{"ID":23551630,"CreatedAt":"2026-09-18T04:14:56.806955139Z","UpdatedAt":"2026-09-18T04:14:56.806955139Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20790","arxiv_id":"2609.20790","title":"Generic Uniqueness of Isoperimetric Regions in Arbitrary Dimension","abstract":"Let $M^{n+1}$ be a closed, connected smooth manifold, where $n\\geq 1$. For each prescribed volume fraction $s\\in(0,1)\\setminus\\{\\frac12\\}$, we prove that the isoperimetric region of volume $s \\operatorname{Vol}_g(M)$ is unique for a generic set of smooth Riemannian metrics $g$. At half volume, a generic metric has exactly two minimizers, a region $E$ and its complement $E^c$. As consequences, for every $m\u003e0$, uniqueness holds for a generic set of metrics $g$ satisfying $m\u003c\\operatorname{Vol}_g(M)$, and it also holds for a generic set of pairs $(g,m)$ with $0\u003cm\u003c\\operatorname{Vol}_g(M)$. The proof is variational and requires neither boundary regularity nor nondegeneracy of the constrained Jacobi operator. In particular, the results apply even when isoperimetric boundaries are singular.","short_abstract":"Let $M^{n+1}$ be a closed, connected smooth manifold, where $n\\geq 1$. For each prescribed volume fraction $s\\in(0,1)\\setminus\\{\\frac12\\}$, we prove that the isoperimetric region of volume $s \\operatorname{Vol}_g(M)$ is unique for a generic set of smooth Riemannian metrics $g$. At half volume, a generic metric has exac...","url_abs":"https://arxiv.org/abs/2609.20790","url_pdf":"https://arxiv.org/pdf/2609.20790v1","authors":"[\"Gongping Niu\"]","published":"2026-09-17T17:53:39Z","proceeding":"math.DG","tasks":"[\"math.DG\"]","methods":"[]","has_code":false}