Existence of Minimizers for the Reifenberg Plateau problem

Abstract

That is, given a compact set $B \subset \mathbb{R}^n$ (the boundary) and a subgroup $L$ of the \v{C}ech homology group $\check{H}_{d-1}(B;G)$ of dimension $d$ over some commutative group $G$, we find a compact set $E \supset B$ such that the image of $L$ by the natural map $\check{H}_{d-1}(B;G)\to\check{H}_{d-1}(S;G)$ induced by the inclusion $B \to E$, is reduced to $\{ 0 \}$, and such that the Hausdorff measure $\mathcal{H}^{d}(E \setminus B)$ is minimal under these constraints. Thus we have no restriction on the group $G$ or the dimensions $0 < d < n$.