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

Abstract

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