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

[math]\displaystyle{ G\subset {\mathbb{C}}^n }[/math]

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

[math]\displaystyle{ \{ z \in G \mid \varphi(z) \lt x \} }[/math]

is a relatively compact subset of [math]\displaystyle{ G }[/math] for all real numbers [math]\displaystyle{ x. }[/math] In other words, a domain is pseudoconvex if [math]\displaystyle{ G }[/math] has a continuous plurisubharmonic exhaustion function. Every (geometrically) convex set is pseudoconvex. However, there are pseudoconvex domains which are not geometrically convex.

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

[math]\displaystyle{ \nabla \rho(p) w = \sum_{i=1}^n \frac{\partial \rho (p)}{ \partial z_j }w_j =0 }[/math], we have

[math]\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. }[/math]

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

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

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

[math]\displaystyle{ G = \bigcup_{k=1}^\infty G_k. }[/math]

This is because once we have a [math]\displaystyle{ \varphi }[/math] 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

• Bremermann, H. J. (1956). "Complex Convexity". Transactions of the American Mathematical Society 82 (1): 17–51. doi:10.1090/S0002-9947-1956-0079100-2. 
• 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.
• Siu, Yum-Tong (1978). "Pseudoconvexity and the problem of Levi". Bulletin of the American Mathematical Society 84 (4): 481–513. doi:10.1090/S0002-9904-1978-14483-8. 
• Catlin, David (1983). "Necessary Conditions for Subellipticity of the [math]\displaystyle{ \bar\partial }[/math]-Neumann Problem". Annals of Mathematics 117 (1): 147–171. doi:10.2307/2006974. https://www.jstor.org/stable/2006974. 
• Zimmer, Andrew (2019). "Characterizing strong pseudoconvexity, obstructions to biholomorphisms, and Lyapunov exponents". Mathematische Annalen 374 (3–4): 1811–1844. doi:10.1007/s00208-018-1715-7. 
• 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. doi:10.2140/pjm.2018.297.79. 

External links

• Range, R. Michael (February 2012), "WHAT IS...a Pseudoconvex Domain?", Notices of the American Mathematical Society 59 (2): 301–303, doi:10.1090/noti798, https://www.ams.org/notices/201202/rtx120200301p.pdf 
• Hazewinkel, Michiel, ed. (2001), "Pseudo-convex and pseudo-concave", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/p075650