{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "2J9GA17YAK"
    },
    {
      "field": "about",
      "target": "3G1CJ6HJMK"
    },
    {
      "field": "uses",
      "target": "3G1CJ6HJMK"
    }
  ],
  "body": "\n## Summary\n\nA harmonic function on a [[Riemannian manifold|7CBT4SM83Y]] is a function killed by the [[Laplace–Beltrami operator|F4NWZNTRMD]] of the metric.\nThis extends the classical [[harmonic functions|0DEB0JMN0W]] of Euclidean space to curved spaces.\nTheir behaviour depends on the geometry: on a connected closed manifold they are constant.\nOn a complete manifold with nonnegative Ricci curvature, the positive ones are constant.\nThey link analysis with geometry, through gradient estimates, Liouville theorems and the study of the manifold at infinity.\n\n## Explanations\n\n### L1 human\n\nOn a curved surface, one can still ask for functions whose value at each point is the average of nearby values, in a sense suited to the curvature.\nThe Laplace–Beltrami operator makes this precise, and harmonic functions are the functions it sends to $0$.\nOn a closed surface such as a sphere, only constant functions are harmonic, because a nonconstant function would have a maximum where it cannot be an average.\nOn the hyperbolic plane there are many bounded harmonic functions, which reflects the large size of the space at infinity.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "31C12",
      "58J05"
    ],
    "name": "Harmonic functions on Riemannian manifolds",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Laplace%E2%80%93Beltrami_operator"
      }
    ],
    "tag": "PK2KNMWNW3"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/PK2KNMWNW3/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/PK2KNMWNW3.md",
  "msc": [
    "31C12",
    "58J05"
  ],
  "name": "Harmonic functions on Riemannian manifolds",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PK2/PK2KNMWNW3.md",
  "status": null,
  "tag": "PK2KNMWNW3",
  "uri": "m2://PK2KNMWNW3",
  "url": "https://tamnd.github.io/math2/wiki/harmonic-functions-on-riemannian-manifolds/",
  "writing": {
    "tag": "PK2KNMWNW3",
    "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
      }
    ]
  }
}