{
  "id": "0804.4211",
  "version": "v1",
  "published": "2008-04-26T07:04:53.000Z",
  "updated": "2008-04-26T07:04:53.000Z",
  "title": "Constant mean curvature surfaces with two ends in hyperbolic space",
  "authors": [
    "Wayne Rossman",
    "Katsunori Sato"
  ],
  "journal": "J. Exp. Math. 7(2) (1998), 101-119",
  "categories": [
    "math.DG"
  ],
  "abstract": "We investigate the close relationship between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. Just as in the case of minimal surfaces in Euclidean 3-space, the only complete connected embedded constant mean curvature 1 surfaces with two ends in hyperbolic space are well-understood surfaces of revolution -- the catenoid cousins. In contrast to this, we show that, unlike the case of minimal surfaces in Euclidean 3-space, there do exist complete connected immersed constant mean curvature 1 surfaces with two ends in hyperbolic space that are not surfaces of revolution -- the genus 1 catenoid cousins. The genus 1 catenoid cousins are of interest because they show that, although minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space are intimately related, there are essential differences between these two sets of surfaces (when the surfaces are considered globally). The proof we give of existence of the genus 1 catenoid cousins is a mathematically rigorous verification that the results of a computer experiment are sufficiently accurate to imply existence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-04-26T07:04:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A10",
      "53A35",
      "53C42"
    ],
    "keywords": [
      "constant mean curvature surfaces",
      "hyperbolic space",
      "minimal surfaces",
      "catenoid cousins",
      "embedded constant mean curvature"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0804.4211R"
    }
  }
}