Pseudoconvexity

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.

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.