On the non-uniqueness of locally minimizing clusters via singular cones

2025/07/18 by Lia Bronsard, Robin Neumayer, Bronsard, Lia +5 · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2507.13995

openalex publication_date 2025/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct partitions of ℝⁿ into three sets \\mathscrX(1),\mathscrX(2),\mathscrX(3)\ that locally minimize interfacial area among compactly supported volume preserving variations and that blow down at infinity to singular area-minimizing cones. As a consequence, we prove the non-uniqueness of the standard lens cluster in a large number of dimensions starting from 8.

Citations

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Cited by

• Multi-Bubble Isoperimetric Problems
• Existence of a non-standard isoperimetric triple partition

Related

• The standard lens cluster in R² uniquely minimizes relative perimeter
• Quantitative estimate on singularities in isoperimetric clusters
• Nonlocal minimal clusters in the plane
• Locally isoperimetric partitions
• Spherical cones: classification and a volume minimization principle