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

2024/06/11 by Antonelli, Gioacchino, Pozzetta, Marco, Semola, Daniele
#Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2406.07509

Abstract

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 exists 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; 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⊂ (0,∞) such that isoperimetric sets with volumes V∈ I_n exist, but they are neither unique nor do they have strictly volume preserving stable boundaries.

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