{
  "id": "0810.5027",
  "version": "v1",
  "published": "2008-10-28T14:15:37.000Z",
  "updated": "2008-10-28T14:15:37.000Z",
  "title": "Prescribing curvatures on three dimensional Riemannian manifolds with boundaries",
  "authors": [
    "Lei Zhang"
  ],
  "comment": "21 pages. Transactions of American Mathematical Society, in press",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "Let $(M,g)$ be a complete three dimensional Riemannian manifold with boundary $\\partial M$. Given smooth functions $K(x)>0$ and $c(x)$ defined on $M$ and $\\partial M$, respectively, it is natural to ask whether there exist metrics conformal to $g$ so that under these new metrics, $K$ is the scalar curvature and $c$ is the boundary mean curvature. All such metrics can be described by a prescribing curvature equation with a boundary condition. With suitable assumptions on $K$,$c$ and $(M,g)$ we show that all the solutions of the equation can only blow up at finite points over each compact subset of $\\bar M$, some of them may appear on $\\partial M$. We describe the asymptotic behavior of the blowup solutions around each blowup point and derive an energy estimate as a consequence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-10-28T14:15:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "53B20"
    ],
    "keywords": [
      "dimensional riemannian manifold",
      "boundary mean curvature",
      "blowup solutions",
      "prescribing curvature equation",
      "metrics conformal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0810.5027Z"
    }
  }
}