{
  "id": "1209.4669",
  "version": "v1",
  "published": "2012-09-20T21:12:02.000Z",
  "updated": "2012-09-20T21:12:02.000Z",
  "title": "Ricci curvature and monotonicity for harmonic functions",
  "authors": [
    "Tobias Holck Colding",
    "William P. Minicozzi II"
  ],
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "In this paper we generalize the monotonicity formulas of [C] for manifolds with nonnegative Ricci curvature. Monotone quantities play a key role in analysis and geometry; see, e.g., [A], [CM1] and [GL] for applications of monotonicity to uniqueness. Among the applications here is that level sets of Green's function on open manifolds with nonnegative Ricci curvature are asymptotically umbilic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-09-20T21:12:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "harmonic functions",
      "nonnegative ricci curvature",
      "monotone quantities play",
      "open manifolds",
      "monotonicity formulas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.4669H"
    }
  }
}