{
  "id": "2108.11705",
  "version": "v1",
  "published": "2021-08-26T11:03:12.000Z",
  "updated": "2021-08-26T11:03:12.000Z",
  "title": "New characterizations of the helicoid in a cylinder",
  "authors": [
    "Eunjoo Lee"
  ],
  "comment": "15 pages, 5 figures",
  "categories": [
    "math.DG"
  ],
  "abstract": "This paper characterizes a compact piece of the helicoid $H_C$ in a solid cylinder $C \\subset \\mathbb{R}^3$ from the following two perspectives. First, under reasonable conditions, $H_C$ has the smallest area among all immersed surfaces $\\Sigma$ with $\\partial \\Sigma \\subset d_1 \\cup d_2 \\cup S$, where $d_1$ and $d_2$ are the diameters of the top and bottom disks of $C$ and $S$ is the side surface of $C$. Second, other than $H_C$, there exists no minimal surface whose boundary consists of $d_1$, $d_2$, and a pair of \\textcolor{black}{rotationally symmetric} curves $\\gamma_1$, $\\gamma_2$ on $S$ along which it meets $S$ orthogonally. We draw the same conclusion when the boundary curves on $S$ are a pair of helices of a certain height.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-26T11:03:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "characterizations",
      "solid cylinder",
      "smallest area",
      "boundary curves",
      "paper characterizes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}