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

TL;DR

This paper demonstrates that in certain noncompact manifolds with nonnegative Ricci curvature, large isoperimetric sets are mostly unique and stable, but exceptions exist at specific large volumes.

Contribution

It establishes the near-uniqueness and stability of large isoperimetric sets in manifolds with specific curvature and volume growth conditions, and constructs examples where these properties fail.

Findings

Existence of a set of volumes with density 1 at infinity where isoperimetric sets are unique.

Most large isoperimetric sets have strictly volume preserving stable boundaries.

Counterexamples show that uniqueness and stability do not hold for all large volumes.

Abstract

Let $(M^n,g)$ be a complete Riemannian manifold which is not isometric to $\mathbb{R}^n$, has nonnegative Ricci curvature, Euclidean volume growth, and quadratic Riemann curvature decay. We prove that there exists a set $\mathcal{G}\subset (0,\infty)$ with density $1$ at infinity such that for every $V\in \mathcal{G}$ 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 $I_n\subset (0,\infty)$ such that isoperimetric sets with volumes $V\in I_n$ exist, but they are neither unique nor do they have strictly volume preserving stable boundaries.