{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "QTHPN6V7XM"
    }
  ],
  "body": "\n## Summary\n\nThe Birkhoff min-max method finds a [[closed geodesic|4MCCP8GG4J]] on a Riemannian sphere by sweeping the sphere with a one-parameter family of closed curves.\nOne takes, over all sweepouts, the length of the longest curve, and then the infimum of this number over all sweepouts.\nGeorge Birkhoff showed that this min-max value, the width, is positive and is the length of a closed geodesic.\nThe method is the prototype of min-max constructions in geometry, including [[Almgren–Pitts min-max theory|6KVJHG1KRZ]] for minimal hypersurfaces.\n\n## Explanations\n\n### L1 human\n\nSlide a rubber band over a sphere from the south pole to the north pole, so that it passes over every point.\nSomewhere along the way the band is longest.\nNow try to choose the sweep so that this longest moment is as short as possible.\nThe best possible longest band cannot be shrunk further without the sweep getting stuck, and it sits on a closed geodesic.\nOn a round unit sphere, sweeping by circles of latitude gives longest band the equator, of length $2\\pi$, and this is the width.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "58E10",
      "53C22",
      "49J35"
    ],
    "name": "Birkhoff min-max method",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Closed_geodesic"
      }
    ],
    "tag": "1R5X9HW6D3"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/1R5X9HW6D3/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/1R5X9HW6D3.md",
  "msc": [
    "58E10",
    "53C22",
    "49J35"
  ],
  "name": "Birkhoff min-max method",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/1R5/1R5X9HW6D3.md",
  "status": null,
  "tag": "1R5X9HW6D3",
  "uri": "m2://1R5X9HW6D3",
  "url": "https://tamnd.github.io/math2/wiki/birkhoff-min-max-method/",
  "writing": {
    "tag": "1R5X9HW6D3",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}