{
  "id": "1307.5764",
  "version": "v3",
  "published": "2013-07-22T16:30:06.000Z",
  "updated": "2015-11-05T17:28:24.000Z",
  "title": "Rigidity results, inverse curvature flows and Alexandrov-Fenchel type inequalities in the sphere",
  "authors": [
    "Matthias Makowski",
    "Julian Scheuer"
  ],
  "comment": "23 pages. Compared to version 2, we removed the statement about the isoperimetric inequality since it turned out to be false. Several typos were corrected",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "We prove a rigidity result in the sphere which allows us to generalize a result about smooth convex hypersurfaces in the sphere by Do Carmo-Warner to convex $C^2$-hypersurfaces. We apply these results to prove $C^{1,\\beta}$-convergence of inverse F-curvature flows in the sphere to an equator in \\mathbb{S}^{n+1} for embedded, closed, strictly convex initial hypersurfaces. The result holds for large classes of curvature functions including the mean curvature and arbitrary powers of the Gauss curvature. We use this result to prove Alexandrov-Fenchel type inequalities in the sphere.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-08-30T09:37:43.000Z",
      "abstract": "We prove a rigidity result in the sphere which allows us to generalize a result about smooth convex hypersurfaces in the sphere by Do Carmo-Warner to convex C2-hypersurfaces. We apply these results to prove C{1,\\beta}-convergence of inverse F-curvature flows in the sphere to an equator in Sn+1 for embedded, closed, orientable, strictly convex initial hypersurfaces. The result holds for large classes of curvature functions including the mean curvature and arbitrary powers of the Gauss curvature. We use this result to prove Alexandrov-Fenchel type inequalities and solve isoperimetric type problems in the sphere.",
      "comment": "27 pages. An error regarding the class of curvature functions was corrected. Extensions of the Alexandrov-Fenchel type inequalities were added",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-11-05T17:28:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "53C21",
      "53C24",
      "53C44",
      "58J05"
    ],
    "keywords": [
      "alexandrov-fenchel type inequalities",
      "inverse curvature flows",
      "rigidity result",
      "isoperimetric type problems",
      "inverse f-curvature flows"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.5764M"
    }
  }
}