Pseudoconvexity

This article is about the notion in several complex variables. For the notion in convex analysis, see 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 C^n. Pseudoconvex sets are important, as they allow for classification of domains of holomorphy.

Let

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

be a domain, that is, an open connected subset. 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 iff 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.

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. JSTOR 1992976.
• 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. MR 0477104.
• Catlin, David (1983). "Necessary Conditions for Subellipticity of the ${\displaystyle {\bar {\partial }}}$-Neumann Problem". Annals of Mathematics. 117 (1): 147–171. doi:10.2307/2006974. JSTOR 2006974.
• 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.

External links

• 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]