On the connectedness of a minimizing cluster's boundary

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.

💡 Research Summary

The paper investigates the topological structure of the boundary of an isoperimetric minimizing cluster in a homogeneous Riemannian manifold. The authors prove that if the ambient manifold (M,g) is smooth, complete, simply connected, homogeneous, and has at most one end (i.e., removing any compact set leaves a single unbounded connected component), then any isoperimetric minimizing cluster has a connected and bounded boundary Σ. In particular, for a single‑bubble (k=1) isoperimetric minimizer the set Ω₁, its complement M\Ω₁, and the interface ∂Ω₁ are all connected.

The paper begins by recalling basic notions: a homogeneous manifold is one on which the isometry group acts transitively; a q‑partition is a collection of pairwise disjoint Borel sets with locally finite perimeter covering M up to a null set; a k‑cluster is a q‑partition with an added “exterior cell”. The perimeter of a cluster is defined via the (n‑1)-dimensional Hausdorff measure of the reduced boundaries of the cells, and an isoperimetric minimizer is a cluster that minimizes this total perimeter under a prescribed volume vector.

The central topological ingredient is Theorem 1.5, which states that in a connected, simply connected, locally path‑connected metric space X, a finite family of open, connected, pairwise disjoint cells covering X cannot have a disconnected union of their boundaries unless one of the cells has a disconnected boundary that splits X into three disjoint pieces: the cell itself and the closures of two complementary unions of the remaining cells. This result is essentially a “2‑connected partition ⇒ connected boundary” principle and fails without simple connectivity (e.g., a cylinder split into vertical strips).

Armed with this, the authors prove Theorem 1.1. Starting from an isoperimetric minimizing cluster Ω, they refine it by replacing each cell with its connected components, obtaining a new cluster ˜Ω. Homogeneity guarantees that ˜Ω still has finitely many cells, and the “one‑end” hypothesis ensures that at most one cell can have infinite volume. Consequently ˜Ω remains a minimizing cluster with the same boundary Σ.

Assuming Σ were disconnected, Theorem 1.5 provides a cell ℓ whose boundary splits into two non‑empty parts ∂¯Ω_I and ∂¯Ω_J. Using the transitive isometry group, the authors construct a continuous family of isometries {φ_t} that moves one piece of ∂Ω_ℓ towards the other while preserving the volumes of all cells (the partition of indices I, ℓ, J guarantees volume preservation). During this motion the boundary pieces approach each other; the first time they touch, a strong maximum principle due to Ilmanen (refined by Wickramasekera) applies. This principle works not only on regular points of the interface but also on a certain class of singular points, yielding a contradiction to the minimality of the cluster. Hence Σ must be connected.

The paper also discusses counterexamples showing the necessity of the hypotheses. In dimension one, the real line (two ends) or the circle S¹ (non‑simply connected) admit k‑bubble minimizers with disconnected boundaries. In higher dimensions, products N^{n‑1}×ℝ with N^{n‑1} compact provide examples where large‑volume single‑bubble minimizers are slabs N^{n‑1}×