On the connectedness of a minimizing cluster's boundary

TL;DR

This paper proves that in certain simply connected homogeneous Riemannian manifolds, isoperimetric minimizing clusters have connected boundaries, ensuring connectedness of minimal sets and their complements, which is not true in more general settings.

Contribution

It establishes the connectedness of minimizing cluster boundaries in simply connected homogeneous Riemannian manifolds with at most one end, a novel geometric result.

Findings

Minimizing clusters have connected boundaries in specified manifolds.

Single-bubble isoperimetric minimizers have connected boundaries.

Connectedness fails without simple connectedness or end restrictions.

Abstract

We verify that an isoperimetric minimizing cluster on a simply connected homogeneous Riemannian manifold with at most one end always has connected boundary. In particular, the boundary of a single-bubble isoperimetric minimizer on such manifolds must be connected, and hence all isoperimetric sets and their complements must be connected. This is demonstrably false without the simple connectedness assumption or the restriction on the number of ends.