Partial regularity of almost minimizing rectifiable G chains in Hilbert space

TL;DR

This paper extends regularity results for almost minimizing rectifiable G chains to infinite-dimensional Hilbert spaces, showing the density of regular points under certain coefficient group conditions.

Contribution

It adapts Reifenberg's epiperimetric inequality and Preiss' second moments to infinite-dimensional settings for the first time.

Findings

Regular points are dense in the support of almost minimizing G chains in Hilbert spaces.

The results depend on the discreteness and closedness of the coefficient group G.

The methods bridge finite and infinite-dimensional geometric measure theory.

Abstract

We adapt to an infinite dimensional ambient space E.R. Reifenberg's epiperimetric inequality and a quantitative version of D. Preiss' second moments computations to establish that the set of regular points of an almost mass minimizing rectifiable $G$ chain in $\ell_2$ is dense in its support, whenever the group $G$ of coefficients is so that $\{\|g\| : g \in G \}$ is discrete and closed.