Pseudoconvexity

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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

<math>G\subset {\mathbb{C}}^n</math>

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

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

is a relatively compact subset of <math>G</math> for all real numbers <math>x.</math> In other words, a domain is pseudoconvex if <math>G</math> has a continuous plurisubharmonic exhaustion function.

When <math>G</math> has a <math>C^2</math> (twice continuously differentiable) boundary, this notion is the same as Levi pseudoconvexity, which is easier to work with. Otherwise, the following approximation result can come in useful.

Proposition 1 If <math>G</math> is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains <math>G_k \subset G</math> with <math>C^\infty</math> (smooth) boundary which are relatively compact in <math>G</math>, such that

<math>G = \bigcup_{k=1}^\infty G_k.</math>

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

[edit] The case n=1

In one complex dimension, every open domain is pseudoconvex. The concept of pseudoconvexity is thus more useful in dimensions higher than 1.

[edit] References

Lars Hörmander, An Introduction to Complex Analysis in Several Variables, North-Holland, 1990. (ISBN 0-444-88446-7).
Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.