Generic Uniqueness of Isoperimetric Regions in Arbitrary Dimension

Abstract

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.

arXiv comment16 pages. All comments are welcome!