基于抽象分解原理的正规可求长G-链的集合分解
Set-decomposition of normal rectifiable G-chains via an abstract decomposition principle
AI总结：
该研究引入正规G-平坦链的集合分解概念，借助新的抽象分解原理，在不假设G有界紧的条件下，证明正规可求长G-平坦链可分解为集合不可分解子链，推广了有限周长集与积分流的相关分解结果。
摘要：
我们引入了正规G-平坦链的集合分解概念。研究表明，任意正规可求长$G$-平坦链都可分解为集合不可分解的子链。这一结论推广了Ambrosio、Caselles、Masnou与Morel提出的有限周长集按其“测度论”连通分量的分解结果，也可视为Federer提出的积分流按不可分解分量分解的一种变体。与此前结果不同的是，我们未假设G是有界紧的，因此无法依赖N-范数一致有界的链序列的紧性，而是通过一个新的抽象分解原理推导出该结论。与早期证明类似，等周不等式的成立是核心要素，我们在此利用某类h-质量的有限性来代替整性，从而得到该不等式。
英文摘要：
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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