{
  "id": "2406.07509",
  "version": "v1",
  "published": "2024-06-11T17:43:18.000Z",
  "updated": "2024-06-11T17:43:18.000Z",
  "title": "Uniqueness on average of large isoperimetric sets in noncompact manifolds with nonnegative Ricci curvature",
  "authors": [
    "Gioacchino Antonelli",
    "Marco Pozzetta",
    "Daniele Semola"
  ],
  "comment": "40 pages. Comments welcome!",
  "categories": [
    "math.DG",
    "math.MG"
  ],
  "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 is 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 $M$, 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 $I_n\\subset (0,\\infty)$ such that isoperimetric sets with volumes $V\\in I_n$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-11T17:43:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonnegative ricci curvature",
      "large isoperimetric sets",
      "noncompact manifolds",
      "volume preserving stable boundaries",
      "uniqueness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}