Plateau's Problem for Integral Currents

In this thesis, we address Plateau’s problem for generalized surfaces in ℝ^N of any dimension. To achieve this, we introduce the notion of currents, which are continuous linear functionals on spaces of differential forms with compact support. Currents were first introduced by Federer and Fleming in the 1960s and have proven to be a natural framework for formulating extremal problems in geometry. Using the direct method of the calculus of variations, the existence of solutions to Plateau’s problem is easily proved within the class of normal currents. However, normal currents are far away from submanifolds of ℝ^N. Therefore, we need to consider Integral currents, which are much closer to smooth surfaces. The existence of solutions to the Plateau’s problem then follows from the deformation theorem and the compactness theorem of Integral currents. Moreover, in the Introduction of the thesis, we briefly mention the currents in metric spaces developed by Ambrosio and Kirchheim, which provide a robust framework for studying surfaces in the broader context of metric spaces.