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

TL;DR

This paper constructs specific partitions of Euclidean space that are locally area-minimizing and resemble singular cones at infinity, demonstrating non-uniqueness of certain minimal clusters in high dimensions.

Contribution

It introduces a method to build locally minimizing partitions that asymptotically resemble singular cones, proving non-uniqueness of the lens cluster in many dimensions.

Findings

Constructed partitions that blow down to singular cones.

Proved non-uniqueness of the lens cluster in dimensions starting from 8.

Showed existence of non-unique minimal clusters in high-dimensional spaces.

Abstract

We construct partitions of $\mathbb{R}^n$ into three sets $\{\mathscr{X}(1),\mathscr{X}(2),\mathscr{X}(3)\}$ 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 $8$.