{
  "id": "2405.08918",
  "version": "v1",
  "published": "2024-05-14T19:15:17.000Z",
  "updated": "2024-05-14T19:15:17.000Z",
  "title": "New spectral Bishop--Gromov and Bonnet--Myers theorems and applications to isoperimetry",
  "authors": [
    "Gioacchino Antonelli",
    "Kai Xu"
  ],
  "comment": "23 pages",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "We show a sharp and rigid spectral generalization of the classical Bishop--Gromov volume comparison theorem: if a closed Riemannian manifold $(M,g)$ of dimension $n\\geq3$ satisfies \\[ \\lambda_1(-\\gamma\\Delta+\\mathrm{Ric})\\geqslant n-1 \\] for some $0\\leqslant\\gamma\\leqslant\\frac{n-1}{n-2}$, then $\\mathrm{vol}(M)\\leqslant\\mathrm{vol}(\\mathbb S^{n})$, and $\\pi_1(M)$ is finite. Moreover, the bound on $\\gamma$ is sharp for this result to hold. A generalization of the Bonnet--Myers theorem is also shown under the same spectral condition. The proofs involve the use of a new unequally weighted isoperimetric problem and unequally warped $\\mu$-bubbles. As an application, in dimensions $3\\leqslant n\\leqslant 5$, we infer sharp results on the isoperimetric structure at infinity of complete manifolds with nonnegative Ricci curvature and uniformly positive biRicci curvature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-14T19:15:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bonnet-myers theorem",
      "spectral bishop-gromov",
      "application",
      "isoperimetry",
      "classical bishop-gromov volume comparison theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}