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

TL;DR

This work analyzes large-volume isoperimetric regions on complete manifolds with nonnegative Ricci curvature, Euclidean volume growth, and quadratic curvature decay, under the nontrivial condition that the manifold is not isometric to $\mathbb{R}^n$. By rescaling volumes and passing to blow-down cones, the authors connect isoperimetric regions to asymptotic cones and use stability and graph-approximation techniques to control perturbations of large isoperimetric boundaries. They prove that there exists a density-1 set $\mathcal{G}$ such that for every $V\in\mathcal{G}$ the isoperimetric region is unique and its boundary is strictly volume-preserving stable; they also show sharpness by constructing 2D counterexamples where large volumes fail uniqueness or strict stability. Central to the argument is an averaging principle for the Ricci curvature in normal directions, which allows effective Taylor expansions of the perimeter near competing isoperimetric boundaries and leads to a contradiction unless the regions coincide. The results illuminate how asymptotic cone structure and averaged curvature govern large-volume isoperimetric geometry in noncompact settings, and they establish a precise sense in which uniqueness holds on average rather than for all sufficiently 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.

Theorems & Definitions (45)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.3
• Theorem 2.1
• Lemma 2.2
• proof : Proof of \ref{['thm:counterexamplesmain']}
• proof : Proof of \ref{['lem:Technical']}
• Theorem 3.1
• Lemma 3.2
• Proposition 3.3