Generic Uniqueness of Isoperimetric Regions in Arbitrary Dimension

Abstract: Let $M<sup>{n+1}$ 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 $E<sup>c$. As consequences, for every $m&gt;0$, uniqueness holds for a generic set of metrics $g$ satisfying $m&lt;\operatorname{Vol}_g(M)$, and it also holds for a generic set of pairs $(g,m)$ with $0&lt;m&lt;\operatorname{Vol}_g(M)$. 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.