CorollaryStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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A domain containing a Hartogs figure but not its hull is not a domain of holomorphy

Statement

Let $Ω \subseteq C^{2}$ be a domain. If there exist $0 < r, s < 1$ such that

$H (r, s) \subseteq Ω \subseteq \hat{H} (r, s) and Ω \neq \hat{H} (r, s),$

then $Ω$ is not a domain of holomorphy.

Facts & Assumptions

Given: A domain $Ω$ with $H (r, s) \subseteq Ω \subseteq \hat{H} (r, s)$ and $Ω \neq \hat{H} (r, s)$ .

[L1]

A domain of holomorphy is defined by the nonexistence of one fixed overlap from which every holomorphic function extends farther (Holomorphic extension and domains of holomorphy in several variables).

[L2]

Every holomorphic function on $H (r, s)$ extends uniquely to the full bidisc $\hat{H} (r, s)$ (A holomorphic function on a Hartogs figure extends to the full bidisc).

Proof

technique · direct

1.1

Let $f \in O (Ω)$ . Its restriction to the open subset $H (r, s)$ is holomorphic, so [L2] gives a holomorphic function $F_{f} \in O (\hat{H} (r, s))$ with $F_{f} = f$ on $H (r, s)$ .

givenL2

2.1

The domain $\hat{H} (r, s)$ is not contained in $Ω$ by hypothesis, and $H (r, s)$ is a nonempty open subset of $Ω \cap \hat{H} (r, s)$ . Thus the same open overlap works for every holomorphic function on $Ω$ , namely the fixed set $U_{1} : = H (r, s)$ and the larger domain $U_{2} : = \hat{H} (r, s)$ .

step 1.1given

3.1

By the definition in [L1], the existence of that common pair $U_{1}, U_{2}$ shows that $Ω$ is not a domain of holomorphy.

L1step 2.1∎

Depends on

Used by

FALSE: every domain in C² is a domain of holomorphy False statement

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Lebl, Tasty Bits of Several Complex Variables, §2.1 (standard reference, not scraped)