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

Let (M-n,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 exists a set G subset of(0,infinity) with density 1 at infinity such that for every V is an element of G there is a unique isoperimetric set of volume V in 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 I-n subset of(0,infinity) such that isoperimetric sets with volumes V is an element of I-n exist, but they are neither unique nor do they have strictly volume preserving stable boundaries.