{
  "id": "2107.13918",
  "version": "v1",
  "published": "2021-07-29T11:49:06.000Z",
  "updated": "2021-07-29T11:49:06.000Z",
  "title": "Uniqueness of the $[\\varphi,\\vec{e}_{3}]$-catenary cylinders by their asymptotic behaviour",
  "authors": [
    "A. L. Martínez-Triviño",
    "J. P. dos Santos"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "We establish a uniqueness result for the $[\\varphi,\\vec{e}_{3}]$-catenary cylinders by their asymptotic behaviour. Well known examples of such cylinders are the grim reaper translating solitons for the mean curvature flow. For such solitons, F. Mart\\'in, J. P\\'erez-Garc\\'ia, A. Savas-Halilaj and K. Smoczyk proved that, if $\\Sigma$ is a properly embedded translating soliton with locally bounded genus, and $\\mathcal{C}^{\\infty}$-asymptotic to two vertical planes outside a cylinder, then $\\Sigma$ must coincide with some grim reaper translating soliton. In this paper, applying the moving plane method of Alexandrov together with a strong maximum principle for elliptic operators, we increase the family of $[\\varphi,\\vec{e}_{3}]$-minimal graphs where these types of results hold under different assumption of asymptotic behaviour.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-29T11:49:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "asymptotic behaviour",
      "catenary cylinders",
      "grim reaper translating soliton",
      "uniqueness",
      "mean curvature flow"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}