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Every proper homologically simply connected plane domain is conformally equivalent to the unit disc
Statement
Assume the Axiom of Choice. Let be a homologically simply connected complex domain and let . Then there is a biholomorphic map such that
Facts & Assumptions
Given: The Axiom of Choice, a proper homologically simply connected complex domain , and a point .
The Axiom of Choice is used by the extremal-attainment lemma (The Axiom of Choice).
Under the Axiom of Choice, the extremal family is nonempty and the supremal derivative is attained by a holomorphic map (A proper homologically simply connected plane domain has a bounded univalent competitor, A maximizing sequence has a locally uniform limit with extremal derivative).
That extremal map is univalent and surjective onto (The extremal limit is univalent, An extremizer onto a proper subdomain of the disc can be enlarged).
An injective holomorphic map on a complex domain is biholomorphic onto its open image (An injective holomorphic map has no critical point and is biholomorphic onto its image).
Proof
By [A1] and [L1], choose a holomorphic map with and extremal derivative .
Fact [L2] makes this map injective and gives . Since is a complex domain by the given data, [L3] makes biholomorphic onto its image, which is exactly .
The map of step 2.1 has the required normalization, so it is the desired conformal equivalence.
Depends on
- The Axiom of Choice
- Homologically simply connected complex domains
- A proper homologically simply connected plane domain has a bounded univalent competitor
- A maximizing sequence has a locally uniform limit with extremal derivative
- The extremal limit is univalent
- An extremizer onto a proper subdomain of the disc can be enlarged
- An injective holomorphic map has no critical point and is biholomorphic onto its image
Used by
- The normalized Riemann map is unique Corollary
- Conformal removability of compact sets Definition
- A normalized Riemann map for a horizontal strip Example
- A normalized Riemann map for a sector with an explicit branch choice Example
- A normalized Riemann map for the slit plane Example
- The normalized Riemann map from the upper half-plane sending i to 0 Example
- FALSE: the Riemann map is unique without normalization False statement
- A homologically simply connected plane domain is either the plane or conformally equivalent to the disc Lemma
- A simply connected Greenian Riemann surface is a disc Lemma
- A simply connected surface without a Green kernel is plane or sphere Lemma
- Analytic quasiconformality gives both quadrilateral modulus bounds Lemma
- Riemann maps of Jordan domains extend to homeomorphisms of the closures Lemma
- Green kernel of a simply connected plane domain from a Riemann map Theorem
Dependency tree · two levels
25 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
- Matthias Weber, Complex Analysis, Theorem 5.2.6 (standard reference, not scraped)
- Walter Rudin, Real and Complex Analysis, Theorem 14.9 (standard reference, not scraped)