DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-28

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Continuous families of analytic discs

Definition

Write $D = {λ \in C : ∣ λ ∣ < 1}$ . A continuous family of analytic discs in $C^{m}$ is a map

$Φ : [0, 1] \times \overline{D} \to C^{m}$

that is continuous on the product and such that, for every $t \in [0, 1]$ , the slice

$Φ_{t} (λ) : = Φ (t, λ)$

is holomorphic on $D$ .

When a domain $Ω \subseteq C^{m}$ is under discussion, saying that the family has boundary in a compact set $K ⋐ Ω$ means $Φ_{t} (\partial D) \subseteq K$ for every $t$ , and saying that the initial disc is compactly contained in $Ω$ means $Φ_{0} (\overline{D}) ⋐ Ω$ .

Used by

Continuity principle for domains of holomorphy Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Jiří Lebl, Tasty Bits of Several Complex Variables, §2.5 (standard reference, not scraped)
Harold P. Boas, Lecture Notes on Several Complex Variables, §3.2.4 (standard reference, not scraped)