http://repositorio.ufc.br/handle/riufc/65 Mon, 06 Apr 2026 12:19:56 GMT 2026-04-06T12:19:56Z http://repositorio.ufc.br/handle/riufc/84370 Título: Desigualdades geométricas para variedades quasi-Einstein Autor(es): Gonçalves, Maria Jaciane Costa Abstract: In this work, we establish geometric inequalities for m-quasi-Einstein manifolds, inspired by classical results such as the isoperimetric, Penrose, and Heintze–Karcher inequalities. In particular, we derive estimates for the area of the boundary of compact m-quasi-Einstein manifolds in terms of the first eigenvalue of the Laplace and Jacobi operators, improving previous result sin [32]. Using the generalized Reilly formula obtained by Qiu and Xia [73], we prove an isoperimetric-type inequality for compact m-quasi-Einstein manifolds with boundary, constant scalar curvature, and m > 1. As an application of the boundary area estimate in terms of the first eigenvalue of the Jacobi operator, we establish a Penrose-type inequality for the Hawking mass of the boundary of a three-dimensional compact m-quasi-Einstein manifold. Finally, by applying a generalization of Reilly’s formula due to Li and Xia [51], we obtain a Heintze–Karcher-type inequality valid for compact domains with connected boundary contained in the interior of an m-quasi-Einstein manifold. Tipo: Tese Wed, 01 Jan 2025 00:00:00 GMT http://repositorio.ufc.br/handle/riufc/84370 2025-01-01T00:00:00Z http://repositorio.ufc.br/handle/riufc/83532 Título: Menger’s theorem and related problems on temporal graphs. Autor(es): Ibiapina, Allen Roossim Passos Ibiapina Abstract: Temporal graphs are an important tool for modeling time-varying relationships in dynamic systems. Among the problems of interest, paths and connectivity problems have been the ones that have attracted the most attention of the community. The famous Menger’s Theorem says that, for every pair of non-adjacent vertices s,z in a graph G, the maximum number of internally vertex disjoint s,z-paths in G is equal to the minimum number of vertices in G − {s,z} whose removal destroys all s,z-paths (called and s,z-cut). This combined with flow techniques also leads to polynomial time algorithms to compute disjoint paths and cuts. A natural question is therefore whether such results carry over to temporal graphs, where the paths/walks between a pair of vertices must respect the flow of time. The first obstacle in such question is the definition of the robustness in context, i.e., what it means for paths/walks to be disjoint. In this thesis, we investigate three possible types of robustness, each of which leading to the definition of two optimization parameters, one concerning the maximum number of disjoint paths, and the other the minimum size of a cut. We give theoretical results about the validity of Menger’s Theorem and computational complexity results for the decision problems related to each parameter. Tipo: Tese Sun, 01 Jan 2023 00:00:00 GMT http://repositorio.ufc.br/handle/riufc/83532 2023-01-01T00:00:00Z http://repositorio.ufc.br/handle/riufc/83414 Título: Symbolic Dynamics for non-uniformly hyperbolic flows in high dimension Autor(es): Nascimento, João Paulo de Sousa Abstract: We construct symbolic dynamics for flows with positive speed in any dimension: for each χ > 0, we code a set that has full measure for every invariant probability measure which is χ–hyperbolic. In particular, the coded set contains all hyperbolic periodic orbits with Lyapunov exponent outside of [ − χ,χ]. This extends the recent work of Buzzi, Crovisier, and Lima for three dimensional flows with positive speed [15]. As an application, we code homoclinic classes of measures by suspensions of irreducible countable Markov shifts, and prove that each such class has at most one probability measure that maximizes the entropy. Tipo: Tese Wed, 01 Jan 2025 00:00:00 GMT http://repositorio.ufc.br/handle/riufc/83414 2025-01-01T00:00:00Z http://repositorio.ufc.br/handle/riufc/83324 Título: Estimativa de Carleson para o g-Laplaciano em domínios NTA Autor(es): Sousa, José Wálisson Vieira de Abstract: This thesis investigates the Carleson estimate, a fundamental result in Harmonic Analysis, Potential Theory, and the Theory of Partial Differential Equations (PDEs), initially established by Lennart Carleson and proven essential for various fields. Generally, the Carleson estimate establishes an upper bound for positive PDE solutions that vanish on a portion of a domain’s boundary. This control is given by the function’s value at a fixed point, distant from the boundary, and a universal constant that depends only on the domain’s geometry and the PDE’s parameters. We discuss the profound relationship between this concept and the Boundary Harnack Inequality (or Boundary Harnack Principle), a result involving the quotient of two functions that, under the same hypotheses as the Carleson estimate, guarantees these functions decay to zero at the same rate as we approach the domain’s boundary. Research in the area has advanced significantly in generalizing these results to nonlinear operators. In this context, our work focuses on the g-Laplacian, an operator that includes a wide class of nonlinear elliptic PDEs. Building on recent developments that obtained the Carleson estimate for inhomogeneous solutions of the g-Laplacian in semi-ball type domains, the main objective of this thesis is to extend this estimate to NTA domains, a substantially more general geometric class, which involve considerable technical challenges. To this end, we will prove essential ingredients such as a refinement of the Harnack Inequality for uniform domains and a De Giorgi-type oscillation lemma up to the boundary. Additionally, this thesis presents the Boundary Harnack Inequality as an application of the obtained Carleson estimate, solidifying the relationship between the two results in our new context. Finally, we explore another important application: the establishment of an exponential growth estimate for solutions of the g-Laplacian in unbounded cylindrical domains with an NTA base, generalizing analogous growth theorems for other domains and operators in the literature. Tipo: Tese Wed, 01 Jan 2025 00:00:00 GMT http://repositorio.ufc.br/handle/riufc/83324 2025-01-01T00:00:00Z