{
  "id": "2008.11888",
  "version": "v1",
  "published": "2020-08-27T02:18:13.000Z",
  "updated": "2020-08-27T02:18:13.000Z",
  "title": "Generalized soap bubbles and the topology of manifolds with positive scalar curvature",
  "authors": [
    "Otis Chodosh",
    "Chao Li"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "We prove that a closed aspherical $4$-manifold does not admit a Riemannian metric with positive scalar curvature. Additionally, we show that for $n\\leq 7$, the connected sum of a $n$-torus with an arbitrary manifold does not admit a complete metric of positive scalar curvature. When combined with forthcoming contributions by Lesourd--Unger--Yau, this proves that the Schoen--Yau Liouville theorem holds for all locally conformally flat manifolds with non-negative scalar curvature. A key tool in these results are generalized soap bubbles---surfaces that are stationary for prescribed-mean-curvature functionals (also called as $\\mu$-bubbles).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-27T02:18:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "positive scalar curvature",
      "generalized soap bubbles",
      "schoen-yau liouville theorem holds",
      "complete metric",
      "riemannian metric"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}