{
  "id": "2107.08521",
  "version": "v1",
  "published": "2021-07-18T19:18:43.000Z",
  "updated": "2021-07-18T19:18:43.000Z",
  "title": "Non-degeneracy and quantitative stability of half-harmonic maps from ${\\mathbb R}$ to ${\\mathbb S}$",
  "authors": [
    "Bin Deng",
    "Liming Sun",
    "Juncheng Wei"
  ],
  "comment": "36 pages; comments welcome",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "We consider half-harmonic maps from $\\mathbb{R}$ (or $\\mathbb{S}$) to $\\mathbb{S}$. We prove that all (finite energy) half-harmonic maps are non-degenerate. In other words, they are integrable critical points of the energy functional. A full description of the kernel of the linearized operator around each half-harmonic map is given. The second part of this paper devotes to studying the quantitative stability of half-harmonic maps. When its degree is $\\pm 1$, we prove that the deviation of any map $\\boldsymbol{u}:\\mathbb{R}\\to \\mathbb{S}$ from M\\\"obius transformations can be controlled uniformly by $\\|\\boldsymbol{u}\\|_{\\dot H^{1/2}(\\mathbb{R})}^2-deg \\boldsymbol{u}$. This result resembles the quantitative rigidity estimate of degree $\\pm 1$ harmonic maps $\\mathbb{R}^2\\to \\mathbb{S}^2$ which is proved recently. Furthermore, we address the quantitative stability for half-harmonic maps of higher degree. We prove that if $\\boldsymbol{u}$ is already near to a Blaschke product, then the deviation of $\\boldsymbol{u}$ to Blaschke products can be controlled by $\\|\\boldsymbol{u}\\|_{\\dot H^{1/2}(\\mathbb{R})}^2-deg \\boldsymbol{u}$. Additionally, a striking example is given to show that such quantitative estimate can not be true uniformly for all $\\boldsymbol{u}$ of degree 2. We conjecture similar things happen for harmonic maps ${\\mathbb R}^2\\to {\\mathbb S}^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-18T19:18:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "half-harmonic map",
      "quantitative stability",
      "non-degeneracy",
      "harmonic maps",
      "blaschke product"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}