{
  "id": "2302.01720",
  "version": "v1",
  "published": "2023-02-03T13:18:33.000Z",
  "updated": "2023-02-03T13:18:33.000Z",
  "title": "Compact surfaces with boundary with prescribed mean curvature depending on the Gauss map",
  "authors": [
    "Antonio Bueno",
    "Rafael López"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "Given a $C^1$ function $\\mathcal{H}$ defined in the unit sphere $\\mathbb{S}^2$, an $\\mathcal{H}$-surface $M$ is a surface in the Euclidean space $\\mathbb{R}^3$ whose mean curvature $H_M$ satisfies $H_M(p)=\\mathcal{H}(N_p)$, $p\\in M$, where $N$ is the Gauss map of $M$. Given a closed simple curve $\\Gamma\\subset\\mathbb{R}^3$ and a function $\\mathcal{H}$, in this paper we investigate the geometry of compact $\\mathcal{H}$-surfaces spanning $\\Gamma$ in terms of $\\Gamma$. Under mild assumptions on $\\mathcal{H}$, we prove non-existence of closed $\\mathcal{H}$-surfaces, in contrast with the classical case of constant mean curvature. We give conditions on $\\mathcal{H}$ that ensure that if $\\Gamma$ is a circle, then $M$ is a rotational surface. We also establish the existence of estimates of the area of $\\mathcal{H}$-surfaces in terms of the height of the surface.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-03T13:18:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A10",
      "53C42",
      "35J93",
      "35B06",
      "35B50"
    ],
    "keywords": [
      "prescribed mean curvature",
      "gauss map",
      "compact surfaces",
      "constant mean curvature",
      "unit sphere"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}