{
  "id": "2211.11715",
  "version": "v1",
  "published": "2022-11-21T18:34:39.000Z",
  "updated": "2022-11-21T18:34:39.000Z",
  "title": "Geometric inequalities on Closed Surfaces with $λ_1(-Δ+β K)\\geq0$",
  "authors": [
    "Kai Xu"
  ],
  "comment": "32 pages. Comments welcome!",
  "categories": [
    "math.DG"
  ],
  "abstract": "Let $\\Sigma$ be a closed orientable surface satisfying the eigenvalue condition $\\lambda_1(-\\Delta+\\beta K)\\geq\\lambda\\geq0$, where $\\beta$ is a positive constant and $K$ is the Gaussian curvature of $\\Sigma$. This eigenvalue condition naturally arises for stable minimal surfaces and $\\mu$-bubbles in 3-manifolds with positive scalar curvature (with $\\beta=1$ there). We prove that when $\\beta>1/2$, the Cheeger constant of $\\Sigma$ is bounded from below by $C(\\beta)/diam(\\Sigma)$. When $\\beta>1/4$, the homogeneous isoperimetric ratio of $\\Sigma$ is bounded from below by $C(\\beta,\\epsilon)\\big(\\frac{Area(\\Sigma)}{diam(\\Sigma)^2}\\big)^{\\frac{1+\\epsilon}{4\\beta-1}}$ ($\\forall\\epsilon>0$). We also show a weak Bonnet-Myers' theorem $diam(\\Sigma)\\leq C(\\beta)\\lambda^{-1/2}$ and a total volume comparison theorem $Area(\\Sigma)\\leq C(\\beta)\\,diam(\\Sigma)^2$, when $\\beta>1/4$. Generalizing Schoen-Yau's observation, associated to each positive supersolution $\\Delta\\varphi\\leq\\beta K\\varphi$ we make a conformal change $\\tilde g=\\varphi^{2/\\beta}g$ with $K_{\\tilde g}\\geq0$. Some of the main theorems are proved by studying the geometric inequalities for $\\tilde g$ and transferring them back to $g$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-21T18:34:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C21",
      "52A40"
    ],
    "keywords": [
      "geometric inequalities",
      "closed surfaces",
      "eigenvalue condition",
      "total volume comparison theorem",
      "generalizing schoen-yaus observation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}