Domain of holomorphy

The sets in the definition.

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

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

In the ${\displaystyle n=1}$ 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 reciprocal. For ${\displaystyle n\geq 2}$ this is no longer true, as it follows from Hartogs' lemma.

Equivalent conditions[]

For a domain ${\displaystyle \Omega }$ the following conditions are equivalent:

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

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

Properties[]

• If ${\displaystyle \Omega _{1},\dots ,\Omega _{n}}$ are domains of holomorphy, then their intersection ${\displaystyle \Omega =\bigcap _{j=1}^{n}\Omega _{j}}$ is also a domain of holomorphy.
• If ${\displaystyle \Omega _{1}\subseteq \Omega _{2}\subseteq \dots }$ is an ascending sequence of domains of holomorphy, then their union ${\displaystyle \Omega =\bigcup _{n=1}^{\infty }\Omega _{n}}$ is also a domain of holomorphy (see Behnke-Stein theorem).
• If ${\displaystyle \Omega _{1}}$ and ${\displaystyle \Omega _{2}}$ are domains of holomorphy, then ${\displaystyle \Omega _{1}\times \Omega _{2}}$ 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[]

• Behnke–Stein theorem
• Levi pseudoconvex
• Stein manifold

References[]

• 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.