{
  "id": "1806.08348",
  "version": "v1",
  "published": "2018-06-21T17:54:00.000Z",
  "updated": "2018-06-21T17:54:00.000Z",
  "title": "Smoothing the Bartnik boundary conditions and other results on Bartnik's quasi-local mass",
  "authors": [
    "Jeffrey L. Jauregui"
  ],
  "comment": "25 pages, 3 figures",
  "categories": [
    "math.DG",
    "gr-qc"
  ],
  "abstract": "Quite a number of distinct versions of Bartnik's definition of quasi-local mass appear in the literature, and it is not a priori clear that any of them produce the same value in general. In this paper we make progress on reconciling these definitions. The source of discrepancies is two-fold: the choice of boundary conditions (of which there are three variants) and the non-degeneracy or \"no-horizon\" condition (at least six variants). To address the boundary conditions, we show that given a 3-dimensional region $\\Omega$ of nonnegative scalar curvature ($R \\geq 0$) extended in a Lipschitz fashion across $\\partial \\Omega$ to an asymptotically flat 3-manifold with $R \\geq 0$ (also holding distributionally along $\\partial \\Omega$), there exists a smoothing, arbitrarily small in $C^0$ norm, such that $R \\geq 0$ and the geometry of $\\Omega$ are preserved, and the ADM mass changes only by a small amount. With this we are able to show that the three boundary conditions yield equivalent Bartnik masses for two reasonable non-degeneracy conditions. We also discuss subtleties pertaining to the various non-degeneracy conditions and produce a nontrivial inequality between a no-horizon version of the Bartnik mass and Bray's replacement of this with the outward-minimizing condition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-21T17:54:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bartniks quasi-local mass",
      "bartnik boundary conditions",
      "boundary conditions yield equivalent bartnik",
      "conditions yield equivalent bartnik masses",
      "non-degeneracy conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}