Levi pseudoconvex

Let $G\subset {ℂ}^n$ be a domain (http://planetmath.org/Domain2) (open connected subset) with $C^2$boundary, that is the boundary is locally the graph of a twice continuouslydifferentiable function. Let $\rho :{ℂ}^n\to ℝ$be a defining function of $G$,that is $\rho$ is a twice continuously differentiable function such that$\mathrm{grad}\rho (z)\ne 0$ for $z\in \partial G$ and (such a function always exists).

Definition.

Let $p\in \partial G$ (boundary of $G$).We call the space of vectors $w=(w_1,\mathrm{\dots },w_n)\in {ℂ}^n$such that

$$\sum _{k=1}^n\frac{\partial \rho }{\partial z_k}(p)w_k=0,$$

the space of holomorphic tangent vectorsat $p$ and denote it$T_p^{1,0}(\partial G)$.

$T_p^{1,0}(\partial G)$ is an $n-1$ dimensional complex vector spaceand is a subspace of the complexified real tangent space (http://planetmath.org/TangentSpace), that is $ℂ{\otimes }_{ℝ}{T}_p(\partial G)$.

Note that when $n=1$ then the complextangent space contains just the zero vector.

Definition.

The point $p\in \partial G$is called Levi pseudoconvex (or just pseudoconvex)if

$$\sum _{j,k=1}^n\frac{{\partial }^2\rho }{\partial z_j\partial {\overline{z}}_k}(p)w_j{\overline{w}}_k\ge 0,$$

for all $w\in T_p^{1,0}(\partial G)$. The point iscalled strongly Levi pseudoconvex (or just strongly pseudoconvex or also strictly pseudoconvex)if the inequality above is strict. The expression on the left iscalled the Levi form.

Note that if a point is not strongly Levi pseudoconvex then it is sometimes called a weakly Levi pseudoconvex point.

The Levi form really acts on an $n-1$ dimensional space, so the expression above may be confusing as it only acts on $T_p^{1,0}(\partial G)$ and not on allof ${ℂ}^n$.

Definition.

The domain $G$is called Levi pseudoconvexif every boundary point isLevi pseudoconvex. Similarly$G$ is called strongly Levi pseudoconvexif every boundary point isstrongly Levi pseudoconvex.

Note that in particular all convex domains are pseudoconvex.

It turns out that $G$ with $C^2$ boundary is a domain of holomorphyif and only if$G$ is Levi pseudoconvex.