http://repositorio.ufc.br/handle/riufc/65 2026-04-12T17:18:37Z 2026-04-12T17:18:37Z Coimbra, Caio Adler Scalser http://repositorio.ufc.br/handle/riufc/85746 2026-04-10T17:17:23Z 2026-01-01T00:00:00Z

Título: ρ-Einstein solitons and critical metrics of the volume functional on complete manifolds Autor(es): Coimbra, Caio Adler Scalser Abstract: The purpose of this work is to investigate analytic and geometric properties of ρ-Einstein solitons and V -static metrics on complete Riemanninan manifolds. In the first part, we study geometric and analytical features of complete non-compact ρ-Einstein solitons, which are self-similar solutions of the Ricci–Bourguignon flow. We study the spectrum of the drifted Laplacian operator for complete gradient shrinking ρ-Einstein solitons. Moreover, similar to classical results due to Calabi–Yau and Bishop for complete Riemannian manifolds with nonnegative Ricci curvature, we prove new volume growth estimates for geodesic balls of complete noncompact ρ-Einstein solitons. In particular, the rigidity case is discussed. In addition, we establish weighted volume growth estimates for geodesic balls of such manifolds. For the second part, we investigate critical metrics of the volume functional (V -static metrics) on complete manifolds without boundary. We prove that every critical metric of the volume functional on a connected complete manifold with parallel Ricci tensor is isometric to one of the standard models. Moreover, we show that a Bach-flat critical metric of the volume functional on a complete, simply connected manifold with proper potential function is isometric to one of the following: the standard sphere Sn, Euclidean space Rn, hyperbolic space Hn, or a warped product R×ϕ Σc, where Σc is a regular level set of the potential function. In particular, we establish classification results in dimensions three and four under weaker assumptions on the Bach tensor. Tipo: Tese

2026-01-01T00:00:00Z Gonçalves, Maria Jaciane Costa http://repositorio.ufc.br/handle/riufc/84370 2026-01-20T17:19:39Z 2025-01-01T00:00:00Z

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

2025-01-01T00:00:00Z Ibiapina, Allen Roossim Passos Ibiapina http://repositorio.ufc.br/handle/riufc/83532 2025-11-24T19:36:09Z 2023-01-01T00:00:00Z

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

2023-01-01T00:00:00Z Nascimento, João Paulo de Sousa http://repositorio.ufc.br/handle/riufc/83414 2025-11-13T17:50:01Z 2025-01-01T00:00:00Z

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

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