{
  "id": "1210.8254",
  "version": "v1",
  "published": "2012-10-31T07:51:53.000Z",
  "updated": "2012-10-31T07:51:53.000Z",
  "title": "Complete stationary surfaces in $\\mathbb{R}^4_1$ with total curvature $-\\int K\\mathrm{d}M=4π$",
  "authors": [
    "Xiang Ma",
    "Peng Wang"
  ],
  "comment": "22 pages",
  "journal": "International Journal of Mathematics Vol. 24, No. 10 (2013)",
  "doi": "10.1142/S0129167X13500882",
  "categories": [
    "math.DG",
    "math.CV",
    "math.GT"
  ],
  "abstract": "Applying the general theory about complete spacelike stationary (i.e. zero mean curvature) surfaces in 4-dimensional Lorentz space $\\mathbb{R}^4_1$, we classify those regular algebraic ones with total Gaussian curvature $-\\int K\\mathrm{d}M=4\\pi$. Such surfaces must be oriented and be congruent to either the generalized catenoids or the generalized enneper surfaces. For non-orientable stationary surfaces, we consider the Weierstrass representation on the oriented double covering $\\widetilde{M}$ (of genus $g$) and generalize Meeks and Oliveira's M\\\"obius bands. The total Gaussian curvature are shown to be at least $2\\pi(g+3)$ when $\\widetilde{M}\\to\\mathbb{R}^4_1$ is algebraic-type. We conjecture that there do not exist non-algebraic examples with $-\\int K\\mathrm{d}M=4\\pi$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-10-31T07:51:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A10",
      "53C42",
      "53C45"
    ],
    "keywords": [
      "complete stationary surfaces",
      "total curvature",
      "total gaussian curvature",
      "zero mean curvature",
      "general theory"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.8254M"
    }
  }
}