{
  "id": "2002.07295",
  "version": "v1",
  "published": "2020-02-17T23:12:58.000Z",
  "updated": "2020-02-17T23:12:58.000Z",
  "title": "Planar minimal surfaces with polynomial growth in the $\\mathrm{Sp}(4, \\mathbb{R})$-symmetric space",
  "authors": [
    "Andrea Tamburelli",
    "Michael Wolf"
  ],
  "comment": "68 pages",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "We study the asymptotic geometry of a family of conformally planar minimal surfaces with polynomial growth in the $\\mathrm{Sp}(4,\\mathbb{R})$-symmetric space. We describe a homeomomorphism between the \"Hitchin component\" of wild $\\mathrm{Sp}(4,\\mathbb{R})$-Higgs bundles over $\\mathbb{CP}^1$ with a single pole at infinity and a component of maximal surfaces with light-like polygonal boundary in $\\mathbb{H}^{2,2}$. Moreover, we identify those surfaces with convex embeddings into the Grassmannian of symplectic planes of $\\mathbb{R}^{4}$. We show, in addition, that our planar maximal surfaces are the local limits of equivariant maximal surfaces in $\\mathbb{H}^{2,2}$ associated to $\\mathrm{Sp}(4,\\mathbb{R})$-Hitchin representations along rays of holomorphic quartic differentials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-17T23:12:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "polynomial growth",
      "symmetric space",
      "equivariant maximal surfaces",
      "planar maximal surfaces",
      "holomorphic quartic differentials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 68,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}