Levi pseudoconvex

Let $G\subset{\mathbb{C}}^{n}$ 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 continuously differentiable function. Let $\rho\colon{\mathbb{C}}^{n}\to{\mathbb{R}}$ be a defining function of $G$, that is $\rho$ is a twice continuously differentiable function such that $\operatorname{grad}\rho(z)\not=0$ for $z\in\partial G$ and $G=\{z\in{\mathbb{C}}^{n}\mid\rho(z)<0\}$ (such a function always exists).

Definition.

Let $p\in\partial G$ (boundary of $G$). We call the space of vectors $w=(w_{1},\ldots,w_{n})\in{\mathbb{C}}^{n}$ such that

 $\sum_{k=1}^{n}\frac{\partial\rho}{\partial z_{k}}(p)w_{k}=0,$

the space of holomorphic tangent vectors at $p$ and denote it $T^{1,0}_{p}(\partial G)$.

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

Definition.

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

 $\sum_{j,k=1}^{n}\frac{\partial^{2}\rho}{\partial z_{j}\partial\bar{z}_{k}}(p)w% _{j}\bar{w}_{k}\geq 0,$

for all $w\in T^{1,0}_{p}(\partial G)$. The point is called strongly Levi pseudoconvex (or just strongly pseudoconvex or also strictly pseudoconvex) if the inequality above is strict. The expression on the left is called 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 $T^{1,0}_{p}(\partial G)$ and not on all of ${\mathbb{C}}^{n}$.

Definition.

The domain $G$ is called Levi pseudoconvex if every boundary point is Levi pseudoconvex. Similarly $G$ is called strongly Levi pseudoconvex if every boundary point is strongly Levi pseudoconvex.

Note that in particular all convex domains are pseudoconvex.

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

References

• 1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.
• 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
 Title Levi pseudoconvex Canonical name LeviPseudoconvex Date of creation 2013-03-22 14:30:37 Last modified on 2013-03-22 14:30:37 Owner jirka (4157) Last modified by jirka (4157) Numerical id 10 Author jirka (4157) Entry type Definition Classification msc 32T15 Classification msc 32T05 Related topic DomainOfHolomorphy Related topic Pseudoconvex Related topic BiholomorphismsOfStronglyPseudoconvexDomainsExtendToTheBoundary Defines Levi form Defines strongly Levi pseudoconvex Defines strongly pseudoconvex Defines strictly pseudoconvex Defines weakly pseudoconvex Defines weakly Levi pseudoconvex Defines holomorphic tangent vector