Rahul Sharma (Editor)

Pseudoconvexity

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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 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 φ on G such that the set

{ z G φ ( 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.

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 ρ : C n R which is C 2 so that G = { ρ < 0 } , and G = { ρ = 0 } . Now, G is pseudoconvex iff for every p G and w in the complex tangent space at p, that is,

ρ ( p ) w = i = 1 n ρ ( p ) z j w j = 0 , we have i , j = 1 n 2 ρ ( p ) z i z j ¯ w i w j ¯ 0.

If G does not have a C 2 boundary, the following approximation result can come in useful.

Proposition 1 If G is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains G k G with C (smooth) boundary which are relatively compact in G , such that

G = k = 1 G k .

This is because once we have a φ 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.

References

Pseudoconvexity Wikipedia


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