{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "QWAZ4DK5AD"
    }
  ],
  "body": "\n## Summary\n\nA domain of finite type is a domain in $\\C^n$ with smooth boundary whose boundary has only finite order of contact with complex curves.\nThe type, introduced by D'Angelo, measures how flat the boundary can be in complex directions.\n[[Strongly pseudoconvex domains|Z7RVXBA2KB]] have type $2$, while a boundary that contains a complex disc has infinite type.\nFor pseudoconvex domains, finite type is the condition under which the $\\bar\\partial$-Neumann problem gains regularity, by work of Kohn and Catlin.\n\n## Explanations\n\n### L1 human\n\nTake a small holomorphic disc that touches the boundary of the domain at a point.\nMeasure how closely the disc hugs the boundary there, as an order of tangency.\nThe type of the boundary point is the largest such order, over all discs.\nOn the unit sphere a complex line touches only to order $2$, as a tangent line touches a circle.\nOn the boundary of $\\set{\\abs{z}^2 + \\abs{w}^4 \u003c 1}$ the disc $t \\mapsto (1, t)$ touches the boundary point $(1, 0)$ to order $4$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "32T25",
      "32F18"
    ],
    "name": "Domain of finite type",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Pseudoconvexity"
      }
    ],
    "tag": "ZJ0GBPREMB"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/ZJ0GBPREMB/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/ZJ0GBPREMB.md",
  "msc": [
    "32T25",
    "32F18"
  ],
  "name": "Domain of finite type",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZJ0/ZJ0GBPREMB.md",
  "status": null,
  "tag": "ZJ0GBPREMB",
  "uri": "m2://ZJ0GBPREMB",
  "url": "https://tamnd.github.io/math2/wiki/domain-of-finite-type/",
  "writing": {
    "tag": "ZJ0GBPREMB",
    "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
      }
    ]
  }
}