{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "HB2K72PSZ6"
    }
  ],
  "body": "\n## Statement\n\nLet $g$ be a [[Riemannian metric|Q61VFHCH2J]] on the torus $T^2$.\nLet $\\operatorname{sys}(g)$ be its [[systole|PZCJV40M4K]] and $\\operatorname{area}(g)$ its area.\nThen\n$$\\operatorname{sys}(g)^2 \\le \\frac{2}{\\sqrt{3}} \\operatorname{area}(g).$$\nLet $\\Lambda_h \\subset \\R^2$ be the hexagonal lattice spanned by $(1, 0)$ and $(\\tfrac{1}{2}, \\tfrac{\\sqrt{3}}{2})$.\nEquality holds exactly when $g$ is flat and isometric to a constant multiple of the metric of $\\R^2 / \\Lambda_h$.\n\n## Remarks\n\nBy the uniformization theorem, $g$ is conformal to a flat metric $g_0$, so $g = f^2 g_0$ for a positive function $f$.\nAveraging $f$ over the translations of the flat torus gives a flat metric.\nBy the Cauchy–Schwarz inequality its area is at most that of $g$, and by averaging lengths over translates of a closed geodesic its systole is at least that of $g$.\nFor a flat torus $\\R^2 / \\Lambda$, the claim is that the shortest nonzero vector $v$ of a lattice $\\Lambda$ of covolume $A$ satisfies $\\abs{v}^2 \\le \\tfrac{2}{\\sqrt{3}} A$.\nThis follows by reduction of a basis of $\\Lambda$, and the hexagonal lattice is the extreme case.\n\nFor the square torus $\\R^2/\\Z^2$ the two sides are $1$ and $2/\\sqrt{3} \\approx 1.155$.\nFor the hexagonal torus above, the systole is $1$ and the area is $\\sqrt{3}/2$, so equality holds.\n\n## Explanations\n\n### L1 human\n\nOn any doughnut-shaped surface, take the shortest loop that goes around a hole.\nThe inequality says that the square of its length is at most about $1.155$ times the area.\nSo a surface whose short loops are all long must have a large area.\nThe best shape is the flat torus glued from a rhombus made of two equilateral triangles.\n\n### L2 human\n\nThis is the first result of [[systolic geometry|PZCJV40M4K]], and the model for [[Gromov's systolic inequality|HB2K72PSZ6]] in higher dimensions.\nThe constant $2/\\sqrt{3}$ is the Hermite constant $\\gamma_2$ of the geometry of numbers.\nThe proof pattern, reduction to the most symmetric metrics in a conformal class by averaging, recurs in other optimal systolic inequalities.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "about": [
      "PZCJV40M4K"
    ],
    "kind": "statement",
    "msc": [
      "53C23",
      "53C20"
    ],
    "name": "Loewner's torus inequality",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q3154012",
      "wikipedia": "Loewner's torus inequality"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Loewner%27s_torus_inequality"
      }
    ],
    "tag": "1Y9X1YW6GH",
    "uses": [
      "PZCJV40M4K",
      "Q61VFHCH2J"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/1Y9X1YW6GH/",
  "links": [
    {
      "field": "about",
      "target": "PZCJV40M4K"
    },
    {
      "field": "uses",
      "target": "PZCJV40M4K"
    },
    {
      "field": "uses",
      "target": "Q61VFHCH2J"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/1Y9X1YW6GH.md",
  "msc": [
    "53C23",
    "53C20"
  ],
  "name": "Loewner's torus inequality",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/1Y9/1Y9X1YW6GH.md",
  "status": null,
  "tag": "1Y9X1YW6GH",
  "uri": "m2://1Y9X1YW6GH",
  "url": "https://math2.tamnd.com/wiki/loewners-torus-inequality/",
  "writing": {
    "tag": "1Y9X1YW6GH",
    "kind": "statement",
    "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
      }
    ]
  }
}