The Plateau problem is a classical problem in the Calculus of Variations, and it is concerned with the existence of surfaces of least area spanning a given boundary. Standard classes of smooth surfaces lack the necessary compactness properties which would guarantee the existence of solutions by direct methods. Therefore, the competition class for the Plateau problem needs to be suitably enlarged in order to contain all possible limits of minimizing sequences for the area functional. Guided by this principle, many alternative classes of generalized surfaces have been proposed within the last century. In this brief lecture, I will focus on the measure-theoretic approach to the problem, pioneered by Federer and Fleming in the 1960s with the introduction of the notion of (integer rectifiable) currents. Allard’s varifolds will also be introduced in order to discuss the notion of “generalized mean curvature” of a surface. These two classes will play a major role in the discussion of the main talk.
Minimal surfaces, i.e. surfaces whose mean curvature vanishes identically, are the classical model used to describe soap films hanging on a wire. The zero mean curvature condition can easily be derived by enforcing the balance of the pressure forces acting on the two sides of the film with the Laplace pressure induced by surface tension, under the critical assumption that gravity effects on the geometry of the resulting surface are negligible. As demonstrated by experiments, this model fails to be accurate when the characteristic length scale of the film is large. In these cases, gravitational forces play a role in determining the shape of macroscopic surfaces, and the film is in fact an almost-minimal surface, that is a surface with small (but non-zero) mean curvature. In this talk, after discussing the “gravity-enforced” physical model leading to the notion of almost-minimal surfaces, I will tackle the following problem: is the theory of minimal surfaces spanning a given boundary wire powerful enough to describe all possible limits of almost-minimal surfaces as the mean curvature of the latter approaches zero? I will discuss the problem both from a qualitative and quantitative point of view, providing sufficient conditions on the boundary wire under which the answer is positive.