{
  "id": "1810.01311",
  "version": "v1",
  "published": "2018-10-02T14:55:10.000Z",
  "updated": "2018-10-02T14:55:10.000Z",
  "title": "Prescribing the curvature of Riemannian manifolds with boundary",
  "authors": [
    "Tiarlos Cruz",
    "Feliciano Vitório"
  ],
  "comment": "21 pages; comments are welcome",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "Let $M$ be a compact connected surface with boundary. We prove that the signal condition given by the Gauss-Bonnet theorem is necessary and sufficient for a given smooth function $f$ on $\\partial M$ (resp. on $M$) to be geodesic curvature of the boundary (resp. the Gauss curvature) of some flat metric on $M$ (resp. metric on $M$ with geodesic boundary). In order to provide analogous results for this problem with $n\\geq 3,$ we prove some topological restrictions which imply, among other things, that any function that is negative somewhere on $\\partial M$ (resp. on $M$) is a mean curvature of a scalar flat metric on $M$ (resp. scalar curvature of a metric on $M$ and minimal boundary with respect to this metric). As an application of our results, we obtain a classification theorem for manifolds with boundary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-02T14:55:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "riemannian manifolds",
      "scalar flat metric",
      "prescribing",
      "signal condition",
      "mean curvature"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}