Solutions to the Reifenberg Plateau problem with cohomological spanning conditions

Abstract

We prove existence and regularity of minimizers for H\"older densities over general surfaces of arbitrary dimension and codimension in \(\R^n \), satisfying a cohomological boundary condition, providing a natural dual to Reifenberg's Plateau problem. We generalize and extend methods of Reifenberg, Besicovitch, and Adams, in particular we generalize a particular type of minimizing sequence used by Reifenberg (whose limits have nice properties, including lower bounds on lower density and finite Hausdorff measure,) prove such minimizing sequences exist, and develop cohomological spanning conditions. Our cohomology lemmas are dual versions of the homology lemmas in the celebrated appendix by Adams found in Reifenberg's 1960 paper.