{
  "id": "math/0512547",
  "version": "v2",
  "published": "2005-12-23T18:04:08.000Z",
  "updated": "2006-04-08T08:31:55.000Z",
  "title": "Area-stationary surfaces in the Heisenberg group H^1",
  "authors": [
    "Manuel Ritoré",
    "César Rosales"
  ],
  "comment": "37 pages, 3 figures; corrected typos, very recent references added",
  "categories": [
    "math.DG",
    "math.MG"
  ],
  "abstract": "We use variational arguments to introduce a notion of mean curvature for surfaces in the Heisenberg group H^1 endowed with its Carnot-Carath\\'eodory distance. By analyzing the first variation of area, we characterize C^2 stationary surfaces for the area as those with mean curvature zero (or constant if a volume-preserving condition is assumed) and such that the characteristic curves meet orthogonally the singular curves. Moreover, a Minkowski type formula relating the area, the mean curvature, and the volume is obtained for volume-preserving area-stationary surfaces enclosing a given region. As a consequence of the characterization of area-stationary surfaces, we refine previous Bernstein type theorems in order to describe entire area-stationary graphs over the xy-plane in H^1. A calibration argument shows that these graphs are globally area-minimizing. Finally, by using the known description of the singular set, the characterization of area-stationary surfaces, and the ruling property of constant mean curvature surfaces, we prove our main results where we classify volume-preserving area-stationary surfaces in H^1 with non-empty singular set. In particular, we deduce the following counterpart to Alexandrov uniqueness theorem in Euclidean space: any compact, connected, C^2 surface in H^1 area-stationary under a volume constraint must be congruent with a rotationally symmetric sphere obtained as the union of all the geodesics of the same curvature joining two points. As a consequence, we solve the isoperimetric problem in H^1 assuming C^2 smoothness of the solutions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-04-08T08:31:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C17",
      "49Q20"
    ],
    "keywords": [
      "heisenberg group",
      "volume-preserving area-stationary surfaces",
      "constant mean curvature surfaces",
      "entire area-stationary graphs",
      "minkowski type formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....12547R"
    }
  }
}