Existence of singular isoperimetric regions in 8-dimensional manifolds. (English) Zbl 07872137
A well-know result states that local minimizers of the perimeter are smooth up to a closed, singular set of codimension \(8\). This is known to be sharp in virtue of the Simons cone example in \(\mathbb{R}^8\).
Consider now sets that minimize the perimeter with a volume constraint (i.e., isoperimetric sets). In the Euclidean setting, it is well-known that the only isoperimetric sets are balls, thus no isoperimetric set with singularities exists. In general in a space form, all isoperimetric sets, regardless of the given dimension, are smooth. In this paper the author exhibits isoperimetric sets with singularities in smooth, closed \(C^\infty\) Riemannian manifolds of dimension \(8\), see Theorem 1.1. It remains open whether such isoperimetric sets with singularities exist for real analytic Riemannian manifolds.
Consider now sets that minimize the perimeter with a volume constraint (i.e., isoperimetric sets). In the Euclidean setting, it is well-known that the only isoperimetric sets are balls, thus no isoperimetric set with singularities exists. In general in a space form, all isoperimetric sets, regardless of the given dimension, are smooth. In this paper the author exhibits isoperimetric sets with singularities in smooth, closed \(C^\infty\) Riemannian manifolds of dimension \(8\), see Theorem 1.1. It remains open whether such isoperimetric sets with singularities exist for real analytic Riemannian manifolds.
Reviewer: Giorgio Saracco (Firenze)
Keywords:
isoperimetric sets; smooth Riemannian manifolds; singular set; volume-constrained area minimizersReferences:
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