Pseudoconvexity Explained

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 { 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: C ^{n} \to 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 ^{n} _{i =1} (\partial \rho (p))/(\partial z _{j}) w _{j} =0 , we have\sum ^{n} _{i,j =1} (\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 = c up ^{inf ty} _{k =1} 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.

See also

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

References

• 1992976. Complex Convexity. Bremermann. H. J.. Transactions of the American Mathematical Society. 1956. 82. 1. 17–51. 10.1090/S0002-9947-1956-0079100-2. free.
• Lars Hörmander, An Introduction to Complex Analysis in Several Variables, North-Holland, 1990. .
• Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
• 10.1090/S0002-9904-1978-14483-8. Pseudoconvexity and the problem of Levi. 1978. Siu. Yum-Tong. Bulletin of the American Mathematical Society. 84. 4. 481–513. 0477104. free.
• Necessary Conditions for Subellipticity of the \bar\partial -Neumann Problem . 2006974 . Catlin . David . Annals of Mathematics . 1983 . 117 . 1 . 147–171 . 10.2307/2006974 .
• 10.1007/s00208-018-1715-7 . Characterizing strong pseudoconvexity, obstructions to biholomorphisms, and Lyapunov exponents . 2019 . Zimmer . Andrew . Mathematische Annalen . 374 . 3–4 . 1811–1844 . 1703.01511 . 253714537 .
• 10.2140/pjm.2018.297.79 . A non-strictly pseudoconvex domain for which the squeezing function tends to 1 towards the boundary . 2018 . Fornæss . John . Wold . Erlend . Pacific Journal of Mathematics . 297 . 79–86 . 1611.04464 . 119149200 .