{
  "id": "2102.05940",
  "version": "v1",
  "published": "2021-02-11T10:58:20.000Z",
  "updated": "2021-02-11T10:58:20.000Z",
  "title": "Limits of manifolds with a Kato bound on the Ricci curvature",
  "authors": [
    "Gilles Carron",
    "Ilaria Mondello",
    "David Tewodrose"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "We study the structure of Gromov-Hausdorff limits of sequences of Riemannian manifolds $\\{(M_\\alpha^n,g_\\alpha)\\}_{\\alpha \\in A}$ whose Ricci curvature satisfies a uniform Kato bound. We first obtain Mosco convergence of the Dirichlet energies to the Cheeger energy and show that tangent cones of such limits satisfy the $\\mathrm{RCD}(0,n)$ condition. When assuming a non-collapsing assumption, we introduce a new family of monotone quantities, which allows us to prove that tangent cones are also metric cones. We then show the existence of a well-defined stratification in terms of splittings of tangent cones. We finally prove volume convergence to the Hausdorff $n$-measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-11T10:58:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tangent cones",
      "ricci curvature satisfies",
      "uniform kato bound",
      "limits satisfy",
      "riemannian manifolds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}