{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n \\ge 2$ be an integer, let $k$ be a real number and let $D \u003e 0$.\nLet $\\mathcal{M}$ be the class of closed [[Riemannian manifolds|7CBT4SM83Y]] $M$ of dimension $n$ whose [[Ricci curvature|4GTNM0WHDH]] satisfies $\\operatorname{Ric} \\ge (n - 1) k$ and whose diameter is at most $D$.\nThen every sequence in $\\mathcal{M}$ has a subsequence that converges in the [[Gromov–Hausdorff sense|GT56RTG9X5]] to a compact metric space.\n\n## Remarks\n\nThe proof rests on a criterion of Gromov.\nA class of compact metric spaces is precompact for $d_{GH}$ when the diameters are bounded and, for each $\\varepsilon \u003e 0$, the number of $\\varepsilon$-balls needed to cover each space is bounded.\nThis is a uniform version of being [[totally bounded|020Q06TYYH]].\nFor the manifolds in $\\mathcal{M}$, the [[Bishop–Gromov inequality|82F5Q4F1S6]] bounds the number of disjoint balls of radius $\\varepsilon/2$, and hence the number of covering balls of radius $\\varepsilon$.\n\nThe limit space need not be a manifold, and its dimension may drop.\nFor example, the flat tori $\\R^2 / (\\Z \\times \\tfrac{1}{m}\\Z)$ have $\\operatorname{Ric} = 0$ and bounded diameter, and they converge to a circle of length $1$ as $m \\to \\infty$.\nWithout the lower curvature bound the statement fails.\nTake spheres with $m$ long thin spikes of length $1$ whose tips are pairwise at distance at least $2$.\nThe diameters stay bounded, but at least $m$ balls of radius $\\tfrac{1}{2}$ are needed to cover, so no subsequence converges.\n\n## Explanations\n\n### L1 human\n\nManifolds whose curvature is not too negative and whose size is bounded cannot be too complicated at any scale.\nThe theorem says that from any sequence of them one can pick a subsequence that settles down to a limit shape.\nThe limit may be rougher than the manifolds, for example lower-dimensional.\n\n### L2 human\n\nThis is the starting point of the study of limit spaces of manifolds with Ricci curvature bounded below.\nUnder a lower bound on sectional curvature the limits are [[Alexandrov spaces|71AWVBG6KH]].\nThe theorem is used to prove finiteness and pinching results by contradiction: take a bad sequence, pass to a limit, and study the limit.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "about": [
      "GT56RTG9X5"
    ],
    "kind": "statement",
    "msc": [
      "53C23",
      "53C21"
    ],
    "name": "Gromov's compactness theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q5610188",
      "wikipedia": "Gromov's compactness theorem (geometry)"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Gromov%27s_compactness_theorem_(geometry)"
      }
    ],
    "tag": "YGSKZFTEX1",
    "uses": [
      "GT56RTG9X5",
      "7CBT4SM83Y",
      "4GTNM0WHDH",
      "82F5Q4F1S6",
      "020Q06TYYH",
      "71AWVBG6KH"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/YGSKZFTEX1/",
  "links": [
    {
      "field": "about",
      "target": "GT56RTG9X5"
    },
    {
      "field": "uses",
      "target": "GT56RTG9X5"
    },
    {
      "field": "uses",
      "target": "7CBT4SM83Y"
    },
    {
      "field": "uses",
      "target": "4GTNM0WHDH"
    },
    {
      "field": "uses",
      "target": "82F5Q4F1S6"
    },
    {
      "field": "uses",
      "target": "020Q06TYYH"
    },
    {
      "field": "uses",
      "target": "71AWVBG6KH"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/YGSKZFTEX1.md",
  "msc": [
    "53C23",
    "53C21"
  ],
  "name": "Gromov's compactness theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YGS/YGSKZFTEX1.md",
  "status": null,
  "tag": "YGSKZFTEX1",
  "uri": "m2://YGSKZFTEX1",
  "url": "https://math2.tamnd.com/wiki/gromovs-compactness-theorem/",
  "writing": {
    "tag": "YGSKZFTEX1",
    "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
      }
    ]
  }
}