{
  "id": "2104.03472",
  "version": "v1",
  "published": "2021-04-08T02:04:53.000Z",
  "updated": "2021-04-08T02:04:53.000Z",
  "title": "Nonnegative scalar curvature and area decreasing maps on complete foliated manifolds",
  "authors": [
    "Guangxiang Su",
    "Xiangsheng Wang",
    "Weiping Zhang"
  ],
  "comment": "26 pages, 3 figures, comments are welcome!",
  "categories": [
    "math.DG"
  ],
  "abstract": "Let $(M,g^{TM})$ be a noncompact complete Riemannian manifold of dimension $n$, and $F\\subseteq TM$ be an integrable subbundle of $TM$. Let $g^F=g^{TM}|_{F}$ be the restricted metric on $F$ and $k^F$ be the associated leafwise scalar curvature. Let $f:M\\to S^n(1)$ be a smooth area decreasing map along $F$, which is locally constant near infinity and of non-zero degree. We show that if $k^F> {\\rm rk}(F)({\\rm rk}(F)-1)$ on the support of ${\\rm d}f$, and either $TM$ or $F$ is spin, then $\\inf (k^F)<0$. As a consequence, we prove Gromov's sharp foliated $\\otimes_\\varepsilon$-twisting conjecture. Using the same method, we also extend two famous non-existence results due to Gromov and Lawson about $\\Lambda^2$-enlargeable metrics (and/or manifolds) to the foliated case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-08T02:04:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J20",
      "53C21",
      "53C12"
    ],
    "keywords": [
      "nonnegative scalar curvature",
      "complete foliated manifolds",
      "noncompact complete riemannian manifold",
      "smooth area decreasing map",
      "famous non-existence results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}