Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30
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FALSE: the Riemann map is unique without normalization

Statement

Every conformal equivalence from a plane domain onto D is unique without any normalization condition.

Facts & Assumptions

Given: The disc automorphisms z↦z and z↦−z.

[L1]

The identity map and every rotated Blaschke factor are automorphisms of the disc (Every automorphism of the disc is a rotated Blaschke factor).

Refutation

technique · direct
1.1

By [L1], both f(z)=z and g(z)=−z are biholomorphic self-maps of D.

L1given
2.1

The maps f and g are distinct, since f(1/2)=1/2 while g(1/2)=−1/2. Thus the disc already has two different conformal self-equivalences.

step 1.1algebra
3.1

Therefore uniqueness fails unless a normalizing condition is imposed.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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