Hartogs's extension theorem

Hartogs's phenomenon

For example, in two variables, consider the interior domain

${\displaystyle H_{\epsilon }=\{z=(z_1,z_2)\in {\mathrm{\Delta }}^2:|z_1|<\epsilon \text{or}1-\epsilon <|z_2|\}}$

in the two-dimensional polydisk ${\displaystyle {\mathrm{\Delta }}^2=\{z\in {\mathbb{C}}^2;|z_1|<1,|z_2|<1\}}$ where ${\displaystyle 0<\epsilon <1.}$

Theorem Hartogs (1906): Any holomorphic function ${\displaystyle f}$ on ${\displaystyle H_{\epsilon }}$ can be analytically continued to ${\displaystyle {\mathrm{\Delta }}^2.}$ Namely, there is a holomorphic function ${\displaystyle F}$ on ${\displaystyle {\mathrm{\Delta }}^2}$ such that ${\displaystyle F=f}$ on ${\displaystyle H_{\epsilon }.}$

Such a phenomenon is called Hartogs's phenomenon, which lead to the notion of this Hartogs's extension theorem and the domain of holomorphy.

Formal statement and proof

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

Ehrenpreis' proof is based on the existence of smooth bump functions, unique continuation of holomorphic functions, and the Poincaré lemma — the last in the form that for any smooth and compactly supported differential (0,1)-form ω on Cⁿ with ∂ω = 0, there exists a smooth and compactly supported function η on Cⁿ with ∂η = ω. The crucial assumption n ≥ 2 is required for the validity of this Poincaré lemma; if n = 1 then it is generally impossible for η to be compactly supported.

The ansatz for F is φ f − v for smooth functions φ and v on G; such an expression is meaningful provided that φ is identically equal to zero where f is undefined (namely on K). Furthermore, given any holomorphic function on G which is equal to f on some open set, unique continuation (based on connectedness of G \ K) shows that it is equal to f on all of G \ K.

The holomorphicity of this function is identical to the condition ∂v = f ∂φ. For any smooth function φ, the differential (0,1)-form f ∂φ is ∂-closed. Choosing φ to be a smooth function which is identically equal to zero on K and identically equal to one on the complement of some compact subset L of G, this (0,1)-form additionally has compact support, so that the Poincaré lemma identifies an appropriate v of compact support. This defines F as a holomorphic function on G; it only remains to show (following the above comments) that it coincides with f on some open set.

On the set Cⁿ \ L, v is holomorphic since φ is identically constant. Since it is zero near infinity, unique continuation applies to show that it is identically zero on some open subset of G \ L. Thus, on this open subset, F equals f and the existence part of Hartog's theorem is proved. Uniqueness is automatic from unique continuation, based on connectedness of 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 a holomorphic function on the whole of C. Therefore, the Hartogs's phenomenon is an elementary phenomenon that highlights the difference between the theory of functions of one and several complex variables.