gradiente condicional são fornecidos para ilustrar a eficácia do método e certificar os resultados teóricos obtidos.Item Fluidos perfeitos estáticos com simetrias(Universidade Federal de Goiás, 2019-04-25) Barboza, Marcelo Bezerra; Pina, Romildo da Silva; http://lattes.cnpq.br/2675728978857991; Corro, Armando Mauro Vasquez; Leandro Neto, Benedito; Manfio, Fernando; Marrocos, Marcus Antônio Mendonça; Pina, Romildo da SilvaThis work presents a two step procedure that is virtually capable of producing an infinite number of exact solutions to Einstein's equation of a perfect fluid on a static manifold. These steps could roughly be described as: 1) classifying the symmetries of the referred equation that convert it into a second order non linear ordinary differential equation of very specific nature -- whose solutions are a whole lot easier to come up with than those of the original problem, and 2) solving this ordinary equation -- which quite explains the need for the word `virtually' above, since not all solutions of the ordinary equation are known to its exact form. Finally, in the last chapter, we utilize a Theorem due to Liouville to determine the rigid motions of Riemannian metrics on euclidean space that do admit symmetries in a translational group and also belong to the conformal class of the flat metric.Item Generalized vector equilibrium problems and algorithms for variational inequality in hadamard manifolds(Universidade Federal de Goiás, 2016-10-20) Batista, Edvaldo Elias de Almeida; Bento, Glaydston de Carvalho; http://lattes.cnpq.br/1089906772427394; Ferreira, Orizon Pereira; http://lattes.cnpq.br/0201145506453251; Ferreira, Orizon Pereira; http://lattes.cnpq.br/0201145506453251; Bento, Glaydston de Carvalho; http://lattes.cnpq.br/1089906772427394; Cruz Neto, João Xavier da; Peréz, Luís Román Lucambio; Alves, Maicon MarquesIn this thesis, we study variational inequalities and generalized vector equilibrium problems. In Chapter 1, several results and basic definitions of Riemannian geometry are listed; we present the concept of the monotone vector field in Hadamard manifolds and many of their properties, besides, we introduce the concept of enlargement of a monotone vector field, and we display its properties in a Riemannian context. In Chapter 2, an inexact proximal point method for variational inequalities in Hadamard manifolds is introduced, and its convergence properties are studied; see [7]. To present our method, we generalize the concept of enlargement of monotone operators, from a linear setting to the Riemannian context. As an application, an inexact proximal point method for constrained optimization problems is obtained. In Chapter 3, we present an extragradient algorithm for variational inequality associated with the point-to-set vector field in Hadamard manifolds and study its convergence properties; see [8]. In order to present our method, the concept of enlargement of maximal monotone vector fields is used and its lower-semicontinuity is established to obtain the convergence of the method in this new context. In Chapter 4, we present a sufficient condition for the existence of a solution to the generalized vector equilibrium problem on Hadamard manifolds using a version of the KnasterKuratowski-Mazurkiewicz Lemma; see [6]. In particular, the existence of solutions to optimization, vector optimization, Nash equilibria, complementarity, and variational inequality is a special case of the existence result for the generalized vector equilibrium problem.Item Sobre estruturas gradiente Einstein-type produto torcido(Universidade Federal de Goiás, 2022-04-13) Batista, Elismar Dias; Adriano, Levi Rosa; http://lattes.cnpq.br/3206466156270217; Adriano, Levi Rosa; Corro, Armando Mauro Vasquez; Santos, João Paulo dos; Gomes, José Nazareno Vieira; Ribeiro Júnior, Ernani de SousaIn this work, we study gradient Einstein-type structures immersed in a warped product space of an interval I and a Riemannian manifold M, with a potential function h given by the height function. First, we obtained the necessary conditions for an Einstein-type gradient structure immersed in a warped product to be minimal, totally umbilic or totally geodesic. In addition, we provide triviality results for the potential function h. Next, we characterize the rotational hypersurfaces having a gradient Einstein-type structure in warped product, where the fiber is a space form. We also study particular cases of gradient Einstein-type structures in warped product spaces, namely, gradient Ricci-harmonic solitons (GRHS). In this case, we prove triviality results for the potential and warped functions when they reach a maximum or minimum. Finally, we provide a family non-enumerable of non-trivial geodesically complete examples of GRHS considering the base and fiber of a warped product conformal to a semi-Euclidean space invariant under the action of a translation group of codimension one.Item Soluções clássicas para um problema de combustão em meios porosos com n camadas(Universidade Federal de Goiás, 2019-08-16) Batista, Marcos Roberto; Mota, Jesus Carlos da; http://lattes.cnpq.br/8457974658695539; Mota, Jesus Carlos da; Silva, Edcarlos Domingos da; Carvalho, Marcos Leandro Mendes; Ercole, Grey; Santos, Marcelo Martins dosIn this work, we study the classical solutions for a parabolic system of reaction-diffusion-convection equations, coupled to a system of ordinary differential equations, with boundary and initial conditions in a bounded domain. The coupled system models the propagation of a combustion front through a porous medium with n layers, where the dependent variables are the temperatures and the fuel concentrations in each layer. Problems for parabolic equations systems coupled with Ordinary Differential Equations (ODEs) system, where the coupling occurs in both the reaction functions and the associated differential operator coefficients, are little known in the literature. In classical theory in general, the coupling appears only in the reaction functions. Initially, using the Monotone Iterative Method, we prove the existence and uniqueness of a global solution in time for the particular case where the fuel concentrations in each layer are known functions. Next, we show the existence of the local solution in time for the complete problem when the concentrations are unknown functions. The proof is obtained by defining an operator in the Continuous Hölder function set and showing that it has a fixed point that is a local solution to the problem. This solution can be extended to a global solution in time for the problem, provided that the spatial derivatives of the temperature in each layer are bounded functions.Item Variedades de Einstein e Ricci solitons(Universidade Federal de Goiás, 2019-03-27) Bezerra, Tatiana Pires Fleury; Pina, Romildo da Silva; http://lattes.cnpq.br/2675728978857991; Pina, Romildo da Silva; Ferraioli, Diego Catalano; Roitman, Pedro; Leandro Neto, Benedito; Adriano, Levi RosaIn this work, we prove that all metrics conformal to the pseudo-Euclidean space ( , ), invariant under the action ( -1)-dimensional translation group and also invariants by a pseudo orthogonal group action are gradient Ricci almost solitons. We also prove that all metrics conformal to = ( , ̅) , invariant under the action ( -1)-dimensional translation group and Ricci flat are gradient Ricci almost soliton. We classify all Einstein manifolds of the type = ( , ̅) , where ̅ , invariant under the action ( -1)-dimensional translation group and Ricci flat with . If is gradiente Ricci soliton of type = ( , ̅) and the fiber is Ricci flat then is teady and we provide all such solutions. Finally we prove that if the warped product = ( , ̅) is a gradient Ricci soliton with Ricci flat, and further, then is steadyItem Hipersuperfícies de tipo Ribaucour(Universidade Federal de Goiás, 2022-09-27) Cárdenas Mendez, Milton Javier; Vasquez Corro, Armando Mauro; http://lattes.cnpq.br/4498595305431615; Vasquez Corro, Armando Mauro; Leandro Neto, Benedito; Carrion Riveros, Carlos Maber; Santos, João Paulo dos; Adriano, Levi RosaIn this work we define the Ribaucour-type surfaces (in short, RT-surfaces),These surfaces satisfy a relationship similar to the Ribaucour surfaces that are related to the Élie Cartan problem. This class furnishes what seems to be the first examples of pairs of noncongruent surfaces in Euclidean space such that, under a diffeomorphism, lines of curvatures are preserved and principal curvatures are switched. We show that every compact and connected RT-surface is the sphere with center at the origin. We obtain present a Weierstrass type representation for RT-surfaces with prescribed Gauss map which depends on two holomorphic functions. We give explicit examples of RT-surfaces. Also, we use this representation to classify the RT-surfaces of rotation. We define the GRT-surfaces which are a generalization of the RT-surfaces, show a local parameterization of this class of surfaces and classify them in the in which case they are of rotation and generalize as RT-surfaces for the case of hypersurfaces in Rn+1, display a parameterizationção for the rotational cases and analyze the general of generatrix curves when hypersurfaces and rotation behavior are 3-dimensional.Item A qualitative study of planar piecewise smooth vector fields(Universidade Federal de Goiás, 2018-05-18) Cardoso Filho, João Lopes; Tonon, Durval José; http://lattes.cnpq.br/3688981956532711; Tonon, Durval José; Medrado, João Carlos da Rocha; Euzébio, Rodrigo Donizete; Buzzi, Cláudio Aguinaldo; Martins, Ricardo MirandaIn this work we exhibit canonical forms for 2D codimension one piecewise smooth vector Fields (PSVF). All possible orientations and codimension one scenarios were covered. Also the intrinsic objects that characterize each one of the canonical forms were presented. Also we present topological distinct canonical forms for a larger class for symmetric PSVF where the set of fixed points is contained in the variety os discontinuity. Finally we analyze the simultaneous occurrence of sliding and crossing limit cycle in the case where the piecewise linear vector fields presents a continuum of periodic orbits.Item Equações diferenciais parciais elípticas multivalentes: crescimento crítico, métodos variacionais(Universidade Federal de Goiás, 2013-09-27) Carvalho, Marcos Leandro Mendes; Gonçalves, José Valdo Abreu; http://lattes.cnpq.br/5148611284176776; Gonçalves, José Valdo Abreu; Mota, Jesus Carlos da; Silca, EdCarlos Domingos da; Alves, Claudianor Oliveira; Santos, Carlos Alberto Pereira dosIn this work we develop arguments on the critical point theory for locally Lipschitz functionals on Orlicz-Sobolev spaces, along with convexity, minimization and compactness techniques to investigate existence of solution of the multivalued equation −∆Φu ∈ ∂ j(.,u) +λh in Ω, where Ω ⊂ RN is a bounded domain with boundary smooth ∂Ω, Φ : R → [0,∞) is a suitable N-function, ∆Φ is the corresponding Φ−Laplacian, λ > 0 is a parameter, h : Ω → R is a measurable and ∂ j(.,u) is a Clarke’s Generalized Gradient of a function u %→ j(x,u), a.e. x ∈ Ω, associated with critical growth. Regularity of the solutions is investigated, as well.Item Soluções invariantes para o tensor de Schouten e tensor curvatura prescritos em variedades localmente conformemente planas(Universidade Federal de Goiás, 2018-06-12) Carvalho, Marcos Tulio Alves de; Pina, Romildo da Silva; http://lattes.cnpq.br/2675728978857991; Pina, Romildo da Silva; Corro, Armando Mauro Vasquez; Pieterzack, Mauricio Donizeti; Xia, Changyu; Lima, Barnabé PessoaIn this work we study two problems: the first one involving the prescribed Schouten tensor and the second one the prescribed curvature operator. The first problem was inspired by the works Deturck and Yang, [6], which consist of: Given a tensor T of order 2 in the pseudo-Euclidean space ( ,g), n ≥ 3, with coordinates x = ( ), and metric g, where , = ±1, find a metric as = g, such that the tensor of Schouten be T. The second problem is the problem of the prescribed curvature tensor consist of: Let Euclidean space ( ;g), n ≥ 3, with coordinates x = ( ), is , the R a tensor of order 4 of the form , where T = , with differentiable functions.We want to find a metric = g, such that = , where is the tensor curvature of the metric . Considering that the solutions are invariant by translation and rotation, we find necessary and sufficient conditions for both problems to have solution.Item Bifurcações de campos vetoriais em duas zonas com simetria(Universidade Federal de Goiás, 2017-11-28) Castro, Ubirajara José Gama de; Medrado, João Carlos da Rocha; http://lattes.cnpq.br/5021927574622286; Tonon, Durval José; Pessoa, Cláudio Gomes; Martins, Ricardo Miranda; Oliveira, Regilene Delazari dos SantosIn this work we study reversible vector fields in two zones and equivariant vector fields in two zones. Our main result is the classification of the symmetric singularities of codimensions 0,1 and 2 of such vector fields. More precisely, in the reversible case in R3, where the dimension of the fixed points variety of the involution associated to the vector field is 2, we present all bifurcation diagram of the codimensions 1 and 2 singularities, describing the changes in the behavior of the symmetric singularities and tangents of the vector field with the transition manifold, S, according to the variation of the bifucartion parameter. We also show the existence of invariant cylinders and, in this case, doing small perturbations we determine invariant manifolds that persisted and we determine the number of limit cycles that were born. When the vector field defined on two zones is equivariant, the dynamic is enriched with the emergence of the sliding vector field and we also do a local study and the classification of singularities (and pseudo-singularities) of codimensions 0,1 and 2. We show the existence of homoclinic sliding orbit and that it is a codimension one phenomenon. Moreover, provided the symmetry we get a double Shilnikov sliding orbit.Item Multiplicidade de soluções para uma classe de problemas elípticos de quarta ordem com condição de contorno de Navier(Universidade Federal de Goiás, 2018-02-27) Cavalcante, Thiago Rodrigues; Silva, Edcarlos Domingos da; http://lattes.cnpq.br/7817014732764711; Silva, Edcarlos Domingos da; Gonçalves, José Valdo; Carvalho, Marcos Leandro; Santos, Carlos Alberto Pereira; Figueiredo, Giovany de Jesus MalcherIn the first two chapters, we consider the following problem \begin{equation*} \left \{ \begin{array}{rcll} \alpha \Delta^{2} u + \beta \Delta u & = & f(x,u)\, & \mbox{in}\,\, \Omega \\ u = \Delta u & = & 0 \, &\mbox{on } \,\,\, \partial \Omega, \end{array} \right. \end{equation*} where $\displaystyle{\Delta^{2} u = \Delta(\Delta u)-\,\mbox{biharmonic (fourth-order operator)}}$, $\alpha > 0$ and $ \beta \in \R.$ The subset $\displaystyle{ \Omega \subset \mathbb{R}^{N}\, (N \geq 4)}$ is as somooth bounded domain and $\displaystyle{ f \in C(\overline{\Omega} \times \mathbb{R},\mathbb{R}) }.$ In each of the results obtained, we will consider different technical hypotheses and characteristics for the nonlinear function $f$ e for the value of the constant $ \beta. $ In the third chapter, we study an equation of the concave type super linear, of the form: \begin{equation} \left \{ \begin{array}{rcll} \alpha \Delta^{2} u + \beta \Delta u & = & a(x)|u|^{s-2}u + f(x,u)\, & \mbox{in}\,\, \Omega \\ u = \Delta u & = & 0 \, &\mbox{on} \,\,\, \partial \Omega, \end{array} \right. \end{equation} where $\beta \in (-\infty, \alpha \lambda_{1}).$ We consider that the function $a \in L^{\infty} (\Omega)$ and $s \in (1,2).$ Finally, in the last chapter we will consider a fourth order problem in which nonlinearity is also of the convex concave type. More precisely, we study the following class of equations: \begin{equation} \left\{ \begin{aligned} \alpha \Delta^{2} u + \beta \Delta u & = \mu a(x)|u|^{q-2}u + b(x)|u|^{p-2}u&\,\,\,\,\ &\mbox{in}\,\, \Omega \\ u = \Delta u & = 0 & \,\,\,\,&\mbox{on} \,\, \partial \Omega, \end{aligned} \right. \end{equation} where the parameter $ \mu > 0 $, the powers $ 1Item Ciclos limite e singularidades típicas de sistemas de equações diferenciais suaves por partes(Universidade Federal de Goiás, 2017-03-07) Cespedes, Oscar Alexander Ramírez; Medrado, João Carlos da Rocha; http://lattes.cnpq.br/5021927574622286; Tonon, Durval José; Buzzi, Claudio Aguinaldo; Silva, Paulo Ricardo da; Martins, Ricardo MirandaIn this work, we analize the version of Hilbert’s 16th problem for a piecewise linear differential system, PWLS, in R2. More precisely,we determinete the maximum number of certain types of limit cycles when the system is define in two zones separated by a straight line. Some results on the maximum number of cycles of a PWLS defined in two sectors were established. In addition, we classify typical singularities of a piecewise smooth systemin R3, taking into account the behavior of the associated sliding field.Item On the classification of Ricci solitons and Yamabe solitons(Universidade Federal de Goiás, 2023-03-20) Contreras, Jeferson Arley Poveda; Leandro Neto, Benedito; http://lattes.cnpq.br/3393448440968708; Leandro Neto, Benedito; Santos, João Paulo do; Ribeiro Júnior, Ernani de Sousa; Tenenblat, Keti; Barboza, Marcelo BezerraIn this work, we will study the self-similar solutions of both Ricci flow and Yamabe flow. These solutions are also known as Ricci and Yamabe soliton, respectively. Inspired by the divergence equation used by Robinson in his demonstration of the uniqueness of static black holes and by Brendle’s classification of steady Ricci solitons, we will make some important characterizations of these solitons. We prove that four-dimensional gradient Yamabe solitons must have a Yamabe metric, provided that an asymptotic condition holds. Inspired by the geometry of the cigar soliton, we demonstrate that a gradient steady Ricci soliton is either Ricci flat with a constant potential function or a quotient of the product steady soliton N n−1×R, where N n−1 is Ricci flat, or isometric to the Bryant soliton. In the final Chapter, we prove some rigidity results for shrinking and expanding Ricci solitons.Item Variedades quasi-Einstein com simetrias e generalizações(Universidade Federal de Goiás, 2021-12-05) Correia, Paula Gonçalves; Pina, Romildo da Silva; http://lattes.cnpq.br/2675728978857991; Leandro Neto, Benedito; Adriano, Levi Rosa; Corro, Armando Mauro Vasquez; Santos, João Paulo dos; Ferraioli, Diego CatalanoIn this work we study quasi-Einstein warped products and generalized quasi-Einstein manifolds locally conformally flat. We prove that in some quasi-Einstein semi-Riemannian warped product the fiber is necessarily an Einstein manifold. We provide all quasi-Einstein manifolds when r-Bakry-Émery tensor is null, the base is conformal to a n-dimensional pseudo-Euclidean space, invariant under the action of the (n-1)-dimensional translation group, the potential function depends only on the base and the fiber is Ricci-flat. As an application, we provide a family of Ricci-flat warped product whose base is not locally conformally flat. Considering generalized m-quasi-Einstein manifolds conformal to the pseudo-Euclidean space, we used an ansatz method to obtain the most general symmetry group of maximal dimension in which there are solutions to the system of equations that describe such manifolds. In addition, using the same method, we show that there is no different invariant of smaller dimension on a generalized m-quasi-Einstein manifold.Item Uma desigualdade de Minkowski e soluções exatas para o espaço-tempo estático de Einstein-Maxwell(Universidade Federal de Goiás, 2023-02-24) Costa, Ana Paula de Melo da; Leandro Neto, Benedito; http://lattes.cnpq.br/3393448440968708; Leandro Neto, Benedito; Cruz, Cícero Tiarlos Nogueira; Adriano, Levi Rosa; Silva, Maria de Andrade Costa e; Batista, Rondinelle MarcolinoWe study the static Einstein-Maxwell space when it is conformal to an n-dimensional pseudo-Euclidean space, which is invariant under the action of an (n−1)-dimensional translation group. We also provide a complete classification of such space. Moreover, we prove a Minkowski-like inequality for an asymptotically flat static Einstein- Maxwell (electrovacuum) space-time using as an approach the inverse mean curvature flow (IMCF).Item On the effect of existence of static horizons(Universidade Federal de Goiás, 2021-12-17) Coutinho, Fernando Soares; Leandro Neto, Benedito; http://lattes.cnpq.br/3393448440968708; Leandro Neto, Benedito; Corro, Armando Mauro Vasquez; Ribeiro Júnior, Ernani de Sousa; Santos, João Paulo dos; Reis, Hiuri Fellipe Santos dosNeste trabalho estudamos os efeitos da existência do horizonte estático. Em geral, o horizonte estático é definido como o conjunto onde a função lapso, para uma variedade estática, é identicamente nula. Este conjunto está fisicamente relacionado com o horizonte de eventos, a fronteira de um buraco negro. No primeiro capítulo, estudamos a conjectura da bola fluida. Construímos uma fórmula de divergência de Robinson para o espaço tempo fluido perfeito estático. Inspirado por esta conjectura, um resultado de rigidez para o fator espacial de um espaço-tempo fluido perfeito estático satisfazendo algumas condições de contorno é provado, desde que uma equação de estado seja válida. No segundo capítulo, o objetivo é investigar a geometria do espaço-tempo fluido perfeito estático em variedades compactas com bordo. Fornecemos uma estimativa de bordo para espaço-tempo fluido perfeito estático e estabelecemos uma fórmula do tipo Böchner para uma grande classe de espaços que incluem o espaço-tempo fluido perfeito estático, métricas críticas do funcional de volume, espaços estáticos e métricas CPE. Mais ainda, como consequência desta fórmula obtemos um resultado do tipo “gap” para um espaço tempo fluído perfeito estático compacto.Item Linhas assintóticas de campos de planos em R3 e de superfícies em R4(Universidade Federal de Goiás, 2017-08-31) Cruz, Douglas Hilário da; Garcia, Ronaldo Alves; http://lattes.cnpq.br/5680428710939826; Silva , Débora Lopes da; Tonon , Durval José; Tari, Farid; Ruas, Maria Aparecida Soares; Tello, Jorge Manuel SotomayorThis work is divided into two parts. The first part deals with asymptotic lines of plane fields in R3. The main results are the analysis of the asymptotic lines in a neighborhood of the regular surface of parabolic points and the analysis of the stability of closed asymptotic lines. We express the first derivative of the Poincar\'e map and show how to make hyperbolic a closed asymptotic line. The second part deals with asymptotic lines of surfaces in R4. The main results are results of structural stability and genericity for asymptotic lines of locally convex surfaces in R4 and a result of structural stability asymptotic lines of surfaces in R4.