Generic Uniqueness of Isoperimetric Regions in Arbitrary Dimension

Let $M^{n+1}$ be a closed, connected smooth manifold, where $n\ge 1$. For each prescribed volume fraction $s\in (0,1)\setminus \{\frac{1}{2}\}$, we prove that the isoperimetric region of volume $s{\mathrm{Vol}}_g(M)$ 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 $m>0$, uniqueness holds for a generic set of metrics $g$ satisfying $m<{\mathrm{Vol}}_g(M)$, and it also holds for a generic set of pairs $(g,m)$ with $0<m<{\mathrm{Vol}}_g(M)$. 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.