{
  "id": "2212.14428",
  "version": "v1",
  "published": "2022-12-29T18:59:55.000Z",
  "updated": "2022-12-29T18:59:55.000Z",
  "title": "Geometry of CMC surfaces of finite index",
  "authors": [
    "William H. Meeks III",
    "Joaquin Perez"
  ],
  "comment": "24 pages, 3 figures. arXiv admin note: text overlap with arXiv:2212.13594",
  "categories": [
    "math.DG"
  ],
  "abstract": "Given $r_0>0$, $I\\in \\mathbb{N}\\cup \\{0\\}$ and $K_0,H_0\\geq 0$, let $X$ be a complete Riemannian $3$-manifold with injectivity radius $\\mbox{Inj}(X)\\geq r_0$ and with the supremum of absolute sectional curvature at most $K_0$, and let $M\\looparrowright X$ be a complete immersed surface with index at most $I$ of constant mean curvature $H\\in [0,H_0]$. We will obtain geometric estimates for such an $M\\looparrowright X$ as a consequence of the Hierarchy Structure Theorem in [3]. The Hierarchy Structure Theorem (see Theorem 2.2 below) will be applied to understand global properties of $M\\looparrowright X$, especially results related to the area and diameter of $M$. By item E of Theorem 2.2, the area of such a non-compact $M\\looparrowright X$ is infinite. We will improve this area result by proving the following when $M$ is connected; here $g(M)$ denotes the genus of the orientable cover of $M$: 1. There exists $C=C(I,r_0,K_0,H_0)>0$ such that Area$(M)\\geq C(g(M)+1)$. 2. There exists $C'>0$ independent of $I,r_0,K_0,H_0$ and $G=G(I,r_0,K_0,H_0)\\in \\mathbb{N}$ such that if $g(M)\\geq G$, then Area$(M)\\geq C'(g(M)+1)$. 3. If the scalar curvature $\\rho$ of $X$ satisfies $3H^2+\\frac{1}{2}\\rho\\geq c$ in $X$ for some $c>0$, then there exist $A,D>0$ depending on $c,I,r_0,K_0,H_0$ such that Area$(M)\\leq A$ and Diameter$(M)\\leq D$. Hence, $M$ is compact and, by item~1, $ g(M)\\leq A/C -1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-29T18:59:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A10",
      "49Q05",
      "53C42"
    ],
    "keywords": [
      "cmc surfaces",
      "finite index",
      "hierarchy structure theorem",
      "absolute sectional curvature",
      "constant mean curvature"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}