---
tag: ZJ0GBPREMB
kind: concept
name: Domain of finite type
msc: [32T25, 32F18]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudoconvexity
---

## Summary

A domain of finite type is a domain in $\C^n$ with smooth boundary whose boundary has only finite order of contact with complex curves.
The type, introduced by D'Angelo, measures how flat the boundary can be in complex directions.
[[Strongly pseudoconvex domains|Z7RVXBA2KB]] have type $2$, while a boundary that contains a complex disc has infinite type.
For pseudoconvex domains, finite type is the condition under which the $\bar\partial$-Neumann problem gains regularity, by work of Kohn and Catlin.

## Explanations

### L1 human

Take a small holomorphic disc that touches the boundary of the domain at a point.
Measure how closely the disc hugs the boundary there, as an order of tangency.
The type of the boundary point is the largest such order, over all discs.
On the unit sphere a complex line touches only to order $2$, as a tangent line touches a circle.
On the boundary of $\set{\abs{z}^2 + \abs{w}^4 < 1}$ the disc $t \mapsto (1, t)$ touches the boundary point $(1, 0)$ to order $4$.