Some new constructions and estimates in the problem of least area

Abstract:

Surfaces of least k dimensional area in ${\textbf {R}^n}$ are constructed by minimization of the n dimensional volume of suitably thickened sets subject to a homological constraint. Specifically, let $1 \leqslant k \leqslant n$ be integers and $B \subset {\textbf {R}^n}$ be compact and $k - 1$ rectifiable. Let G be a compact abelian group and L be a subgroup of the Čech homology group ${H_{k - 1}}\left ( {B; G} \right )$ (in case $k = 1$, 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 $\textrm {X} \subset {\textbf {R}^n}$ to span L. Using also a natural notion of what it means for a compact set to be $\varepsilon$-thick, we show that, for each $\varepsilon > 0$, there exists an $\varepsilon$-thick set which minimizes n dimensional volume subject to the requirement that it span L. Our main result is that as $\varepsilon$ 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 ${\textbf {R}^n}$.