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

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

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]