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arXiv · 2212.04752

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

Michael Goldman·Benoît Merlet
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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.

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Michael Goldman, Benoît Merlet. 2024-11-04. Set-decomposition of normal rectifiable G-chains via an abstract decomposition principle. https://doi.org/10.4171/rmi%2F1504

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