A topological characterization of indecomposable sets of finite perimeter

We prove that a set of finite perimeter is indecomposable if and only if it is, up to a choice of suitable representative, connected in the 1-fine topology. This gives a topological characterization of indecomposability which is new even in Euclidean spaces. Our approach relies crucially on the metric space theory of functions of bounded variation, and we are able to prove our main result in a complete, doubling metric measure space supporting a $1$-Poincaré inequality and having the two-sidedness property (this class includes all Riemannian manifolds, Carnot groups, and ${\sf RCD}(K,N)$ spaces with $K\in\mathbb R$ and $N<\infty$). As an immediate corollary, we obtain an alternative proof of the decomposition theorem for sets of finite perimeter into maximal indecomposable components.

💡 Research Summary

The paper establishes a precise topological characterization of indecomposable sets of finite perimeter in a very general metric‑measure setting. Working on a complete, doubling space (X,d,μ) that supports a 1‑Poincaré inequality (a PI space) and satisfies the two‑sidedness property, the authors prove that a set E with finite perimeter P(E,X) is indecomposable if and only if, after choosing the appropriate representative (the measure‑theoretic interior I E), its fine interior fine‑int I E is connected with respect to the 1‑fine topology. The fine topology is defined via 1‑capacity: a set A is 1‑thin at x if the relative capacity of A∩B(x,r) inside B(x,2r) becomes negligible compared with the capacity of the whole ball as r→0. The authors first develop the basic properties of this topology, showing it is locally connected and that fine‑open balls are “fine‑domains”.

Key technical tools include the theory of functions of bounded variation (BV) on metric spaces, the coarea formula linking total variation to perimeters of level sets, and several isoperimetric and Sobolev‑Poincaré inequalities that hold in PI spaces. The two‑sidedness property guarantees that for any two disjoint finite‑perimeter sets E and F, the common part of their reduced boundaries carries no “third” region; this is verified for Riemannian manifolds, Carnot groups, and RCD(K,N) spaces.

The proof of the main equivalence proceeds in two directions.

1. If E is indecomposable, any decomposition of fine‑int I E into two non‑empty 1‑fine open subsets would induce a decomposition of E into two sets whose perimeters add up to P(E,X), contradicting indecomposability. This uses the subadditivity of perimeter (P(F∩G)+P(F∪G)≤P(F)+P(G)) and the fact that the difference I E \ fine‑int I E is μ‑null (even H¹‑null).
2. Conversely, if fine‑int I E is 1‑fine connected, assume a non‑trivial decomposition E=F∪G with disjoint F,G of positive measure. By the fine‑connectedness, the fine interiors of I F and I G must be 1‑thin, which forces the capacities of the separating sets to vanish, leading to P(E,X) < P(F,X)+P(G,X), a contradiction.

An important corollary is a new proof of the classical decomposition theorem: every finite‑perimeter set can be uniquely written as a union of maximal indecomposable components, which are precisely the connected components of fine‑int I E in the 1‑fine topology. The paper also establishes a Cartan property for p=1, showing that points which are not 1‑thin admit fine‑open neighborhoods, reinforcing the robustness of the fine topology.

Overall, the work bridges measure‑theoretic concepts (indecomposability, BV, perimeter) with a refined topological framework (1‑fine connectivity), extending classical Euclidean results to a broad class of non‑smooth spaces. This provides a powerful tool for future investigations of geometric measure theory, minimal surfaces, and variational problems in metric‑measure settings where traditional topological notions are insufficient.