{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "M03K2ZWNST"
    }
  ],
  "body": "\n## Summary\n\nLet $\\Omega \\subseteq \\C^n$ be a domain and $w \\in \\Omega$.\nThe pluricomplex Green function $g_\\Omega(\\cdot, w)$ is the largest negative [[plurisubharmonic function|FF2P7ER3QT]] on $\\Omega$ with a logarithmic pole at $w$.\nIt was introduced by Maciej Klimek as the analogue of the classical [[Green's function|M36SD56MWQ]] of a planar domain.\nOn bounded hyperconvex domains it is continuous and solves a homogeneous [[complex Monge–Ampère|MXRC891TQS]] equation off the pole, with a Dirac mass at the pole.\nOn bounded convex domains it is expressed through the [[Kobayashi distance|DY8QQ5GTTX]], by a theorem of Lempert.\nThe version with pole at infinity for a compact set is the [[Siciak extremal function|QBJ9R06GYE]].\n\n## Explanations\n\n### L1 human\n\nIn the plane, the Green function of a domain with pole at $w$ is harmonic away from $w$.\nIt tends to $-\\infty$ like $\\log\\abs{z - w}$ at the pole and tends to $0$ at the boundary.\nIn several complex variables, harmonic is replaced by plurisubharmonic, and one takes the largest such function that is negative and has the logarithmic pole.\nFor the unit ball and the pole at the origin, the pluricomplex Green function is simply $\\log\\abs{z}$.\nIt measures how far a point is from the pole in a way that holomorphic maps cannot increase.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "32U35"
    ],
    "name": "Pluricomplex Green function",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Plurisubharmonic_function"
      }
    ],
    "tag": "3MQVKVGYYY"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/3MQVKVGYYY/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/3MQVKVGYYY.md",
  "msc": [
    "32U35"
  ],
  "name": "Pluricomplex Green function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3MQ/3MQVKVGYYY.md",
  "status": null,
  "tag": "3MQVKVGYYY",
  "uri": "m2://3MQVKVGYYY",
  "url": "https://tamnd.github.io/math2/wiki/pluricomplex-green-function/",
  "writing": {
    "tag": "3MQVKVGYYY",
    "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
      }
    ]
  }
}