{
  "id": "2408.07823",
  "version": "v1",
  "published": "2024-08-14T21:33:10.000Z",
  "updated": "2024-08-14T21:33:10.000Z",
  "title": "Nonexistence of minimizers for the second conformal eigenvalue near the round sphere in low dimensions",
  "authors": [
    "Bruno Premoselli",
    "Jérôme Vétois"
  ],
  "comment": "Comments welcome",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "We consider the problem of minimizing the second conformal eigenvalue of the conformal Laplacian in a conformal class of metrics with renormalized volume. We prove, in dimensions $n\\in\\left\\{3,\\dotsc,10\\right\\}$, that a minimizer for this problem does not exist for metrics sufficiently close to the round metric on the sphere. This is in striking contrast with the situation in dimensions $n \\ge 11$, where Ammann and Humbert obtained the existence of minimizers for the second conformal eigenvalue on any smooth closed non-locally conformally flat manifold. As a byproduct of our techniques, we also obtain a lower bound on the energy of sign-changing solutions of the \\nobreak Yamabe equation in dimensions 3, 4 and 5, which extends a result obtained by Weth in the case of the round sphere.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-14T21:33:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58C40",
      "47J10",
      "35J60",
      "35P99"
    ],
    "keywords": [
      "second conformal eigenvalue",
      "round sphere",
      "low dimensions",
      "non-locally conformally flat manifold",
      "closed non-locally conformally flat"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}