A topological characterization of indecomposable sets of finite perimeter

TL;DR

This paper characterizes indecomposable sets of finite perimeter using the 1-fine topology, providing a new topological perspective that extends to various metric measure spaces including Riemannian manifolds and Carnot groups.

Contribution

It introduces a novel topological characterization of indecomposable sets of finite perimeter applicable in general metric measure spaces.

Findings

Characterization of indecomposability via 1-fine topology

Applicable to Riemannian manifolds, Carnot groups, and ${ m RCD}(K,N)$ spaces

Provides an alternative proof of the decomposition theorem

Abstract

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 $1$-Poincar\'{e} 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.