Introdução às superfícies mínimas

Abstract

The theory of minimal surfaces is a fascinating subject that, despite being quite old, is able to combine several areas of mathematics. Initial studies date back to the 1760s with investigations done by Joseph-Louis Lagrange. In short, he studied the problem of finding the surface with the smallest possible area whose edge was a given Jordan curve. We start with a review of basic concepts such as the tangent plane, first and second fundamental forms of parameterized surfaces. Then, we will enter the central object of study of this work, the minimal surfaces. We will develop the Ennerper-Weierstrass representation and it’s geometric consequences thatthat complete the tools to approach global aspects of this theory such as Bernstein’s theorem. Therefore, the objective of this work is to present and investigate several important examples, in addition to exposing and demonstrating the most classic results of the theory.