{
  "id": "2408.15278",
  "version": "v1",
  "published": "2024-08-17T02:37:18.000Z",
  "updated": "2024-08-17T02:37:18.000Z",
  "title": "Harmonic metrics of $\\mathrm{SO}_{0}(n,n)$-Higgs bundles in the Hitchin section on non-compact hyperbolic surfaces",
  "authors": [
    "Weihan Ma"
  ],
  "comment": "34 pages. arXiv admin note: text overlap with arXiv:2307.03365 by other authors",
  "categories": [
    "math.DG"
  ],
  "abstract": "Let $X$ be a Riemann surface. Using the canonical line bundle $K$ and some holomorphic differentials $\\boldsymbol{q}$, Hitchin constructed the $G$-Higgs bundles in the Hitchin section for a split real form $G$ of a complex simple Lie group. We study the ${\\mathrm{SO}_0(n,n)}$ case. In our work, we establish the existence of harmonic metrics for these Higgs bundles, which are compatible with the ${\\mathrm{SO}_0(n,n)}$-structure for any non-compact hyperbolic Riemann surface. Moreover, these harmonic metrics also weakly dominate $h_X$ which is the natural diagonal harmonic metric induced by the unique complete K\\\"ahler hyperbolic metric $g_X$ on $X$. Assuming that these holomorphic differentials are all bounded with respect to the metric $g_X$, we are able to prove the uniqueness of such a harmonic metric.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-17T02:37:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C07",
      "58E15",
      "14D21",
      "81T13"
    ],
    "keywords": [
      "higgs bundles",
      "non-compact hyperbolic surfaces",
      "hitchin section",
      "complex simple lie group",
      "non-compact hyperbolic riemann surface"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}