{
  "id": "2008.13531",
  "version": "v1",
  "published": "2020-08-31T12:26:16.000Z",
  "updated": "2020-08-31T12:26:16.000Z",
  "title": "On the non-existence of compact surfaces of genus one with prescribed, almost constant mean curvature, close to the singular limit",
  "authors": [
    "Paolo Caldiroli",
    "Alessandro Iacopetti"
  ],
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "In Euclidean 3-space endowed with a Cartesian reference system we consider a class of surfaces, called Delaunay tori, constructed by bending segments of Delaunay cylinders with neck-size $a$ and $n$ lobes along circumferences centered at the origin. Such surfaces are complete and compact, have genus one and almost constant, say 1, mean curvature, when $n$ is large. Considering a class of mappings $H\\colon\\mathbb{R}^{3}\\to\\mathbb{R}$ such that $H(X)\\to 1$ as $|X|\\to\\infty$ with some decay of inverse-power type, we show that for $n$ large and $|a|$ small, in a suitable neighborhood of any Delaunay torus with $n$ lobes and neck-size $a$ there is no parametric surface constructed as normal graph over the Delaunay torus and whose mean curvature equals $H$ at every point.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-31T12:26:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A10",
      "53A05",
      "53C42",
      "53C21"
    ],
    "keywords": [
      "constant mean curvature",
      "singular limit",
      "compact surfaces",
      "delaunay torus",
      "non-existence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}