Uniqueness on average of large isoperimetric sets in noncompact manifolds with nonnegative Ricci curvature

• Published in: Communications on pure and applied mathematics Vol. 78; no. 9; pp. 1656 - 1702
• Main Authors: Antonelli, Gioacchino, Pozzetta, Marco, Semola, Daniele
• Format: Journal Article
• Language: English
• Published: New York John Wiley and Sons, Limited 01.09.2025
• Subjects: Curvature; Riemann manifold; Riemann surfaces; Uniqueness
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.22252

Let (Mn,g)$(M^n,g)$ be a complete Riemannian manifold which is not isometric to Rn$\mathbb {R}^n$, has nonnegative Ricci curvature, Euclidean volume growth, and quadratic Riemann curvature decay. We prove that there exists a set G⊂(0,∞)$\mathcal {G}\subset (0,\infty)$ with density 1 at infinity such that for every V∈G$V\in \mathcal {G}$ there is a unique isoperimetric set of volume V$V$ in M$M$; moreover, its boundary is strictly volume preserving stable. The latter result cannot be improved to uniqueness or strict stability for every large volume. Indeed, we construct a complete Riemannian surface satisfying the previous assumptions and with the following additional property: there exist arbitrarily large and diverging intervals In⊂(0,∞)$I_n\subset (0,\infty)$ such that isoperimetric sets with volumes V∈In$V\in I_n$ exist, but they are neither unique nor do they have strictly volume preserving stable boundaries.