Pseudoconvexity information

In mathematics, more precisely in the theory of functions of several complex variables, a pseudoconvex set is a special type of open set in the n-dimensional complex space Cⁿ. Pseudoconvex sets are important, as they allow for classification of domains of holomorphy.

Let

G\subset {\mathbb {C} }^{n}

be a domain, that is, an open connected subset. One says that $G$ is pseudoconvex (or Hartogs pseudoconvex) if there exists a continuous plurisubharmonic function $\varphi$ on $G$ such that the set

\{z\in G\mid \varphi (z)<x\}

is a relatively compact subset of $G$ for all real numbers $x.$ In other words, a domain is pseudoconvex if $G$ has a continuous plurisubharmonic exhaustion function. Every (geometrically) convex set is pseudoconvex. However, there are pseudoconvex domains which are not geometrically convex.

When $G$ has a $C^{2}$ (twice continuously differentiable) boundary, this notion is the same as Levi pseudoconvexity, which is easier to work with. More specifically, with a $C^{2}$ boundary, it can be shown that $G$ has a defining function, i.e., that there exists $\rho :\mathbb {C} ^{n}\to \mathbb {R}$ which is $C^{2}$ so that $G=\{\rho <0\}$ , and $\partial G=\{\rho =0\}$ . Now, $G$ is pseudoconvex iff for every $p\in \partial G$ and $w$ in the complex tangent space at p, that is,

\nabla \rho (p)w=\sum _{i=1}^{n}{\frac {\partial \rho (p)}{\partial z_{j}}}w_{j}=0

, we have

\sum _{i,j=1}^{n}{\frac {\partial ^{2}\rho (p)}{\partial z_{i}\partial {\bar {z_{j}}}}}w_{i}{\bar {w_{j}}}\geq 0.

The definition above is analogous to definitions of convexity in Real Analysis.

If $G$ does not have a $C^{2}$ boundary, the following approximation result can be useful.

Proposition 1 If $G$ is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains $G_{k}\subset G$ with $C^{\infty }$ (smooth) boundary which are relatively compact in $G$ , such that

G=\bigcup _{k=1}^{\infty }G_{k}.

This is because once we have a $\varphi$ as in the definition we can actually find a C^∞ exhaustion function.

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