2023/02/14 by Gongping Niu, Niu, Gongping
Computer Science · Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2302.06910
openalex publication_date 2023/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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≥ 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 §n-7.