Hartogs domain

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 is a Hartogs domain that is not complete. A Reinhardt domain is invariant under rotation of each coordinate separately, so it is a Hartogs domain in particular.