{
  "id": "2405.16249",
  "version": "v1",
  "published": "2024-05-25T14:14:52.000Z",
  "updated": "2024-05-25T14:14:52.000Z",
  "title": "On the fundamental theorem of submanifold theory and isometric immersions with supercritical low regularity",
  "authors": [
    "Siran Li",
    "Xiangxiang Su"
  ],
  "comment": "This paper corrects and supercedes 2003.05595",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "A fundamental result in global analysis and nonlinear elasticity asserts that given a solution $\\mathfrak{S}$ to the Gauss--Codazzi--Ricci equations over a simply-connected closed manifold $(\\mathcal{M}^n,g)$, one may find an isometric immersion $\\iota$ of $(\\mathcal{M}^n,g)$ into the Euclidean space $\\mathbb{R}^{n+k}$ whose extrinsic geometry coincides with $\\mathfrak{S}$. Here the dimension $n$ and the codimension $k$ are arbitrary. Abundant literature has been devoted to relaxing the regularity assumptions on $\\mathfrak{S}$ and $\\iota$. The best result up to date is $\\mathfrak{S} \\in L^p$ and $\\iota \\in W^{2,p}$ for $p>n \\geq 3$ or $p=n=2$. In this paper, we extend the above result to $\\iota \\in \\mathcal{X}$ whose topology is strictly weaker than $W^{2,n}$ for $n \\geq 3$. Indeed, $\\mathcal{X}$ is the weak Morrey space $L^{p, n-p}_{2,w}$ with arbitrary $p \\in ]2,n]$. This appears to be first supercritical result in the literature on the existence of isometric immersions with low regularity, given the solubility of the Gauss--Codazzi--Ricci equations. Our proof essentially utilises the theory of Uhlenbeck gauges -- in particular, Rivi\\`{e}re--Struwe's work [Partial regularity for harmonic maps and related problems, Comm. Pure Appl. Math. 61 (2008)] on harmonic maps in arbitrary dimensions and codimensions -- and compensated compactness.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-25T14:14:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "isometric immersion",
      "supercritical low regularity",
      "submanifold theory",
      "fundamental theorem",
      "gauss-codazzi-ricci equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}