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 ${\displaystyle G\subset {\mathbb {C} }^{n}}$ be a domain. One says that ${\displaystyle G}$ is pseudoconvex (or Hartogs pseudoconvex) if there exists a continuous plurisubharmonic function ${\displaystyle \varphi }$ on ${\displaystyle G}$ such that the set ${\displaystyle \{z\in G\mid \varphi (z)<x\}}$ is a relatively compact subset of ${\displaystyle G}$ for all real numbers ${\displaystyle x.}$ In other words, a domain is pseudoconvex if ${\displaystyle G}$ has a continuous plurisubharmonic exhaustion function. Every (geometrically) convex set is pseudoconvex. However, there are pseudoconvex domains which are not geometrically convex.

When ${\displaystyle G}$ has a ${\displaystyle C^{2}}$ (twice continuously differentiable) boundary, this notion is the same as Levi pseudoconvexity, which is easier to work with. More specifically, with a ${\displaystyle C^{2}}$ boundary, it can be shown that ${\displaystyle G}$ has a defining function, i.e., that there exists ${\displaystyle \rho :\mathbb {C} ^{n}\to \mathbb {R} }$ which is ${\displaystyle C^{2}}$ so that ${\displaystyle G=\{\rho <0\}}$, and ${\displaystyle \partial G=\{\rho =0\}}$. Now, ${\displaystyle G}$ is pseudoconvex if and only if for every ${\displaystyle p\in \partial G}$ and ${\displaystyle w}$ in the complex tangent space at p, that is,

${\displaystyle \nabla \rho (p)w=\sum _{i=1}^{n}{\frac {\partial \rho (p)}{\partial z_{j}}}w_{j}=0}$, we have

${\displaystyle \sum _{i,j=1}^{n}{\frac {\partial ^{2}\rho (p)}{\partial z_{i}\partial {\bar {z_{j}}}}}w_{i}{\bar {w_{j}}}\geq 0.}$

The definition above is analogous to definitions of convexity in Real Analysis.

If ${\displaystyle G}$ does not have a ${\displaystyle C^{2}}$ boundary, the following approximation result can be useful.

Proposition 1 If ${\displaystyle G}$ is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains ${\displaystyle G_{k}\subset G}$ with ${\displaystyle C^{\infty }}$ (smooth) boundary which are relatively compact in ${\displaystyle G}$, such that

${\displaystyle G=\bigcup _{k=1}^{\infty }G_{k}.}$

This is because once we have a ${\displaystyle \varphi }$ as in the definition we can actually find a C^∞ exhaustion function.

In one complex dimension, every open domain is pseudoconvex.