{
  "id": "2107.12344",
  "version": "v1",
  "published": "2021-07-26T17:34:39.000Z",
  "updated": "2021-07-26T17:34:39.000Z",
  "title": "Weak Laplacian bounds and minimal boundaries in non-smooth spaces with Ricci curvature lower bounds",
  "authors": [
    "Andrea Mondino",
    "Daniele Semola"
  ],
  "comment": "72 pages",
  "categories": [
    "math.DG",
    "math.AP",
    "math.MG"
  ],
  "abstract": "The goal of the paper is four-fold. In the setting of non-smooth spaces with Ricci curvature lower bounds (more precisely $RCD(K,N)$ metric measure spaces): - We establish a new principle relating lower Ricci curvature bounds to the preservation of Laplacian lower bounds under the evolution via the $p$-Hopf-Lax semigroup, for general exponents $p\\in[1,\\infty)$. - We develop an instrinsic viscosity theory of Laplacian bounds. - We prove Laplacian bounds on the distance function from a set (locally) minimizing the perimeter; this corresponds to vanishing mean curvature in the smooth setting. - We initiate a regularity theory for boundaries of sets (locally) minimizing the perimeter. The class of $RCD(K,N)$ metric measure spaces includes as remarkable sub-classes: measured Gromov-Hausdorff limits of smooth manifolds with lower Ricci curvature bounds and finite dimensional Alexandrov spaces with lower sectional curvature bounds. Most of the results are new also in these frameworks.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-26T17:34:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ricci curvature lower bounds",
      "weak laplacian bounds",
      "lower ricci curvature bounds",
      "non-smooth spaces",
      "minimal boundaries"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 72,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}