Levi form

Definition

Let $U \subseteq \C^n$ be an open set and let $\rho \colon U \to \R$ be a function of class $C^2$ (see smooth function). The Levi form of $\rho$ at a point $p \in U$ is the Hermitian form $$L_\rho(p; w) = \sum_{j,k=1}^{n} \frac{\partial^2 \rho}{\partial z_j \partial \bar z_k}(p)\, w_j \bar w_k, \qquad w \in \C^n.$$ It is also called the complex Hessian of $\rho$.

Now let $\Omega \subseteq \C^n$ be a domain with $C^2$ boundary. A defining function of $\Omega$ is a $C^2$ function $\rho$ on a neighbourhood $U$ of $\partial\Omega$ with $\Omega \cap U = \set{z \in U : \rho(z) < 0}$ and $d\rho \ne 0$ on $\partial\Omega$. For $p \in \partial\Omega$, the complex tangent space of $\partial\Omega$ at $p$ is $$H_p = \set{w \in \C^n : \sum_{j=1}^{n} \frac{\partial \rho}{\partial z_j}(p)\, w_j = 0}.$$ The Levi form of $\partial\Omega$ at $p$ is the restriction of $L_\rho(p; \cdot)$ to $H_p$.