On the connectedness of a minimizing cluster's boundary

2025/12/17 by Milman, Emanuel, Neeman, Joe
Mathematics · Computer Science · Engineering · #Geometric Analysis and Curvature Flows #Topological and Geometric Data Analysis #3D Shape Modeling and Analysis

paper · doi:10.48550/arxiv.2512.15403

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.

Citations

• Isoperimetric Inequalities on Slabs with applications to Cubes and Gaussian Slabs
• Isoperimetric inequalities in cylinders with density
• The Gaussian Double-Bubble and Multi-Bubble Conjectures
• The singular set of minimal surfaces near polyhedral cones
• Soap bubbles and isoperimetric regions in the product of a closed manifold with Euclidean space
• A general regularity theory for stable codimension 1 integral varifolds
• Some isoperimetric comparison theorems for convex bodies in Riemannian manifolds

Related

• Connectedness properties of small minimal clusters in Riemannian or Finsler manifolds
• Quantitative estimate on singularities in isoperimetric clusters
• Improved convergence theorems for bubble clusters. I. The planar case
• Isoperimetric Regions in Nonpositively Curved Manifolds
• Minimising hulls, p-capacity and isoperimetric inequality on complete Riemannian manifolds