{
  "id": "2401.13377",
  "version": "v1",
  "published": "2024-01-24T11:10:06.000Z",
  "updated": "2024-01-24T11:10:06.000Z",
  "title": "The prescribed curvature flow on the disc",
  "authors": [
    "Michael Struwe"
  ],
  "comment": "53 pages",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "For given functions $f$ and $j$ on the disc $B$ and its boundary $\\partial B=S^1$, we study the existence of conformal metrics $g=e^{2u}$ with prescribed Gauss curvature $K_g=f$ and boundary geodesic curvature $k_g=j$. Using the variational characterization of such metrics obtained by Cruz-Blazquez and Ruiz (2018), we show that there is a canonical negative gradient flow of such metrics, either converging to a solution of the prescribed curvature problem, or blowing up to a spherical cap. In the latter case, similar to our work Struwe (2005) on the prescribed curvature problem on the sphere, we are able to exhibit a $2$-dimensional shadow flow for the center of mass of the evolving metrics from which we obtain existence results complementing the results recently obtained by Ruiz (2021) by degree-theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-24T11:10:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K55",
      "53E99"
    ],
    "keywords": [
      "prescribed curvature flow",
      "prescribed curvature problem",
      "dimensional shadow flow",
      "boundary geodesic curvature",
      "canonical negative gradient flow"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}