{
  "id": "2407.06344",
  "version": "v1",
  "published": "2024-07-08T19:29:12.000Z",
  "updated": "2024-07-08T19:29:12.000Z",
  "title": "Corrections to a paper of Allard and Almgren on the uniqueness of tangent cones",
  "authors": [
    "William K. Allard"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "The paper referred to in the title is Allard/Almgren 1981 [AA81]. Several months ago Francesco Maggi emailed me saying that the inequalities 5.3(4),(5) of \\cite{AA81} were wrong. In fact, as he pointed out, their incorrectness is immediately apparent if one takes $Z=0$ there. Maggi and his coauthor wanted to use these inequalities in their paper \\cite{MN}. They were able to obtain a version of these inequalities which suffice for the carrying out the work in \\cite{MN}. I started writing this paper in order to provide a version of these inequalities as needed in [AA81]. In thinking about this material I began to realize there were other problems with the paper. As a result I ended up {\\em completely rewriting 5.1-5.4 on pages 243-248 of \\cite{AA81}}; this rewrite is the contents of this paper. In addition to many annoying misprints many of the needed definitions and proofs in 5.1-5.4 are incomplete or absent. For example, it is not said where $z$ in 5.1(2) comes from; this omission completely surprised me since I remember doing a lot of work to come up with $z$ when \\cite{AA81} was being written. Also, much of the necessary material about the reach of a submanifold as in \\cite{FE2} is not provided in \\cite{AA81}. The table of contents can serve as an index. In particular one sees there where the constants $\\epsilon_1$ through $\\epsilon_6$ are introduced. This material is extremely technical. One way to navigate this paper would be to start looking at Proposition 9.5 and work backwards.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-08T19:29:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49Q20"
    ],
    "keywords": [
      "tangent cones",
      "uniqueness",
      "inequalities",
      "corrections",
      "francesco maggi"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}