pseudoconvex

Definition.

Let $G\subset {ℂ}^n$ be a domain (open connected subset).We say $G$ is pseudoconvex (or Hartogs pseudoconvex) if there exists a continuous plurisubharmonic function $\phi$ on $G$ such thatthe sets are relatively compactsubsets of $G$ for all $x\in ℝ$. That is we say that$G$ has a continuous plurisubharmonic exhaustion function.

When $G$ has a $C^2$ (twice continuously differentiable) boundary then thisnotion is the same as Levi pseudoconvexity (http://planetmath.org/LeviPseudoconvex), whichis easier to work with if you have such nice boundaries. If you don’t havenice boundaries then the following approximation result can come in useful.

Proposition.

If $G\mathrm{\subset }{\mathrm{C}}^n$ is pseudoconvex then there exist bounded,strongly Levi pseudoconvex domains $G_k\mathrm{\subset }G$ with $C^{\mathrm{\infty }}$ (smooth)boundary which are relatively compactin $G$, such that $G\mathrm{=}{\mathrm{\bigcup }}_{k\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}{G}_k$.

This is because once we have a $\phi$ as in the definition we can actually find a $C^{\mathrm{\infty }}$ exhaustion function.

The reason for the definition of pseudoconvexity is that it classifies domains of holomorphy. One thing to note then is that every open domain in one complexdimension (in the complex plane $ℂ$) is then pseudoconvex.