Superfícies de Curvatura Média Constante no Espaço Euclidiano

• Ano de defesa: 2019
• Autor(a) principal: Santos, José Ramos Araujo dos
• Orientador(a): Barreto, Alexandre Paiva
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: por
• Instituição de defesa: Universidade Federal de São Carlos; Câmpus São Carlos
• Programa de Pós-Graduação: Programa de Pós-Graduação em Matemática - PPGM
• Palavras-chave em Português: Superfícies mínimas; Superfícies de Curvatura Média Constante; Teorema de Representação de Enneper-Weierstrass; Teorema de Jorge-Xavier; Teorema Rosenberg-Toubiana; Teorema de Osserman; Teorema do Semi-espaço; Curvatura Total Finita; Teorema de Heinz; Teorema de Delaunay; Teorema de Estabilidade da Esfera; Superfícies de Weingarten
• Palavras-chave em Inglês: Minimal Surfaces; Surfaces of Constant Mean Curvature; Enneper-Weirstrass Representation Theorem; Jorge-Xavier's Theorem; Rosenberg-Toubiana Theorem; Osserman's Theorem; Semi-space Theorem; Finite Total Curvature; Heinz's Theorem; Delaunay's Theorem; Sphere Stability Theorem; Weingarten Surfaces
• Área do conhecimento CNPq: CIENCIAS EXATAS E DA TERRA::MATEMATICA::GEOMETRIA E TOPOLOGIA
• Link de acesso: https://repositorio.ufscar.br/handle/20.500.14289/11145

This paper deals with the surfaces of constant mean curvature in the Euclidean space. The first part of the text is devoted to minimal surfaces. We begin our studies with the Enneper-Weirstrass Representation Theorem and discuss some of its most important applications such as Jorge-Xavier, Rosenberg-Toubiana, and Osserman Theorems. Next, we present the Principle of Tangency of Fontenele-Silva and use it to demonstrate the classical half-space Theorem. We close this part by discussing the topological constraints imposed by the hypothesis of finite total curvature. In the second part of the manuscript we studied the surfaces of constant mean curvature, possibly non-zero. We start with Heinz's Theorem and its applications, we present the classification theorem of the surfaces of rotation with constant mean curvature made by Delaunay, and we conclude with the concept of stability where we demonstrate the classical Sphere Stability Theorem. We close the text with a succinct presentation of recent results on the surfaces of Weingarten in the Euclidean space.