{
  "id": "2407.06934",
  "version": "v1",
  "published": "2024-07-09T15:10:20.000Z",
  "updated": "2024-07-09T15:10:20.000Z",
  "title": "Quantitative stability of the total $Q$-curvature near minimizing metrics",
  "authors": [
    "João Henrique Andrade",
    "Tobias König",
    "Jesse Ratzkin",
    "Juncheng Wei"
  ],
  "comment": "43 pages. Comments welcome!",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "Under appropriate positivity hypotheses, we prove quantitative estimates for the total $k$-th order $Q$-curvature functional near minimizing metrics on any smooth, closed $n$-dimensional Riemannian manifold for every integer $1 \\leq k < \\frac{n}{2}$. More precisely, we show that on a generic closed Riemannian manifold the distance to the minimizing set of metrics is controlled quadratically by the $Q$-curvature energy deficit, extending recent work by Engelstein, Neumayer and Spolaor in the case $k=1$. Next we prove, for any integer $1 \\leq k< \\frac{n}{2}$, the existence of an $n$-dimensional Riemannian manifold such that the $k$-th order $Q$-curvature deficit controls a higher power of the distance to the minimizing set. We believe that these degenerate examples are of independent interest and can be used for further development in the field.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-09T15:10:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "minimizing metrics",
      "quantitative stability",
      "dimensional riemannian manifold",
      "th order",
      "curvature deficit controls"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}