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

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.

When G has a C2 (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 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.

[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.

[edit] See also

  • Levi pseudoconvex
  • solution of the Levi problem
  • exhaustion function
  • Stein manifold

This article incorporates material from Pseudoconvex on PlanetMath, which is licensed under the GFDL.