{
  "id": "2307.03912",
  "version": "v1",
  "published": "2023-07-08T06:10:52.000Z",
  "updated": "2023-07-08T06:10:52.000Z",
  "title": "Convergence of the volume preserving fractional mean curvature flow for convex sets",
  "authors": [
    "Vesa Julin",
    "Domenico Angelo La Manna"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove that the volume preserving fractional mean curvature flow starting from a convex set does not develop singularities along the flow. By the recent result of Cesaroni-Novaga \\cite{CN} this then implies that the flow converges to a ball exponentially fast. In the proof we show that the apriori estimates due to Cinti-Sinestrari-Valdinoci \\cite{CSV2} imply the $C^{1+\\alpha}$-regularity of the flow and then provide a regularity argument which improves this into $C^{2+\\alpha}$-regularity of the flow. The regularity step from $C^{1+\\alpha}$ into $C^{2+\\alpha}$ does not rely on convexity and can probably be adopted to more general setting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-08T06:10:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "volume preserving fractional mean curvature",
      "preserving fractional mean curvature flow",
      "convex set",
      "mean curvature flow starting",
      "convergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}