Set-decomposition of normal rectifiable G-chains via an abstract decomposition principle

TL;DR

This paper introduces a new set-decomposition method for normal G-flat chains, extending previous decompositions of sets and currents without assuming G is boundedly compact, using an abstract principle and isoperimetric inequality.

Contribution

It generalizes the decomposition of rectifiable G-chains without the boundedly compact assumption on G, via a novel abstract decomposition principle.

Findings

Decomposition of normal G-flat chains into set-indecomposable sub-chains.

Extension of measure-theoretic connected components to G-chains.

Use of a new isoperimetric inequality based on h-mass.

Abstract

We introduce the notion of set-decomposition of a normal G-flat chain. We show that any normal rectifiable $G$-flat chain admits a decomposition in set-indecomposable sub-chains. This generalizes the decomposition of sets of finite perimeter in their ``measure theoretic'' connected components due to Ambrosio, Caselles, Masnou and Morel. It can also be seen as a variant of the decomposition of integral currents in indecomposable components by Federer.As opposed to previous results, we do not assume that G is boundedly compact. Therefore we cannot rely on the compactness of sequences of chains with uniformly bounded N-norms. We deduce instead the result from a new abstract decomposition principle. As in earlier proofs a central ingredient is the validity of an isoperimetric inequality. We obtain it here using the finiteness of some h-mass to replace integrality.