Hartogs' extension theorem

Historical note

The original proof was given by Friedrich Hartogs in 1906, using Cauchy's integral formula for functions of several complex variables. Today, usual proofs rely on either Bochner–Martinelli–Koppelman formula or the solution of the inhomogeneous Cauchy–Riemann equations with compact support. The latter approach is due to Leon Ehrenpreis who initiated it in the paper (Ehrenpreis 1961). Yet another very simple proof of this result was given by Gaetano Fichera in the paper (Fichera 1957), by using his solution of the Dirichlet problem for holomorphic functions of several variables and the related concept of CR-function: later he extended the theorem to a certain class of partial differential operators in the paper (Fichera 1983), and his ideas were later further explored by Giuliano Bratti. Also the Japanese school of the theory of partial differential operators worked much on this topic, with notable contributions by Akira Kaneko. Their approach is to use Ehrenpreis' fundamental principle.

Hartogs' phenomenon

A phenomenon that holds in several variables but does not hold in one variable is called Hartogs' phenomenon, which lead to the notion of this Hartogs' extension theorem and the domain of holomorphy, hence the theory of several complex variables.

For example, in two variables, consider the interior domain

H ε = { z = ( z 1 , z 2 ) ∈ Δ 2 : | z 1 | < ε     or     1 − ε < | z 2 | }

in the two-dimensional polydisk Δ 2 = { z ∈ Z ; | z 1 | < 1 , | z 2 | < 1 } where 0 < ε < 1 .

Theorem Hartogs (1906): any holomorphic functions f on H ε are analytically continued to Δ 2 . Namely, there is a holomorphic function F on Δ 2 such that F = f on H ε .

In fact, using the Cauchy integral formula we obtain the extended function F , All holomorphic functions are analytically continued to the polydisk, which is strictly larger than the domain on which the original holomorphic function is defined. Such phenomena never happen in the case of one variable.

Formal statement

Let f be a holomorphic function on a set GK, where G is an open subset of Cⁿ (n ≥ 2) and K is a compact subset of G. If the complement GK is connected, then f can be extended to a unique holomorphic function on G.

Counterexamples in dimension one

The theorem does not hold when n = 1. To see this, it suffices to consider the function f(z) = z⁻¹, which is clearly holomorphic in C{0}, but cannot be continued as an holomorphic function on the whole C. Therefore, the Hartogs' phenomenon constitutes one elementary phenomenon that emphasizes the difference between the theory of functions of one and several complex variables.