{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n \\ge 1$ and give $\\R^{2n}$ the coordinates $(x_1, y_1, \\ldots, x_n, y_n)$ and the standard symplectic form $\\omega_0 = \\sum_{i=1}^n dx_i \\wedge dy_i$.\nFor $r \u003e 0$, let $B^{2n}(r)$ be the open ball of radius $r$ centered at $0$.\nFor $R \u003e 0$, let\n$$Z^{2n}(R) = \\set{(x, y) \\in \\R^{2n} : x_1^2 + y_1^2 \u003c R^2}$$\nbe the open cylinder of radius $R$ over a disc in the $(x_1, y_1)$-plane.\nIf there is a symplectic embedding $\\varphi \\colon B^{2n}(r) \\to Z^{2n}(R)$, that is, a smooth embedding with $\\varphi^* \\omega_0 = \\omega_0$, then $r \\le R$.\n\n## Remarks\n\nGromov proved the theorem in 1985 with [[pseudoholomorphic curves|N469E47V75]].\nAssume $r \u003e R$ and an embedding exists; compactify the cylinder to $S^2 \\times T^{2n-2}$ with $S^2$ of area slightly more than $\\pi R^2$.\nFor a suitable almost complex structure, there is a pseudoholomorphic sphere in the class of $S^2$ through the image of the center of the ball.\nIts area is less than $\\pi R^2 + \\varepsilon$.\nPulled back to the ball, it gives a proper minimal surface through the center with area less than $\\pi r^2$.\nThis contradicts the monotonicity formula, which says that such a surface has area at least $\\pi r^2$.\n\nThe hypothesis that the disc lies in a symplectic plane is needed.\nThe map $(x_1, y_1, x_2, y_2) \\mapsto (\\lambda x_1, \\lambda^{-1} y_1, \\lambda x_2, \\lambda^{-1} y_2)$ with small $\\lambda \u003e 0$ is a linear symplectomorphism that squeezes any ball into the cylinder $\\set{x_1^2 + x_2^2 \u003c R^2}$ over the $(x_1, x_2)$-plane, which is Lagrangian.\nFor $n = 1$ the theorem is just the statement that area-preserving maps preserve area.\n\n## Explanations\n\n### L1 human\n\nA symplectic map preserves volume, but in dimension four and higher it is much more rigid than that.\nA ball cannot be squeezed by a symplectic map through a round hole whose radius is smaller than the ball's, even though the cylinder over the hole has infinite volume.\nThis is sometimes called the principle of the symplectic camel.\n\n### L2 human\n\nThe theorem marks the birth of symplectic topology: it shows that symplectic maps are not just volume-preserving maps.\nIt leads to the notion of a [[symplectic capacity|8KCB5KJKNT]], a size of a symplectic set that is invariant under symplectic embeddings.\nBy Liouville's theorem, Hamiltonian flows preserve volume; non-squeezing gives a finer constraint on them, a classical analogue of the uncertainty principle.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "CAN490EPRT"
    ],
    "kind": "statement",
    "msc": [
      "53D35",
      "57R17"
    ],
    "name": "Gromov's non-squeezing theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q3527214",
      "wikipedia": "Non-squeezing theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Non-squeezing_theorem"
      }
    ],
    "tag": "RDX55HQ3XT",
    "uses": [
      "CAN490EPRT",
      "N469E47V75"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/RDX55HQ3XT/",
  "links": [
    {
      "field": "about",
      "target": "CAN490EPRT"
    },
    {
      "field": "uses",
      "target": "CAN490EPRT"
    },
    {
      "field": "uses",
      "target": "N469E47V75"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/RDX55HQ3XT.md",
  "msc": [
    "53D35",
    "57R17"
  ],
  "name": "Gromov's non-squeezing theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/RDX/RDX55HQ3XT.md",
  "status": null,
  "tag": "RDX55HQ3XT",
  "uri": "m2://RDX55HQ3XT",
  "url": "https://math2.tamnd.com/wiki/gromovs-non-squeezing-theorem/",
  "writing": {
    "tag": "RDX55HQ3XT",
    "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
      }
    ]
  }
}