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

Let (Mⁿ, g) be a complete Riemannian manifold which is not isometric to Rⁿ, has nonnegative Ricci curvature, Euclidean volume growth, and quadratic Riemann curvature decay. We prove that there is a set G (0, ) with density 1 at infinity such that for every V G there is a unique isoperimetric set of volume V in M, and 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 that satisfies the previous assumptions, and with the following property: there are arbitrarily large and diverging intervals Iₙ (0, ) such that isoperimetric sets with volumes V Iₙ exist, but they are neither unique nor they have strictly volume preserving stable boundaries. The proof relies on a set of new ideas, as the present setting goes beyond the range of applicability of the methods based on the implicit function theorem, and no symmetry is assumed.