Existence of singular isoperimetric regions in 8-dimensional manifolds

Abstract

It is well known that isoperimetric regions in a smooth compact \((n+1)\)-manifold are themselves smooth, up to a closed set of codimension at most 8. In this note, we construct an 8-dimensional compact smooth manifold whose unique isoperimetric region with half volume that of the manifold exhibits two isolated singularities. This stands in contrast with the situation in which a manifold is a space form, where isoperimetric regions are smooth in every dimension.