Domain of holomorphy

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. There exists a function holomorphic on ${\displaystyle \Omega }$ that cannot be analytically continued beyond ⁠${\displaystyle \Omega }$⁠. That is, its domain of existence is ⁠${\displaystyle \Omega }$⁠.
4. ${\displaystyle \Omega }$ is pseudoconvex.
5. ${\displaystyle \Omega }$ 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).
6. ${\displaystyle \Omega }$ has local Levi property – for every point ${\displaystyle x\in \partial \Omega }$ there exists 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 ${\displaystyle 1\Leftrightarrow 2\Leftrightarrow 3,4\Leftrightarrow 5,1\Rightarrow 5,4\Rightarrow 6}$ are standard results (for ⁠${\displaystyle 1\Rightarrow 4}$⁠, see Oka's lemma). The equivalence of 1, 2, 3 is the Cartan–Thullen theorem.^[1] The main difficulty lies in proving ⁠${\displaystyle 6\Rightarrow 1}$⁠, i.e. constructing a global holomorphic function that 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 Hörmander using methods from functional analysis and partial differential equations (a consequence of ${\displaystyle {\bar {\partial }}}$-problem).

• If ${\displaystyle \Omega _{1},\dots ,\Omega _{n}}$ are domains of holomorphy, then their intersection ${\displaystyle \textstyle \Omega =\bigcap _{j=1}^{n}\Omega _{j}}$ 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 ${\displaystyle \textstyle \Omega =\bigcup _{n=1}^{\infty }\Omega _{n}}$ 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.

${\displaystyle {\mathbb {C} }^{n}}$ is trivially a domain of holomorphy.

In the ${\displaystyle n=1}$ case, every open set is a domain of holomorphy. A particular example is the open unit disk. Define the lacunary function ⁠${\displaystyle \textstyle \sum _{n=0}^{\infty }z^{2^{n}}}$⁠.

it is holomorphic on the open unit disk by the Weierstrass M-test, and singular at all ⁠${\displaystyle \{e^{2i\pi k/2^{n}}:n,k\in \mathbb {N} \}}$⁠, which is dense on the unit circle, and therefore it cannot be analytically extended beyond the unit disk.

In the ${\displaystyle n\geq 2}$ case, let ${\displaystyle K\subset U\subset \mathbb {C} ^{n}}$ where ${\displaystyle U}$ is open and ${\displaystyle K}$ is nonempty and compact. If ${\displaystyle U\setminus K}$ is connected, then by the Hartogs's extension theorem, any function holomorphic on ${\displaystyle U\setminus K}$ can be analytically continued to ⁠${\displaystyle U}$⁠, which means ${\displaystyle U\setminus K}$ is an open set that is not a domain of holomorphy. Thus, domain of holomorphy becomes a nontrivial concept in the case ⁠${\displaystyle n\geq 2}$⁠.

• Behnke–Stein theorem
• Levi pseudoconvex
• Stein manifold