The regularity of minimal surfaces in higher codimension

Abstract

The Plateau’s problem investigates those surfaces of least area spanning a given contour. It is one of the most classical problems in the calculus of variations, it lies at the crossroad of several branches of mathematics and it has generated a large amount of mathematical theory in the last one hundred years. The problem itself and its various generalizations have found fundamental applications in several mathematical and scientific branches. Since it is a prototype of a vast family of questions with geometric and physical significance, the techniques developed to analyze it have proved to be very useful in a variety of other situations. The original formulation is attributed to the Belgian physicist Plateau, although it was considered earlier by Lagrange, and it regards 2-dimensional surfaces spanning a given onedimensional contour γ in the 3-dimensional space: among these surfaces one is interested in those which minimize the area (and, more in general, on the critical points of the area, although in this survey we will restrict our attention to “absolute” minimizers). Plateau considered such “minimal surfaces” to model soap films. However it is very natural to generalize the question and look for surfaces of dimension m which minimize the mdimensional volume among those spanning a given contour of dimensionm−1 in R, or in more general ambient spaces. Such generalizations have not only an intrinsic mathematical beauty, but they have proved to be very fruitful. In this note we will restrict ourselves to ambient spaces which are complete oriented Riemannian manifolds Σ and since all the considerations will be of a local nature we will often assume that Σ itself is isometrically embedded in some euclidean space (of dimensionm+n). In this way the competitor surfaces (classical or generalized) spanning the contour γ will always be (suitable generalizations of) subsets of the standard euclidean space, constrained to be subsets of Σ. Although this is not very elegant from a geometric point of view, it allows us to avoid a lot of technicalities.