PlanetMath: pseudoconvex

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pseudoconvex

(Definition)

Definition 1 Let $G \subset {\mathbb{C}}^n$ be a domain (open connected subset). We say

is pseudoconvex (or Hartogs pseudoconvex) if there exists a continuous plurisubharmonic function $\varphi$ on

such that the sets $\{ z \in G \mid \varphi(z) < x \}$ are relatively compact subsets of

for all $x \in {\mathbb{R}}$ . That is we say that

has a continuous plurisubharmonic exhaustion function.

When has a (twice continuously differentiable) boundary then this notion is the same as Levi pseudoconvexity, which is easier to work with if you have such nice boundaries. If you don't have nice boundaries then the following approximation result can come in useful.

Proposition 1 If $G \subset {\mathbb{C}}^n$ is pseudoconvex then there exist bounded, strongly Levi pseudoconvex domains $G_k \subset G$ with $C^\infty$ (smooth) boundary which are relatively compact in , 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 $C^\infty$ 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 complex dimension (in the complex plane ${\mathbb{C}}$ ) is then pseudoconvex.

Bibliography

1: Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.

"pseudoconvex" is owned by jirka.

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Other names:

Hartogs pseudoconvex

Attachments:: Levi pseudoconvex (Definition) by jirka

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Cross-references: complex plane, dimension, complex, domains of holomorphy, smooth, strongly Levi pseudoconvex, bounded, approximation, boundary, continuously differentiable, exhaustion function, relatively compact, plurisubharmonic function, continuous, subset, connected, open, domain
There are 12 references to this entry.

This is version 2 of pseudoconvex, born on 2004-08-02, modified 2005-03-07.
Object id is 6056, canonical name is Pseudoconvex.
Accessed 3857 times total.

Classification:

AMS MSC:	32T05 (Several complex variables and analytic spaces :: Pseudoconvex domains :: Domains of holomorphy)
	32T15 (Several complex variables and analytic spaces :: Pseudoconvex domains :: Strongly pseudoconvex domains)

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