ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The holomorphic hull of a product torus in the bidisc is the closed polydisc it bounds

Example

Fix radii $0 < r_{1}, r_{2} < 1$ in the bidisc $D^{2}$ , and let

$T = {(z_{1}, z_{2}) \in C^{2} : ∣ z_{1} ∣ = r_{1}, ∣ z_{2} ∣ = r_{2}} .$

Then the holomorphic hull of $T$ in $D^{2}$ is

${\hat{T}}_{D^{2}} = {(z_{1}, z_{2}) \in C^{2} : ∣ z_{1} ∣ \leq r_{1}, ∣ z_{2} ∣ \leq r_{2}} .$

Facts & Assumptions

Given: The product torus $T$ in the bidisc $D^{2}$ .

[L1]

Hull membership is tested against all holomorphic functions on the ambient domain (Holomorphic hulls and holomorphic convexity).

[L2]

On a closed polydisc, the supremum of a holomorphic function is attained on the distinguished boundary (The modulus of a holomorphic function on a closed polydisc is bounded by its supremum on the distinguished boundary).

Verification

technique · direct

1.1

Let $P = {(z_{1}, z_{2}) : ∣ z_{1} ∣ \leq r_{1}, ∣ z_{2} ∣ \leq r_{2}}$ . If $a \in P$ and $f$ is holomorphic on $D^{2}$ , then $f$ is continuous on the closed polydisc $P$ , and the distinguished boundary of $P$ is exactly $T$ . Therefore [L2] gives $∣ f (a) ∣ \leq \sup_{T} ∣ f ∣$ . By [L1], every point of $P$ lies in ${\hat{T}}_{D^{2}}$ .

L1L2given

2.1

If $a = (a_{1}, a_{2})$ lies in $D^{2} ∖ P$ , then either $∣ a_{1} ∣ > r_{1}$ or $∣ a_{2} ∣ > r_{2}$ . In the first case the holomorphic coordinate function $f (z_{1}, z_{2}) = z_{1}$ satisfies $∣ f (a) ∣ > \sup_{T} ∣ f ∣$ , and in the second case $f (z_{1}, z_{2}) = z_{2}$ does the same. Hence [L1] excludes every point outside $P$ from the hull. Together with step 1.1, this identifies ${\hat{T}}_{D^{2}}$ exactly.

L1step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Jiří Lebl, Tasty Bits of Several Complex Variables, §1.2 and §2.6 (standard reference, not scraped)
Harold P. Boas, Lecture Notes on Several Complex Variables, §3.2.3 (standard reference, not scraped)