{
  "id": "1505.06183",
  "version": "v1",
  "published": "2015-05-22T19:18:34.000Z",
  "updated": "2015-05-22T19:18:34.000Z",
  "title": "A non-compactness result on the fractional Yamabe problem in large dimensions",
  "authors": [
    "Seunghyeok Kim",
    "Monica Musso",
    "Juncheng Wei"
  ],
  "comment": "47 pages. Any comment is welcome",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "Let $(X^{n+1}, g^+)$ be an $(n+1)$-dimensional asymptotically hyperbolic manifold with a conformal infinity $(M^n, [\\hat{h}])$. The fractional Yamabe problem addresses to solve \\[P^{\\gamma}[g^+,\\hat{h}] (u) = cu^{n+2\\gamma \\over n-2\\gamma}, \\quad u > 0 \\quad \\text{on } M\\] where $c \\in \\mathbb{R}$ and $P^{\\gamma}[g^+,\\hat{h}]$ is the fractional conformal Laplacian whose principal symbol is $(-\\Delta)^{\\gamma}$. In this paper, we construct a metric on the half space $X = \\mathbb{R}^{n+1}_+$, which is conformally equivalent to the unit ball, for which the solution set of the fractional Yamabe equation is non-compact provided that $n \\ge 24$ for $\\gamma \\in (0, \\gamma^*)$ and $n \\ge 25$ for $\\gamma \\in [\\gamma^*,1)$ where $\\gamma^* \\in (0, 1)$ is a certain transition exponent. The value of $\\gamma^*$ turns out to be approximately 0.940197.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-22T19:18:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "large dimensions",
      "non-compactness result",
      "fractional yamabe problem addresses",
      "fractional conformal laplacian",
      "dimensional asymptotically hyperbolic manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150506183K"
    }
  }
}