{
  "id": "2301.05933",
  "version": "v1",
  "published": "2023-01-14T15:22:23.000Z",
  "updated": "2023-01-14T15:22:23.000Z",
  "title": "On the ergodicity of unitary frame flows on Kähler manifolds",
  "authors": [
    "Mihajlo Cekić",
    "Thibault Lefeuvre",
    "Andrei Moroianu",
    "Uwe Semmelmann"
  ],
  "categories": [
    "math.DS",
    "math.AT",
    "math.DG"
  ],
  "abstract": "Let $(M,g,J)$ be a closed K\\\"ahler manifold with negative sectional curvature and complex dimension $m := \\dim_{\\mathbb{C}} M \\geq 2$. In this article, we study the unitary frame flow, that is, the restriction of the frame flow to the principal $\\mathrm{U}(m)$-bundle $F_{\\mathbb{C}}M$ of unitary frames. We show that if $m \\geq 6$ is even, and $m \\neq 28$, there exists $\\lambda(m) \\in (0, 1)$ such that if $(M, g, J)$ has negative $\\lambda(m)$-pinched holomorphic sectional curvature, then the unitary frame flow is ergodic and mixing. The constants $\\lambda(m)$ satisfy $\\lambda(6) = 0.9330...$, $\\lim_{m \\to +\\infty} \\lambda(m) = \\tfrac{11}{12} = 0.9166...$, and $m \\mapsto \\lambda(m)$ is decreasing. This extends to the even-dimensional case the results of Brin-Gromov who proved ergodicity of the unitary frame flow on negatively-curved compact K\\\"ahler manifolds of odd complex dimension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-14T15:22:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A05",
      "37A20",
      "37A25",
      "32Q15",
      "53C10",
      "53C22"
    ],
    "keywords": [
      "unitary frame flow",
      "kähler manifolds",
      "ergodicity",
      "odd complex dimension",
      "pinched holomorphic sectional curvature"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}