{
  "id": "2101.11780",
  "version": "v1",
  "published": "2021-01-28T02:26:10.000Z",
  "updated": "2021-01-28T02:26:10.000Z",
  "title": "A characterization of $p$-minimal surfaces in the Heisenberg group $H_1$",
  "authors": [
    "Hung-Lin Chiu",
    "Hsiao-Fan Liu"
  ],
  "comment": "39 pages, 6 figures and submitted",
  "categories": [
    "math.DG",
    "math.CA"
  ],
  "abstract": "In Euclidean $3$-space, it is well known that the Sine-Gordon equation was considered in the nineteenth century in the course of investigation of surfaces of constant Gaussian curvature $K=-1$. Such a surface can be constructed from a solution to the Sine-Gordon equation, and vice versa. With this as motivation, by means of the fundamental theorem of surfaces in the Heisenberg group $H_{1}$, we show in this paper that the existence of a constant $p$-mean curvature surface (without singular points) is equivalent to the existence of a solution to a nonlinear second order ODE, which is a kind of {\\bf Li\\'{e}nard equations}. Therefore, we turn to investigate this equation. It is a surprise that we give a complete set of solutions to such ODEs in the $p$-minimal case, and hence use the types of the solution to divide $p$-minimal surfaces into several classes. As a result, we obtain a representation of $p$-minimal surfaces and classify further all $p$-minimal surfaces. In Section 9, we provide an approach to construct $p$-minimal surfaces. It turns out that, in some sense, generic $p$-minimal surfaces can be constructed via this approach. Finally, as a derivation, we recover the Bernstein-type theorem which was first shown in 2005.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-28T02:26:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A10",
      "53C42",
      "53C22",
      "34A26"
    ],
    "keywords": [
      "minimal surfaces",
      "heisenberg group",
      "characterization",
      "sine-gordon equation",
      "nonlinear second order ode"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}