Domain of holomorphy

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The sets in the definition.

In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a set which is maximal in the sense that there exists a holomorphic function on this set which cannot be extended to a bigger set.

Formally, an open set \Omega in the n-dimensional complex space {\mathbb{C}}^n is called a domain of holomorphy if there do not exist non-empty open sets U \subset \Omega and V \subset {\mathbb{C}}^n where V is connected, V \not\subset \Omega and U \subset \Omega \cap V such that for every holomorphic function f on \Omega there exists a holomorphic function g on V with f = g on U

In the n=1 case, every open set is a domain of holomorphy: we can define a holomorphic function with zeros accumulating everywhere on the boundary of the domain, which must then be a natural boundary for a domain of definition of its inverse. For n \geq 2 this is no longer true, as it follows from Hartogs' lemma.

Equivalent conditions[edit]

For a domain \Omega the following conditions are equivalent:

  1. \Omega is a domain of holomorphy
  2. \Omega is holomorphically convex
  3. \Omega is pseudoconvex
  4. \Omega is Levi convex - for every sequence S_{n} \subseteq \Omega of analytic compact surfaces such that S_{n} \rightarrow S, \partial S_{n} \rightarrow \Gamma for some set \Gamma we have S \subseteq \Omega (\partial \Omega cannot be "touched from inside" by a sequence of analytic surfaces)
  5. \Omega has local Levi property - for every point x \in \partial \Omega there exist a neighbourhood U of x and f holomorphic on U \cap \Omega such that f cannot be extended to any neighbourhood of x

Implications 1 \Leftrightarrow 2, 3 \Leftrightarrow 4, 1 \Rightarrow 4, 3 \Rightarrow 5 are standard results (for 1\Rightarrow 3, see Oka's lemma). The main difficulty lies in proving 5 \Rightarrow 1, i.e. constructing a global holomorphic function which admits no extension from non-extendable functions defined only locally. This is called the Levi problem and was first solved by Kiyoshi Oka, and then by Lars Hörmander using methods from functional analysis and partial differential equations (a consequence of \bar{\partial}-problem).

Properties[edit]

  • if \Omega_1, \dots, \Omega_{n} are domains of holomorphy, then their intersection \Omega = \bigcap_{j=1}^{n} \Omega_j is also a domain of holomorphy
  • if \Omega_{1} \subseteq \Omega_{2} \subseteq \dots is an ascending sequence of domains of holomorphy, then their union \Omega = \bigcup_{n=1}^{\infty}\Omega_{n} is also a domain of holomorphy (see Behnke-Stein theorem)
  • product \Omega = \Omega_{1} \times \Omega_{2} of domains of holomorphy \Omega_{1}, \Omega_{2} is a domain of holomorphy
  • the first Cousin problem is always solvable in a domain of holomorphy; this is also true, with additional topological assumptions, for the second Cousin problem

See also[edit]

References[edit]

  • Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
  • Boris Vladimirovich Shabat, Introduction to Complex Analysis, AMS, 1992

This article incorporates material from Domain of holomorphy on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.