{
  "id": "2112.14816",
  "version": "v2",
  "published": "2021-12-29T20:26:14.000Z",
  "updated": "2022-03-26T15:23:02.000Z",
  "title": "On stability of determination of Riemann surface from its DN-map",
  "authors": [
    "M. I. Belishev",
    "D. V. Korikov"
  ],
  "comment": "20 pages, 0 figures",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "Suppose that $M$ is a Riemann surface with boundary $\\partial M$, $\\Lambda$ is its DN-map, and $\\mathscr E:M\\to\\mathbb{C}^{n}$ % $\\mathfrak{J}_{M}$ is a holomorphic immersion. Let $M'$ be diffeomorphic to $M$, $\\partial M=\\partial M'$; let $\\Lambda'$ be the DN map of $M'$. Let us write $M'\\in\\mathbb M_t$ if $\\parallel\\Lambda'-\\Lambda\\parallel_{H^{1}(\\partial M)\\to L_{2}(\\partial M)}\\leqslant t$ holds. We show that, for any holomorphic immersion $\\mathscr{E}: M \\to \\mathbb C^n$ ($n\\geqslant 1$), the relation \\begin{equation*} \\sup_{M'\\in \\mathbb{M}_{t}}\\inf_{\\mathscr{E}'}d_{H}(\\mathscr E'(M'),\\mathscr{E}(M))\\underset{t\\to 0}{\\longrightarrow}0, \\end{equation*} holds, where $d_H$ is the Haussdorf distance in $\\mathbb C^n$ and the infimum is taken over all holomorphic immersions $\\mathscr E': M'\\mapsto\\mathbb C^n$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-03-26T15:23:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R30",
      "46J15",
      "46J20",
      "30F15"
    ],
    "keywords": [
      "riemann surface",
      "holomorphic immersion",
      "determination",
      "dn map",
      "haussdorf distance"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}