Surfaces of least k dimensional area in R n {\textbf {R}^n} are constructed by minimization of the n dimensional volume of suitably thickened sets subject to a homological constraint. Specifically, let 1 ⩽ k ⩽ n 1 \,\, \leqslant \,\,k\,\, \leqslant \,n be integers and B ⊂ R n B\, \subset \,{\textbf {R}^n} be compact and k − 1 k\, - \,1 rectifiable. Let G be a compact abelian group and L be a subgroup of the Čech homology group H k − 1 ( B ; G ) {H_{k - 1}}\left ( {B;\,\,G} \right ) (in case k = 1 k = \,1 , suppose, additionally, L is contained in the kernel of the usual augmentation map). J. F. Adams has defined what it means for a compact set X ⊂ R n \textrm {X}\, \subset \,{\textbf {R}^n} to span L. Using also a natural notion of what it means for a compact set to be ε \varepsilon -thick, we show that, for each ε > 0 \varepsilon \, > \,0 , there exists an ε \varepsilon -thick set which minimizes n dimensional volume subject to the requirement that it span L. Our main result is that as ε \varepsilon approaches 0 a subsequence of the above volume minimizing sets converges in the Hausdorff distance topology to a set, X, which minimizes k dimensional area subject to the requirement that it span L. It follows, of course, from the regularity results of Reifenberg or Almgren that, except for a compact singular set of zero k dimensional measure, X is a real analytic minimal submanifold of R n {\textbf {R}^n} .
Read full abstract