{
  "id": "1208.1020",
  "version": "v2",
  "published": "2012-08-05T15:15:18.000Z",
  "updated": "2016-06-06T08:25:46.000Z",
  "title": "$\\cF$-functional and geodesic stability",
  "authors": [
    "Weiyong He"
  ],
  "comment": "Comments are welcome; published version by Asian J. of Math",
  "categories": [
    "math.DG",
    "math.AG"
  ],
  "abstract": "We consider canonical metrics on Fano manifolds. First we introduce a norm-type functional on Fano manifolds, which has Kahler-Einstein or Kahler-Ricci soliton as its critical point and the Kahler-Ricci flow can be viewed as its (reduced) gradient flow. We then obtain a natural lower bound of this functional. As an application, we prove that Kahler-Ricci soliton, if exists, maximizes Perelman's $\\mu$-functional without extra assumptions. Second we consider a conjecture proposed by S.K. Donaldson in terms of $\\cK$-energy. Our simple observation is that $\\cF$-functional, as $\\cK$-energy, also integrates Futaki invariant. We then restate geodesic stability conjecture on Fano manifolds in terms of $\\cF$-functional. Similar pictures can also be extended to Kahler-Ricci soliton and modified $\\cF$-functional.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-08-05T15:15:18.000Z",
      "comment": "Comments are welcome",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2016-06-06T08:25:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "functional",
      "fano manifolds",
      "kahler-ricci soliton",
      "restate geodesic stability conjecture",
      "natural lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.1020H"
    }
  }
}