Domain of holomorphy

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

Equivalently, for any such ⁠${\displaystyle U}$⁠ and ⁠${\displaystyle V}$⁠, there exists a holomorphic ${\displaystyle f}$ on ⁠${\displaystyle \mathrm{\Omega }}$⁠, such that ${\displaystyle f{|}_U}$ cannot be analytically continued to ⁠${\displaystyle V}$⁠.

In the case ⁠${\displaystyle n=1}$⁠, every open set is a domain of holomorphy: we can define a holomorphic function that is not identically zero, but whose zeros accumulate everywhere on the boundary of the domain, which must then be a natural boundary for a domain of definition of its reciprocal. For ${\displaystyle n\ge 2}$ this is no longer true, as it follows from Hartogs's extension theorem.

Equivalent conditions

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

1. ${\displaystyle \mathrm{\Omega }}$ is a domain of holomorphy.
2. ${\displaystyle \mathrm{\Omega }}$ is holomorphically convex.
3. There exists a function holomorphic on ${\displaystyle \mathrm{\Omega }}$ that cannot be analytically continued beyond ⁠${\displaystyle \mathrm{\Omega }}$⁠. That is, its domain of existence is ⁠${\displaystyle \mathrm{\Omega }}$⁠.
4. ${\displaystyle \mathrm{\Omega }}$ is pseudoconvex.
5. ${\displaystyle \mathrm{\Omega }}$ is Levi convex – for every sequence ${\displaystyle S_n\subseteq \mathrm{\Omega }}$ of analytic compact surfaces such that ${\displaystyle S_n\to S,\mathrm{\partial }{S}_n\to \mathrm{\Gamma }}$ for some set ${\displaystyle \mathrm{\Gamma }}$ we have ${\displaystyle S\subseteq \mathrm{\Omega }}$ (⁠${\displaystyle \mathrm{\partial }\mathrm{\Omega }}$⁠ cannot be "touched from inside" by a sequence of analytic surfaces).
6. ${\displaystyle \mathrm{\Omega }}$ has local Levi property – for every point ${\displaystyle x\in \mathrm{\partial }\mathrm{\Omega }}$ there exists a neighbourhood ${\displaystyle U}$ of ${\displaystyle x}$ and ${\displaystyle f}$ holomorphic on ${\displaystyle U\cap \mathrm{\Omega }}$ such that ${\displaystyle f}$ cannot be extended to any neighbourhood of ${\displaystyle x}$

Implications ${\displaystyle 1\iff 2\iff 3,4\iff 5,1\Rightarrow 5,4\Rightarrow 6}$ are standard results (for ⁠${\displaystyle 1\Rightarrow 4}$⁠, see Oka's lemma). The equivalence of 1, 2, 3 is the Cartan–Thullen theorem. 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 ${\displaystyle \overline{\mathrm{\partial }}}$-problem).