{
  "id": "math/0511387",
  "version": "v1",
  "published": "2005-11-15T15:04:15.000Z",
  "updated": "2005-11-15T15:04:15.000Z",
  "title": "Bending the Helicoid",
  "authors": [
    "William H. Meeks III",
    "Matthias Weber"
  ],
  "comment": "17 pages, 4 figures",
  "categories": [
    "math.DG"
  ],
  "abstract": "We construct Colding-Minicozzi limit minimal laminations in open domains in $\\rth$ with the singular set of $C^1$-convergence being any properly embedded $C^{1,1}$-curve. By Meeks' $C^{1,1}$-regularity theorem, the singular set of convergence of a Colding-Minicozzi limit minimal lamination ${\\cal L}$ is a locally finite collection $S({\\cal L})$ of $C^{1,1}$-curves that are orthogonal to the leaves of the lamination. Thus, our existence theorem gives a complete answer as to which curves appear as the singular set of a Colding-Minicozzi limit minimal lamination. In the case the curve is the unit circle $\\esf^1(1)$ in the $(x_1, x_2)$-plane, the classical Bj\\\"orling theorem produces an infinite sequence of complete minimal annuli $H_n$ of finite total curvature which contain the circle. The complete minimal surfaces $H_n$ contain embedded compact minimal annuli $\\bar{H}_n$ in closed compact neighborhoods $N_n$ of the circle that converge as $n \\to \\infty$ to $\\rth - x_3$-axis. In this case, we prove that the $\\bar{H}_n$ converge on compact sets to the foliation of $\\rth - x_3$-axis by vertical half planes with boundary the $x_3$-axis and with $\\esf^1(1)$ as the singular set of $C^1$-convergence. The $\\bar{H}_n$ have the appearance of highly spinning helicoids with the circle as their axis and are named {\\em bent helicoids}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-11-15T15:04:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A10",
      "49Q05",
      "53C42"
    ],
    "keywords": [
      "singular set",
      "embedded compact minimal annuli",
      "construct colding-minicozzi limit minimal laminations",
      "complete minimal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....11387M"
    }
  }
}