pseudoconvexity

{{about|the notion in several complex variables|the notion in convex analysis|pseudoconvex function}}

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.

References

{{cite journal |jstor=1992976|title=Complex Convexity|last1=Bremermann|first1=H. J.|journal=Transactions of the American Mathematical Society|year=1956|volume=82|issue=1|pages=17–51|doi=10.1090/S0002-9947-1956-0079100-2|doi-access=free}}
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.
{{cite journal |doi=10.1090/S0002-9904-1978-14483-8|title=Pseudoconvexity and the problem of Levi|year=1978|last1=Siu|first1=Yum-Tong|journal=Bulletin of the American Mathematical Society|volume=84|issue=4|pages=481–513|mr=0477104|doi-access=free}}
{{cite journal |url=https://www.jstor.org/stable/2006974|jstor=2006974 |last1=Catlin |first1=David |title=Necessary Conditions for Subellipticity of the $\bar\partial$ -Neumann Problem |journal=Annals of Mathematics |year=1983 |volume=117 |issue=1 |pages=147–171 |doi=10.2307/2006974 }}
{{cite journal |doi=10.1007/s00208-018-1715-7 |title=Characterizing strong pseudoconvexity, obstructions to biholomorphisms, and Lyapunov exponents |year=2019 |last1=Zimmer |first1=Andrew |journal=Mathematische Annalen |volume=374 |issue=3–4 |pages=1811–1844 |arxiv=1703.01511 |s2cid=253714537 }}
{{cite journal |doi=10.2140/pjm.2018.297.79 |title=A non-strictly pseudoconvex domain for which the squeezing function tends to 1 towards the boundary |year=2018 |last1=Fornæss |first1=John |last2=Wold |first2=Erlend |journal=Pacific Journal of Mathematics |volume=297 |pages=79–86 |arxiv=1611.04464 |s2cid=119149200 }}

External links

{{Citation |last=Range |first= R. Michael |date=February 2012 |title=WHAT IS...a Pseudoconvex Domain? |journal=Notices of the American Mathematical Society |volume=59 |issue=2 |pages=301–303 |url=https://www.ams.org/notices/201202/rtx120200301p.pdf |doi=10.1090/noti798|doi-access=free }}
{{springer|title=Pseudo-convex and pseudo-concave|id=p/p075650}}

Category:Several complex variables

pseudoconvexity

The case ''n'' = 1

See also

References

External links