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Dissertations/Thesis

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2024
Dissertations
1
  • TALITA SANTOS DE ARAUJO
  • Stable Intersections of Cantor sets

  • Advisor : DAVI DOS SANTOS LIMA
  • COMMITTEE MEMBERS :
  • DAVI DOS SANTOS LIMA
  • RAFAEL NOBREGA DE OLIVEIRA LUCENA
  • SERGIO AUGUSTO ROMAÑA IBARRA
  • CARLOS GUSTAVO TAMM DE ARAÚJO MOREIRA
  • Data: Feb 16, 2024


  • Show Abstract
  • In this work, we study stable intersections and arithmetic differences of regular Cantor sets.

    We present techniques that help us to detect the existence of stable intersections or extremals stable intersections between pairs of Cantor sets. We start with the Gap Lemma of Newhouse, which uses the thickness of the Cantor sets to this conclusion, the Generalized Thickness Test, which studies the intersection of Markov domains and finally, the existence of a recurrent compact in space of the relative configurations of these Cantor sets.

    We also present some examples that show how such techniques are applied registered. In certain examples, we highlight that one of the techniques works, while the others may fail, thus showing the importance of studying each one from them.

    We finish with a brief study of the topology of the arithmetic difference of affine Cantor sets, presenting a family of Cantor sets whose the arithmetic difference is almost always an R-Cantorval.

2
  • GLEYDSON SANTOS DA SILVA
  • QUADRATIC RESPONSE OF RANDOM AND DETERMINISTIC DYNAMICAL
    SYSTEMS

  • Advisor : RAFAEL NOBREGA DE OLIVEIRA LUCENA
  • COMMITTEE MEMBERS :
  • DAVI DOS SANTOS LIMA
  • RAFAEL JOSE ALVAREZ BILBAO
  • RAFAEL NOBREGA DE OLIVEIRA LUCENA
  • Data: Feb 16, 2024


  • Show Abstract

  • The present work involves the study of the article "Quadratic Response of Random and Deterministic Dynamical Systems" [1], which investigates sufficient conditions for dynamic systems to exhibit a linear response. Additionally, if, in addition to such conditions, other conditions are also observed, the system may exhibit a quadratic response in its stationary measures. It is noteworthy that, although the original article also deals with random dynamical systems, the focus of the current work is on the study of deterministic dynamical systems, emphatically on the study of deterministic dynamical systems with expanding applications on the unit circle. This constitutes a wide range of deterministic dynamical systems and, as we will see, they manifest the desired conditions and, therefore, exhibit linear and/or quadratic responses in their invariant measures when subjected to small perturbations.

3
  • VINICIUS GUARDIANO SOUZA
  • Area and spectrum estimates for stable minimal surfaces

  • Advisor : MARCOS PETRUCIO DE ALMEIDA CAVALCANTE
  • COMMITTEE MEMBERS :
  • MARCIO HENRIQUE BATISTA DA SILVA
  • MARCOS PETRUCIO DE ALMEIDA CAVALCANTE
  • DAVI MÁXIMO ALEXANDRINO NOGUEIRA
  • Data: Feb 26, 2024


  • Show Abstract
  • This dissertation is based on the recent results of O. Munteanu, C.-J. Sung, and J. Wang, published in 2023 in the referenced article [MSW23]. Our main motivation lies in the study of the area growth of geodesic balls and estimates for the bottom of the spectrum of the Laplacian operator on stable minimal surfaces in a three-dimensional manifold with scalar curvature bounded from below. After a review of the topics and techniques involved, we initially focus on the case of Euclidean space R3. In this case, we obtain an optimal area estimate that allows us to assert that stable minimal surfaces have area growth exactly like the Euclidean plane. This is sufficient to prove that complete stable minimal surfaces in Rare planes. This conclusion is already known from the contributions of Do Carmo and Peng [dCP79], Fisher-Colbrie and Schoen [FCS80], and Pogorelov [Pog81]. The technique for proving the area estimate can be adapted to obtain area estimates in the case of a more general ambient manifold.

    In the second part of the work, we focus on upper estimates for the bottom of the spectrum of stable minimal hypersurfaces. Initially, we recall that the bottom of the spectrum is closely related to the growth of the volume of geodesic balls. Motivated by this fact, we obtain our estimates using test functions constructed in terms of the Green’s function. Due to technical reasons, these estimates are only valid for stable minimal hypersurfaces in complete manifolds with dimension up to six.

2023
Dissertations
1
  • CICERO CALHEIROS DOS SANTOS FILHO
  • Global-Local Mixing

  • Advisor : DAVI DOS SANTOS LIMA
  • COMMITTEE MEMBERS :
  • DAVI DOS SANTOS LIMA
  • RAFAEL NOBREGA DE OLIVEIRA LUCENA
  • Ricardo Turolla Bortolotti
  • Data: Feb 27, 2023


  • Show Abstract
  • This dissertation aims to present consolidated and recent concepts, examples and techniques in Ergodic Theory. The main new concept is Global Mixing, which we will continue to call Global-Local Mixing, a recent idea, dating back to 2010, to understand the behavior of a dynamical system that preserves an infinite measure that is sigma-finite. We will show that the famous Boole and Maneville-Pomeau maps are both Global-Local Mixing.

2
  • ERIC ALBERTO DE SOUZA SANTOS
  • Control theory for the Korteweg-de Vries equation in a bounded domain

  • Advisor : MARCIO CAVALCANTE DE MELO
  • COMMITTEE MEMBERS :
  • MARCIO CAVALCANTE DE MELO
  • RENAN DANTAS MEDRADO
  • VICTOR HUGO GONZALEZ MARTINEZ
  • Data: May 31, 2023


  • Show Abstract
  • This dissertation addresses the study of control for the transport equation, the linearized Korteweg-de Vries equation, and the nonlinear Korteweg-de Vries equation in bounded domains. The aim of this work is to prove the controllability of these systems through the Hilbert’s uniqueness principle.

3
  • JOSAFÁ JOAQUIM DA SILVA JÚNIOR
  • Conformal methods applied to stable minimal hypersurfaces in R^4

  • Advisor : MARCIO HENRIQUE BATISTA DA SILVA
  • COMMITTEE MEMBERS :
  • GREGORIO PACELLI FEITOSA BESSA
  • MARCIO HENRIQUE BATISTA DA SILVA
  • MARCOS PETRUCIO DE ALMEIDA CAVALCANTE
  • Data: Aug 14, 2023


  • Show Abstract
  • In this work, we will develop the calculations made by [3] in which techniques involving conformal metrics and estimates are used to obtain a result on minimal hypersurfaces, namely, an immersed complete, orientable and stable minimal hypersurface R^4 must be (isometric a) a hyperplane.

2022
Dissertations
1
  • RODRIGO COSTA
  • Classification of umbilical submanifolds of Sn x R

  • Advisor : CARLOS GONCALVES DO REI FILHO
  • COMMITTEE MEMBERS :
  • CARLOS GONCALVES DO REI FILHO
  • FELICIANO MARCILIO AGUIAR VITORIO
  • SAMUEL DA CRUZ CANEVARI
  • Data: Mar 24, 2022


  • Show Abstract
  • In this work, we present a classification of the arbitrary dimension and codimension umbilic submanifolds of Sn × R, based on the work of Bruno Mendonça and Ruy Tojeiro (2013). The classification that will be presented extends the classification of umbilical surfaces in S2 × R by Souam and Toubiana (2009), as well as the local description of umbilical hypersurfaces in Sn × R by Van der Veken and Vrancken (2008), to a codimension greater than one. In both works, the umbilical surfaces and the umbilical hypersurfaces are rotational. It will be shown that the submanifolds studied in this work are also rotational and we will present explicit parameterizations.

2
  • ANTONIO DEÍGERSON DA COSTA LOPES
  • On Ricci's Conditions for Immersions of
    Constant Mean Curvature of Free Boundary in
    Balls of Space Shapes.

  • Advisor : FELICIANO MARCILIO AGUIAR VITORIO
  • COMMITTEE MEMBERS :
  • NEWTON LUIS SANTOS
  • FELICIANO MARCILIO AGUIAR VITORIO
  • MARCIO HENRIQUE BATISTA DA SILVA
  • Data: Aug 26, 2022


  • Show Abstract
  • In this dissertation, we will investigate the Ricci conditions for immersions of constant mean curvature of free boundary in balls of space forms. Given a two-dimensional Riemannian manifold $\Sigma^{2}$ with a metric $ds^{2}$ whose Gaussian curvature is $K_{s}< H^{2}+c$, the Gregory Ricci-Curbastro condition is a necessary and sufficient condition for such an immersion to be isometrically minimally immersed or of constant mean curvature in a space form is that the new metric $d\Tilde{s}^{2}=\sqrt{-K_{s}+H^ {2}+c}ds^{2}$ be flat. If the condition for performing this immersion is that $\Sigma^{2}$ is simply connected, then we can perform the immersion induced by the metric $ds^{2}$ on balls of space forms. In the same vein, we saw that the existence of minimal immersions in $\mathbb{R}^{3}$, the Simons Type equation, for the three-dimensional case, is equivalent to the differential equation $K\bigtriangleup K - ||\nabla K ||^{2} - 4K^{3}=0$ with $K<0$. Adding such a condition to a minimal isometric immersion $f: {\Sigma}^{2} \rightarrow B^{n} $, with possible branch points and no umbility points, we show that after a possible codimension reduction, $f (\Sigma^{2})$ essentially arrives at $\mathbb{R}^{3}$ or essentially $\mathbb{R}^{6}$. We thus obtain an analytic version for the minimal immersion $f$ , with possible branch points, where $f(\Sigma^{2})$ meets $\partial B $ orthogonally, so that $f(\Sigma^{ 2})$ is totally umbilical.

3
  • CARLOS EDUARDO SOARES DE MARIA
  • f-extremal domains for the Laplacian operator

  • Advisor : MARCOS PETRUCIO DE ALMEIDA CAVALCANTE
  • COMMITTEE MEMBERS :
  • MARCOS PETRUCIO DE ALMEIDA CAVALCANTE
  • MARCIO HENRIQUE BATISTA DA SILVA
  • JOSÉ NAZARENO VIEIRA GOMES
  • Data: Aug 26, 2022


  • Show Abstract
  • In this dissertation, we investigate geometric properties of extremal domains, bounded or not. The extremal domains are the domains that are critical points for the first eigenvalue functional for volume-preserving variations.
    We show that these domains are characterized by admitting a non-trivial solution to an overdetermined Serrin-type problem.
    This motivates us to define the f-extremal domains, when we use an arbitrary function f as the nonlinearity of the overdetermined problem.
    The main tool used is the maximum principle in the format of the moving planes method and the main result is the Ros-Sicbaldi Theorem on the proof of Berestycki-Caffarelli-Nirenberg Conjecture in dimension two when the nonlinearity grows at least as a function linear.

4
  • JANDIR GOMES DE SOUZA TAVARES
  • Decay of Correlations for the Manneville-Pommeau Map

     
  • Advisor : RAFAEL NOBREGA DE OLIVEIRA LUCENA
  • COMMITTEE MEMBERS :
  • RAFAEL NOBREGA DE OLIVEIRA LUCENA
  • MARIA JOSÉ PACÍFICO
  • RAFAEL JOSE ALVAREZ BILBAO
  • Data: Dec 20, 2022


  • Show Abstract
  • In this work, we will study ergodic properties of the Maneville-Poumeau Map. More precisely, we will prove that such dynamics has an invariant probability, equivalent to the Lebesgue measure, whose density is locally Lipschitz. We will also prove that such a transformation has polynomial decay of correlations over the space L∞ and C1. To obtain the first result, we will build cones, with compactness properties, invariant by the action of the Transfer operator. For the second, we will use operator perturbation techniques. The results obtained in this work were developed by C. Liverani, B. Saussol and S. Vaienti in [5].

     
2021
Dissertations
1
  • MANUEL VINICIUS RIBEIRO LOPES LIMA
  • Index estimates for free boundary minimal hypersurfaces

  • Advisor : ABRAAO MENDES DO REGO GOUVEIA
  • COMMITTEE MEMBERS :
  • ABRAAO MENDES DO REGO GOUVEIA
  • MARCOS PETRUCIO DE ALMEIDA CAVALCANTE
  • DARLAN FERREIRA DE OLIVEIRA
  • Data: Mar 19, 2021


  • Show Abstract
  • In this dissertation, we present some results on index estimates for properly embedded free boundary minimal hypersurfaces in strictly mean convex domains of Euclidean space. The estimates described in this work were obtained by L. Ambrozio, A. Carlotto and B. Sharp in [2], in which they showed that the index of a minimal hypersurface with the properties described above is bounded from below by a linear function of the dimension of its first relative homology group. In three-dimensional ambients, the index of a free boundary minimal surface is bounded from below by a linear function of its genus and the number of boundary components.

2
  • FRANCISCO CLEONE NERES DE LIMA
  • ESTIMATING THE FIRST EIGENVALUE OF THE LAPLACIAN IN MINIMAL HYPERSURFACES

  • Advisor : CICERO TIARLOS NOGUEIRA CRUZ
  • COMMITTEE MEMBERS :
  • ABDENAGO ALVES DE BARROS
  • CICERO TIARLOS NOGUEIRA CRUZ
  • MARCIO HENRIQUE BATISTA DA SILVA
  • Data: Mar 30, 2021


  • Show Abstract
  • In the article "A first eigenvalue estimate for minimal hypersurfaces" H. Choi e A. N. Wang obtained a lower bound for the first eigenvalue of the  Laplacian of a compact orientable embedded minimal hypersurface  in an compact orientable manifold with  Positive Ricci curvature. In this dissertation, using covering space argument we prove this result dropping the orientability assumption. Moreover,  we use Reilly's Formula was used, which is actually a version obtained by integrating of Bochner's Formula.  Combining Choi and Wang's estimate  with  Yang and Yau's Theorem,  we  found a upper bounds estimate for the area of an embedded minimal surface in S3 only depending on its topology, more precisely only in terms of the genus of the surface.

3
  • ALLAN KENEDY SANTOS SILVA
  • Estimates of volume of minimal submanifolds in some symmetric spaces of rank one.

  • Advisor : MARCIO HENRIQUE BATISTA DA SILVA
  • COMMITTEE MEMBERS :
  • FÁBIO REIS DOS SANTOS
  • HENRIQUE FERNANDES DE LIMA
  • MARCIO HENRIQUE BATISTA DA SILVA
  • Data: Mar 31, 2021


  • Show Abstract
  • The theory of minimal surfaces came up with a problem proposed by Lagrange, which consisted

    of the following: given a closed curve without self-intersections, find the surface of

    smallest area that has that curve as boundary. Such a problem became known as

    the Plateau Problem. It took about 16 years from Lagrange's work to discover non-trivial

    examples of minimal surfaces due to Meusnier. The theory was stagnant for 60 years until

    Scherk found new examples of minimal surfaces. With the work of Weierstrass it was possible to

    obtain more examples of these surfaces. Thereafter there were major developments in theory,

    becoming one of the most fertile fields of Differential Geometry. One class of problems studied

    is that of estimate the volume of minimal submanifolds immersed in certain ambient

    manifolds, such as spheres, hyperplanes, projective spaces, etc. The objective of the present

    work is to provide lower bounds of volume of minimal compact submanifolds immersed

    in certain symmetrical spaces of rank 1, namely: the unitary sphere Sn, and the real projective space RPn,

    complex projective space CPn and quaternionic projective space HPn. It will be shown that if Mm is a

    minimal submanifold of Sn, then volM >=V_c(n;M) where V_c(n;M) is the n-conformal volume

    of M. Another estimate for this is volM>=c_n, where c_n = vol(Sn). In the case of

    M being immersed in projective spaces, we have the lower bounds: c_n/2 in RPm, c_{n+1}/2\pi in

    CPm and c_{n+2}=2\pi^2 in HPm.

4
  • CARLOS ALBERTO SANTOS BARBOSA
  • PARABOLICITY, SPACES OF HARMONIC FUNCTIONS AND TOPOLOGY AT INFINITY OF A COMPLETE MANIFOLD

  • Advisor : MARCOS RANIERI DA SILVA
  • COMMITTEE MEMBERS :
  • MARCOS RANIERI DA SILVA
  • FELICIANO MARCILIO AGUIAR VITORIO
  • MARCIO SILVA SANTOS
  • Data: Apr 16, 2021


  • Show Abstract
  • The aim of this work is to investigate the intrinsic relationship between certain spaces of harmonic functions on a complete manifold and their topology in the infinite. Assuming appropriate bounds on the Ricci curvature, we obtain estimates for the solutions of the Laplace and heat equations. This theory has important applications to geometry and topology of manifolds, some of which are presented here. In a manifold with more than one end, we have built a space of harmonic functions with remarkable properties. In turn, estimating the dimension of this space through geometric considerations will help us to understand the topology in the infinite of the manifold. More specifically, we will show a generalization of the celebrated Cheeger-Gromoll Splitting theorem.

5
  • CARLOS HENRIQUE DA SILVA
  • Characterizations of hypersurfaces with constant r-mean curvature

  • Advisor : GREGORIO MANOEL DA SILVA NETO
  • COMMITTEE MEMBERS :
  • GREGORIO MANOEL DA SILVA NETO
  • HILARIO ALENCAR DA SILVA
  • GREGORIO PACELLI FEITOSA BESSA
  • Data: Aug 26, 2021


  • Show Abstract
  • This thesis aims to present a proof of a theorem of Sebastián Montiel and Antonio Ros which characterizes the compact hypersurfaces, without boundary, with constant r-mean curvature embedded in the space forms. This theorem generalizes the classical theorem of Alexandrov and states that

    "The only compact, without boundary, hypersurfaces with constant r-mean curvature for some r=1,...,n, embedded in the Euclidean space, in the open hemisphere of the Euclidean sphere, or in the hyperbolic space, are the geodesic hyperspheres."


    We remark that, despite we follow the ideas of Montiel and Ros, we give a new approach for some steps of the proof presented here.

2020
Dissertations
1
  • DEIVID SANTOS DE ALMEIDA
  • Unicidade de Hipersuperfícies Capilares Estáveis em uma Bola

  • Advisor : ABRAAO MENDES DO REGO GOUVEIA
  • COMMITTEE MEMBERS :
  • ABRAAO MENDES DO REGO GOUVEIA
  • CICERO TIARLOS NOGUEIRA CRUZ
  • FRANCISCO VANDERSON MOREIRA DE LIMA
  • MARCOS PETRUCIO DE ALMEIDA CAVALCANTE
  • Data: Mar 13, 2020


  • Show Abstract
  • A. Ros e E. S. Vergasta, juntamente com I. Nunes em um trabalho independente, classificaram as superfícies CMC estáveis com bordo livre na bola euclidiana tridimensional. Neste trabalho iremos apresentar uma generalização feita por G. Wang e C. Xia para hipersuperfícies capilares estáveis em bolas geodésicas do espaço euclidiano, do espaço hiperbólico e da esfera unitária. A ideia principal da demonstração é a utilização de uma fórmula do tipo Minkowski que nos fornece uma família de funções testes importantes que são chaves na prova do teorema, pois anulam o termo do bordo na fórmula da segunda variação da energia. Concluímos a dissertação mostrando as alterações necessárias para resolver o problema exterior em um dos três modelos citados. Tomaremos como exemplo o caso hiperbólico.

2
  • NEMUEL ROCHA LIMA
  • Estabilidade de soluções do tipo Solitons para a equação generalizada de Korteweg-de Vries

  • Advisor : MARCIO CAVALCANTE DE MELO
  • COMMITTEE MEMBERS :
  • ADAN JOSE CORCHO FERNANDEZ
  • ISNALDO ISAAC BARBOSA
  • MARCIO CAVALCANTE DE MELO
  • RENAN DANTAS MEDRADO
  • ROBERTO DE ALMEIDA CAPISTRANO FILHO
  • Data: Mar 20, 2020


  • Show Abstract
  • Neste trabalho fazemos um estudo de estabilidade global das soluções especiais, chamadas
    solitons, com respeito ao problema de Cauchy (PVI) associado à equação não-linear de Korteweg-de
    Vries generalizada (gKdV), para p = 2; 3; 4.
    Para a prova do resultado de estabilidade, usamos a abordagem de Weinstein que foi revisada
    recentemente no artigo de Muñoz.

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