{
  "id": "2404.08410",
  "version": "v1",
  "published": "2024-04-12T11:44:03.000Z",
  "updated": "2024-04-12T11:44:03.000Z",
  "title": "On weak inverse mean curvature flow and Minkowski-type inequalities in hyperbolic space",
  "authors": [
    "Brian Harvie"
  ],
  "comment": "24 pages, 2 figures, comments welcome!",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "We prove that a proper weak solution $\\{ \\Omega_{t} \\}_{0 \\leq t < \\infty}$ to inverse mean curvature flow in $\\mathbb{H}^{n}$, $3\\leq n \\leq 7$, is smooth and star-shaped by the time \\begin{equation*} T= (n-1) \\log \\left( \\frac{\\text{sinh} \\left( r_{+} \\right)}{ \\text{sinh} \\left( r_{-} \\right)} \\right), \\end{equation*} where $r_{+}$ and $r_{-}$ are the geodesic out-radius and in-radius of the initial domain $\\Omega_{0}$. The argument is inspired by the Alexandrov reflection method for extrinsic curvature flows in $\\mathbb{R}^{n}$ due to Chow-Gulliver and uses a result of Li-Wei. As applications, we extend the Minkowski inequalities of Brendle-Hung-Wang and De Lima-Girao to outer-minimizing domains $\\Omega_{0} \\subset \\mathbb{H}^{n}$ in these dimensions. From this, we also extend the asymptotically hyperbolic Riemannian Penrose inequality to balanced asymptotically hyperbolic graphs over the exteriors of outer-minimizing domains of $\\mathbb{H}^{n}$, $3 \\leq n \\leq 7$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-12T11:44:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weak inverse mean curvature flow",
      "minkowski-type inequalities",
      "hyperbolic space",
      "hyperbolic riemannian penrose inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}