https://ri.ufs.br/jspui/handle/riufs/2423 2026-04-08T06:35:47Z https://ri.ufs.br/jspui/handle/riufs/24695 Título: Prescrição de curvaturas gaussiana e geodésica em superfícies compactas com característica de Euler não positiva Autor(es): Santana, Junior Tavares de Abstract: This dissertation, based on the article [30], addresses the problem of prescribing Gaussian and geodesic curvatures on a compact Riemannian surface with boundary (Σ, g). The main goal of this work is to prove, in broad terms, the existence of a conformal metric g = e u g such that the Gaussian and geodesic curvatures with respect to g are prescribed. The problem is reduced to finding a solution for a second-order elliptic partial differential equation with boundary conditions. Furthermore, the work explores the energy functional associated with this problem and its variational properties, addressing issues of coercivity and the existence of minimizers. The study focuses on different scenarios, including cases where the Euler characteristic of Σ is negative or zero. Additionally, there exists a function defined on ∂Σ, denoted by D, which plays a fundamental role in the study. When D(q) > 1 for some point q E ∂Σ, the analysis becomes more delicate. The work employs concepts from Riemannian Geometry, Functional Analysis, and Partial Differential Equations to develop the proofs of the main theorems. Moreover, advanced methods such as blow-up analysis and the Morse index of solutions are used to handle more refined situations. 2025-03-07T00:00:00Z https://ri.ufs.br/jspui/handle/riufs/23343 Título: Um estudo sobre a transcendência de períodos de funções elípticas Autor(es): Santiago, Angelina Rodrigues Abstract: This work is based on the book ”Transcendental Numbers”by M. Ram Murty and Puru- sottam Rath and aims to present results that intertwine studies on elliptic functions and the theory of transcendental numbers, proving that it is possible to obtain transcendental numbers from the periods of elliptic functions. Initially, we review some preliminary results from complex analysis, with emphasis on properties of holomorphic functions, power series, and the residue theorem, as well as fundamentals of algebraic number theory, including algebraic numbers, symmetric polynomials, and Siegel’s lemma. Subsequently, we address the emergence of the theory of transcendental numbers, highlighting historical milestones such as Liouville’s Theorem, and later results (such as Hermite’s and Lindemann’s proofs of the transcendence of the number e and π, respectively). Finally, we explore the deep connection between the theory of transcendental numbers and the theory of elliptic functions, evidenced by the study of the periods of these functions. With this, the work seeks to illustrate how objects of an analytical nature, such as elliptic functions and their periods, contribute significantly to advancing the understanding of the transcendence of certain numbers, unifying ideas from different areas of mathematics. 2025-08-29T00:00:00Z https://ri.ufs.br/jspui/handle/riufs/23249 Título: Um problema de fronteira livre para equações elípticas na forma divergente Autor(es): Ramos, Samorane de Jesus Abstract: Based on the article Cavity Problems in Discontinuous Media, we investigate regularity properties along the free boundary for minimizers of the functional associated with the cavity problem. Our main goal is to show that such functions are Lipschitz continuous on this set. The approach adopted consists in establishing this property for uniform limits of sequences of minimizers of regularized functionals, so that the result for arbitrary minimizers follows from an analogous argument. To this end, we obtain a regularity estimate that also implies geometric properties of the free boundary of the limit functions. 2025-07-31T00:00:00Z https://ri.ufs.br/jspui/handle/riufs/23173 Título: Estimativa do tensor de Ricci sobre dados discretos Autor(es): Rodrigues, Lays Vanessa Santana Abstract: The study of geometry in high-dimensional spaces, particularly in point clouds, presents challenges in the calculation and interpretation of geometric properties, such as the metric tensor and Ricci curvature. This work proposes an approach that integrates concepts from Riemannian geometry with machine learning tools to estimate these tensors in discrete data, building upon a finite difference-based method and extending it through an adaptation employing neural networks. The methodology was experimentally validated on two-dimensional surfaces, yielding results consistent with those of the reference work—that is, accurate estimates on surfaces with positive and constant curvature, but with limitations on surfaces with negative curvature. Meanwhile, experiments with the neural network-based adaptation showed inferior results when compared to the original method, though they were comparable, in certain aspects, to other exis- ting approaches, such as Ollivier-Ricci and Forman-Ricci curvatures. These results demonstrate the feasibility of the proposed approach and highlight important challenges to be addressed in future work. 2025-08-28T00:00:00Z