{
  "id": "1903.07927",
  "version": "v1",
  "published": "2019-03-19T10:41:40.000Z",
  "updated": "2019-03-19T10:41:40.000Z",
  "title": "$α$-Dirac-harmonic maps from closed surfaces",
  "authors": [
    "Jürgen Jost",
    "Jingyong Zhu"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "$\\alpha$-Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to $\\alpha$-harmonic maps that were introduced by Sacks-Uhlenbeck to attack the existence problem for harmonic maps from surfaces. For $\\alpha >1$, the latter are known to satisfy a Palais-Smale condtion, and so, the technique of Sacks-Uhlenbeck consists in constructing $\\alpha$-harmonic maps for $\\alpha >1$ and then letting $\\alpha \\to 1$. The extension of this scheme to Dirac-harmonic maps meets with several difficulties, and in this paper, we start attacking those. We first prove the existence of nontrivial perturbed $\\alpha$-Dirac-harmonic maps when the target manifold has nonpositive curvature. The regularity theorem then shows that they are actually smooth. By $\\varepsilon$-regularity and suitable perturbations, we can then show that such a sequence of perturbed $\\alpha$-Dirac-harmonic maps converges to a smooth nontrivial $\\alpha$-Dirac-harmonic map.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-19T10:41:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "closed surfaces",
      "dirac-harmonic maps meets",
      "dirac-harmonic maps converges",
      "palais-smale condtion",
      "sacks-uhlenbeck consists"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}