{
  "id": "1103.3838",
  "version": "v1",
  "published": "2011-03-20T09:06:31.000Z",
  "updated": "2011-03-20T09:06:31.000Z",
  "title": "A new conformal invariant on 3-dimensional manifolds",
  "authors": [
    "Yuxin Ge",
    "Guofang Wang"
  ],
  "comment": "23 pages",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "By improving the analysis developed in the study of $\\s_k$-Yamabe problem, we prove in this paper that the De Lellis-Topping inequality is true on 3-dimensional Riemannian manifolds of nonnegative scalar curvature. More precisely, if $(M^3, g)$ is a 3-dimensional closed Riemannian manifold with non-negative scalar curvature, then \\[\\int_M |Ric-\\frac{\\bar R} 3 g|^2 dv (g)\\le 9\\int_M |Ric-\\frac{R} 3 g|^2dv(g), \\] where $\\bar R=vol (g)^{-1} \\int_M R dv(g)$ is the average of the scalar curvature $R$ of $g$. Equality holds if and only if $(M^3,g)$ is a space form. We in fact study the following new conformal invariant \\[\\ds \\widetilde Y([g_0]):=\\sup_{g\\in {\\cal C}_1([g_0])}\\frac {\\ds vol(g)\\int_M \\s_2(g) dv(g)} {\\ds (\\int_M \\s_1(g) dv(g))^2}, \\] where ${\\cal C}_1([g_0]):=\\{g=e^{-2u}g_0\\,|\\, R>0\\}$ and prove that $\\widetilde Y([g_0])\\le 1/3$, which implies the above inequality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-03-20T09:06:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C21",
      "53C20",
      "58E11"
    ],
    "keywords": [
      "conformal invariant",
      "fact study",
      "nonnegative scalar curvature",
      "yamabe problem",
      "closed riemannian manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.3838G"
    }
  }
}