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

${\displaystyle G\subset {\mathbb {C} }^{n}}${\displaystyle G\subset {\mathbb {C} }^{n}}

be a domain, that is, an open connected subset. One says that ${\displaystyle G}${\displaystyle G} is pseudoconvex (or Hartogs pseudoconvex) if there exists a continuous plurisubharmonic function ${\displaystyle \varphi }${\displaystyle \varphi } on ${\displaystyle G}${\displaystyle G} such that the set

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

is a relatively compact subset of ${\displaystyle G}${\displaystyle G} for all real numbers ${\displaystyle x.}${\displaystyle x.} In other words, a domain is pseudoconvex if ${\displaystyle G}${\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 ${\displaystyle G}${\displaystyle G} has a ${\displaystyle C^{2}}${\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 ${\displaystyle C^{2}}${\displaystyle C^{2}} boundary, it can be shown that ${\displaystyle G}${\displaystyle G} has a defining function, i.e., that there exists ${\displaystyle \rho :\mathbb {C} ^{n}\to \mathbb {R} }${\displaystyle \rho :\mathbb {C} ^{n}\to \mathbb {R} } which is ${\displaystyle C^{2}}${\displaystyle C^{2}} so that ${\displaystyle G=\{\rho <0\}}${\displaystyle G=\{\rho <0\}}, and ${\displaystyle \partial G=\{\rho =0\}}${\displaystyle \partial G=\{\rho =0\}}. Now, ${\displaystyle G}${\displaystyle G} is pseudoconvex iff for every ${\displaystyle p\in \partial G}${\displaystyle p\in \partial G} and ${\displaystyle w}${\displaystyle w} in the complex tangent space at p, that is,

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

${\displaystyle \sum _{i,j=1}^{n}{\frac {\partial ^{2}\rho (p)}{\partial z_{i}\partial {\bar {z_{j}}}}}w_{i}{\bar {w_{j}}}\geq 0.}${\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 ${\displaystyle G}${\displaystyle G} does not have a ${\displaystyle C^{2}}${\displaystyle C^{2}} boundary, the following approximation result can be useful.

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

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

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

In one complex dimension, every open domain is pseudoconvex. The concept of pseudoconvexity is thus more useful in dimensions higher than 1.

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

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• Zimmer, Andrew (2019). "Characterizing strong pseudoconvexity, obstructions to biholomorphisms, and Lyapunov exponents". Mathematische Annalen. 374 (3–4): 1811–1844. arXiv:1703.01511 . doi:10.1007/s00208-018-1715-7. S2CID 253714537.
• Fornæss, John; Wold, Erlend (2018). "A non-strictly pseudoconvex domain for which the squeezing function tends to 1 towards the boundary". Pacific Journal of Mathematics. 297: 79–86. arXiv:1611.04464 . doi:10.2140/pjm.2018.297.79. S2CID 119149200.

This article incorporates material from Pseudoconvex on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

• Range, R. Michael (February 2012), "WHAT IS...a Pseudoconvex Domain?" (PDF), Notices of the American Mathematical Society, 59 (2): 301–303, doi:10.1090/noti798
• "Pseudo-convex and pseudo-concave", Encyclopedia of Mathematics , EMS Press, 2001 [1994]

• Several complex variables

• Wikipedia articles incorporating text from PlanetMath