Minimal surfaces

Abstract

[en] In the present work we define the concept of Minimal Surface and prove some important results related to it. To begin with, we review some elementary definitions and results of differential geometry. Then, we study normal variations of curves and surfaces and solve some optimisation problems as examples of this techniques. Afterwards, we define Minimal Surface and prove a theorem relating Minimal Surfaces and normal variations of surfaces. The next section is dedicated to graph surfaces and in it we prove Jörgen’s Theorem and Bernstein’s Theorem. Finally, we extend the definitions introduced to a higher number of dimensions, study the cone in three and four dimensions and give a brief account of the history of Bernstein’s theorem and its generalization to higher dimensions.