Strongly pseudoconvex domain

Remarks

A strongly pseudoconvex domain is Levi pseudoconvex. Many authors also ask $\Omega$ to be bounded. Example: the unit ball, with $\rho(z) = \abs{z}^2 - 1$, has Levi form $\abs{w}^2$, which is positive for $w \ne 0$. Non-example: the domain $\set{z \in \C^2 : \abs{z_1}^2 + \abs{z_2}^4 < 1}$ is Levi pseudoconvex but not strongly pseudoconvex. At $p = (1, 0)$, with $\rho(z) = \abs{z_1}^2 + \abs{z_2}^4 - 1$, the space $H_p$ is $\set{w : w_1 = 0}$ and $L_\rho(p; (0, w_2)) = 4 \abs{p_2}^2 \abs{w_2}^2 = 0$. A bounded domain $\Omega$ with $C^2$ boundary is strongly pseudoconvex if and only if it has a defining function that is strictly plurisubharmonic near $\partial\Omega$. This means that $L_\rho(p; \cdot)$ is positive definite on all of $\C^n$, not only on $H_p$.