---
tag: 5BW600RVX5
kind: definition
name: Hartogs domain
msc: [32Q02, 32D05]
defines: THH2YHW8G5
uses: [02BH3MCBQB, 0SPE8WNWFV]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudoconvexity
---

## Definition

Let $n \ge 2$ and write a point of $\C^n$ as $(z', z_n)$ with $z' \in \C^{n-1}$ and $z_n \in \C$.
A *Hartogs domain* is a [[connected|0SPE8WNWFV]] [[open set|02BH3MCBQB]] $D \subseteq \C^n$ such that $(z', e^{i\theta} z_n) \in D$ for all $(z', z_n) \in D$ and all $\theta \in \R$.
It is a *complete Hartogs domain* if moreover $(z', \lambda z_n) \in D$ for all $(z', z_n) \in D$ and all $\lambda \in \C$ with $\abs{\lambda} \le 1$.

## Remarks

Example: $D = \set{(z_1, z_2) \in \C^2 : \abs{z_1} < 1, \ \abs{z_2} < 1 - \abs{z_1}^2}$.
It is open and connected, and the condition on $z_2$ involves only $\abs{z_2}$.
This number does not change under $z_2 \mapsto e^{i\theta} z_2$, and it does not increase under $z_2 \mapsto \lambda z_2$ with $\abs{\lambda} \le 1$.
So $D$ is a complete Hartogs domain.

Non-example: $D = \set{(z_1, z_2) \in \C^2 : \abs{z_1 - z_2} < 1}$.
The point $(1, 1/2)$ lies in $D$, but its rotation $(1, -1/2)$ does not, since $\abs{1 + 1/2} > 1$.

The [[Hartogs figure|GHJC53RFBT]] is a Hartogs domain that is not complete.
A [[Reinhardt domain|WA0SE56FVH]] is invariant under rotation of each coordinate separately, so it is a Hartogs domain in particular.