Holomorphically convex hull

In mathematics, more precisely in complex analysis, the holomorphically convex hull of a given compact set in the n-dimensional complex space Cⁿ is defined as follows. Let <math>G \subset {\mathbb{C}}^n</math> be a domain (an open and connected set), or alternatively for a more general definition, let <math>G</math> be an <math>n</math> dimensional complex analytic manifold. Further let <math>{\mathcal{O}}(G)</math> stand for the set of holomorphic functions on <math>G.</math> For <math>K \subset G</math> a compact set, we define the holomorphically convex hull of <math>K</math> as

<math> \hat{K}_G := \{ z \in G \big| \left| f(z) \right| \leq \sup_{w \in K} \left| f(w) \right| \mbox{ for all } f \in {\mathcal{O}}(G) \} .</math>

(One obtains the related concept of polynomially convex hull by requiring in the above definition that f be a polynomial, which is more than just asking for a holomorphic function.) The domain <math>G</math> is called holomorphically convex if for every <math>K \subset G</math> compact in <math>G</math>, <math>\hat{K}_G</math> is also compact in <math>G</math>. Sometimes this is just abbreviated as holomorph-convex. When <math>n=1</math>, any domain <math>G</math> is holomorphically convex since then <math>\hat{K}_G = K</math> for all compact <math>K \subset G.</math> Also note that for n=1 being holomorphically convex is the same as being a domain of holomorphy. These concepts are more important in the case n > 1 of several complex variables.

See also

• Stein manifold

References

• Lars Hörmander. An Introduction to Complex Analysis in Several Variables, North-Holland Publishing Company, New York, New York, 1973.
• Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.

This article incorporates material from Holomorphically convex on PlanetMath, which is licensed under the GFDL.