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 Čech 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 $0<d<n$.