Pseudoconvexity

Template:About 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

$G\subset {ℂ}^n$

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 $\phi$ on $G$ such that the set

$\{z\in G\mid \phi (z)<x\}$

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 $\rho :{ℂ}^n\to ℝ$ 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,

$\mathrm{\nabla }\rho (p)w=\sum _{i=1}^n\frac{\partial \rho (p)}{\partial z_j}{w}_j=0$, we have

$\sum _{i,j=1}^n\frac{{\partial }^2\rho (p)}{\partial z_i\partial \overline{z_j}}{w}_i\overline{w_j}\ge 0.$

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 $C^{\mathrm{\infty }}$ (smooth) boundary which are relatively compact in $G$, such that

$G=\bigcup _{k=1}^{\mathrm{\infty }}{G}_k.$

This is because once we have a $\phi$ 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.

See also

• Analytic polyhedron
• Eugenio Elia Levi
• Holomorphically convex hull
• Stein manifold

References

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• Lars Hörmander, An Introduction to Complex Analysis in Several Variables, North-Holland, 1990. (Template:ISBN).
• Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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External links

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