Domain of holomorphy

A domain $ D $ in a complex space $ \mathbf C ^ {n} $ for which there exists a function $ f( z) $, holomorphic in $ D $, that is not holomorphically extendable to a larger domain; the domain $ D $ is then called the natural domain of definition of $ f( z) $. For example, the natural domain of definition of the function

$$ \sum _ {k = 1 } ^ \infty z ^ {k!} $$

is the unit disc, which is thus a domain of holomorphy in $ \mathbf C ^ {1} $. Any domain in $ \mathbf C ^ {1} $ is a domain of holomorphy. In $ \mathbf C ^ {n} $, $ n \geq 2 $, on the contrary, not all domains are domains of holomorphy. E.g., no domain of the form $ D \setminus K $, where $ K $ is a compactum contained in $ D $, is a domain of holomorphy.

A domain $ D \subset \mathbf C ^ {n} $ is said to be holomorphically convex if for each compact set $ A \subset D $ there exists a compact set $ F _ {A} \subset D $ containing $ A $ such that for any point $ z _ {0} \in D \setminus F _ {A} $ there exists a holomorphic function $ f( z) $ in $ D $ such that

$$ \sup _ {z \in A } | f ( z) | < | f ( z _ {0} ) | . $$

A domain $ D $ is a domain of holomorphy if and only if it is holomorphically convex (the Cartan–Thullen theorem). A domain $ D $ is a domain of holomorphy if and only if each point $ z _ {0} \in \partial D $ has a barrier — a holomorphic function $ f _ {z _ {0} } ( z) $ in $ D $ that cannot be holomorphically continued to $ z _ {0} $. For example, if $ D $ is an arbitrary domain in $ \mathbf C ^ {1} $, then the function $ ( z - z _ {0} ) ^ {-} 1 $ is a barrier at any point $ z _ {0} \in \partial D $, so that $ D $ is a domain of holomorphy; if $ D $ is a convex domain in $ \mathbf C ^ {n} $ and if

$$ \mathop{\rm Re} ( a, z - z _ {0} ) = \ \mathop{\rm Re} \sum _ {i = 1 } ^ { n } a _ {i} ( z _ {i} - z _ {0i} ) = 0 $$

is the supporting plane at a point $ z _ {0} \in \partial D $, then the function $ ( a, z - z _ {0} ) ^ {-} 1 $ is a barrier at $ z _ {0} $, and for this reason any convex domain in $ \mathbf C ^ {n} $ is a domain of holomorphy.

The intersection of domains of holomorphy is a domain of holomorphy; any biholomorphic mapping maps a domain of holomorphy onto a domain of holomorphy (the Behnke–Stein theorem).

A domain $ D \subset \mathbf C ^ {n} $ is said to be pseudo-convex if the function $ - \mathop{\rm ln} \Delta _ {D} ( z) $ is a plurisubharmonic function in $ D $, where $ \Delta _ {D} ( z) $ is the distance from the point $ z \in D $ to $ \partial D $. A domain is a domain of holomorphy if and only if it is pseudo-convex (Oka's theorem). That, in Oka's theorem, this condition is sufficient, forms the content of the Levi problem, formulated by E. Levi in 1911. It was solved by K. Oka in 1942 for $ n = 2 $; it was solved independently by Oka, F. Norguet and H. Bremermann in 1953–1954 for $ n \geq 2 $.

A domain of holomorphy with a sufficiently smooth boundary can be locally described. A domain $ D \subset \mathbf C ^ {n} $ is said to be pseudo-convex at a point $ z _ {0} \in \partial D $ if there exists a neighbourhood $ V $ of $ z _ {0} $ and a real-valued function $ \phi ( z) $ of class $ C ^ {2} $ such that: a) $ D \cap V = \{ {z } : {\phi ( z) < 0, z \in V } \} $; and b) on the plane

$$ \sum _ {i = 1 } ^ { n } a _ {i} \frac{\partial \phi ( z _ {0} ) }{\partial z _ {i} } = 0 $$

the Hessian form

$$ \sum _ {i, k = 1 } ^ { n } \frac{\partial ^ {2} \phi ( z _ {0} ) }{\partial z _ {i} \partial \overline{ {z _ {k} }}\; } a _ {i} \overline{ {a _ {k} }}\; \geq 0. $$

If in condition b) strict inequality holds for all vectors $ a \neq 0 $ under consideration, the domain $ D $ is said to be strictly pseudo-convex at the point $ z _ {0} $. A domain $ D $ is said to be (strictly) pseudo-convex in the sense of Levi if it is (strictly) pseudo-convex at all points $ z _ {0} \in \partial D $.

If a domain is strictly pseudo-convex in the sense of Levi, it is pseudo-convex (Levi's theorem).

The domain of holomorphy of a function $ f( z) $, defined in an initial neighbourhood $ V $, can be constructed by expansions into Taylor series using the principle of holomorphic continuation; it may then turn out that in the domain thus constructed the holomorphic function $ f( z) $ is not single-valued. In order to make the function single-valued, the concept of a domain must be widened. This is done by the introduction of a Riemann (Riemannian) domain (a covering domain, a multi-sheeted domain) over $ \mathbf C ^ {n} $( a Riemann domain over $ \mathbf C ^ {1} $ is known as a Riemann surface). The concept of a domain of holomorphy is generalized to Riemann domains and even to objects of a more general structure — complex manifolds and complex spaces. The generalization of the concept of a domain of holomorphy leads to Stein spaces (cf. Stein space).

References

[1]	V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)
[2]	B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian)
[3]	L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973)

Comments

The following is usually also considered as part of the above-mentioned Behnke–Stein theorem: The (countable) union of an increasing sequence of domains of holomorphy is a domain of holomorphy.

See Riemannian domain for the notion of "domain of holomorphy" on Riemann surfaces. For pseudo-convex domains, etc. see also Pseudo-convex and pseudo-concave.