Solutions to the Reifenberg Plateau problem with cohomological spanning conditions

TL;DR

This paper establishes existence and regularity results for minimizers of a generalized Plateau problem in higher dimensions, using cohomological boundary conditions to extend classical methods and provide a dual perspective.

Contribution

It introduces cohomological spanning conditions and generalizes previous methods to prove existence and regularity of minimizers for the Reifenberg Plateau problem.

Findings

Existence of minimizers with regularity properties.

Development of cohomological spanning conditions.

Extension of classical methods to higher dimensions.

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.