Superfícies de Curvatura Média Constante no Espaço Euclidiano

Abstract

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.