Existence of singular isoperimetric regions

TL;DR

This paper constructs examples of high-dimensional smooth manifolds where the unique isoperimetric regions have isolated singularities, challenging the typical smoothness results in geometric measure theory.

Contribution

It provides explicit constructions of manifolds with isoperimetric regions exhibiting singularities, extending understanding of regularity limits in high dimensions.

Findings

Existence of 8-dimensional manifolds with singular isoperimetric regions

Construction of manifolds with isoperimetric regions having singular sets of positive dimension

Examples demonstrating singularities in isoperimetric regions in dimensions ≥7

Abstract

It is well known that isoperimetric regions in a smooth compact $(n+1)$-manifold are smooth, up to a closed set of codimension at most $6$. In this note, we first construct an $8$-dimensional compact smooth manifold whose unique isoperimetric region with half volume that of the manifold exhibits two isolated singularities. And then, for $n\geq 7$, using Smale's construction of singular homological area minimizers for higher dimensions, we construct a Riemannian manifold such that the unique isoperimetric region of half volume, with singular set the submanifold $\S^{n-7}$.