Levi pseudoconvex domains

Fix an integer $m\ge 1$. Use the one-based coordinate, vector-component and Wirtinger-operator aliases of The Levi form and strict plurisubharmonicity: for $1\le j\le m$, $z_j$ and $v_j$ denote the canonical coordinate and vector component with index $j-1$, and ${\mathrm{\partial }}_{z_j}$ denotes the canonical Wirtinger operator with index $j-1$. All coordinates, vector components and derivatives below use these aliases.

Let $\mathrm{\Omega }\subseteq {\mathbb{C}}^m$ be a domain with $C^2$ boundary. We say that $\mathrm{\Omega }$ is Levi pseudoconvex when for every boundary point $p\in \mathrm{\partial }\mathrm{\Omega }$ there are a neighbourhood $U$ of $p$ and a function $\rho \in C^2(U,\mathbb{R})$ such that

$\mathrm{\Omega }\cap U=\{z\in U:\rho (z)<0\},d\rho (p)\mathrm{\ne }0,$

and

${\mathcal{L}}_{\rho }(p;v)\ge 0$

for every complex tangent vector $v\in {\mathbb{C}}^m$ satisfying

${\sum }_{j=1}^m\frac{\mathrm{\partial }\rho }{\mathrm{\partial }{z}_j}(p)v_j=0.$