Biholomorphic maps between complex domains

Depends on

• A complex domain is a nonempty connected open subset of $\mathbb C$
• Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions
• Injection, surjection, bijection

Used by

• A local degree-m holomorphic map has m nearby sheets Corollary
• An injective holomorphic map has no critical point and is biholomorphic onto its image Corollary
• Every Riemann surface is a quotient of a simply connected model Corollary
• Conformal equivalence and the automorphism group of a domain Definition
• Poincaré metric on a hyperbolic Riemann surface Definition
• Ramification index, ramification order and branch value Definition
• Spherical, parabolic and hyperbolic universal-covering types Definition
• A genus-two compact surface gives a cocompact Fuchsian group Example
• A nonsingular affine conic is a punctured-plane Riemann surface Example
• Annulus and punctured disc have hyperbolic universal covers Example
• Compactness and Liouville distinguish the three models Example
• Green kernel of a slit plane via the square-root map Example
• Riemann–Hurwitz for the sphere power map Example
• The modular lambda function: Y(2) biholomorphic to the twice-punctured plane, and the slit-plane quadrilateral Example
• FALSE: conformal maps preserve Euclidean lengths False statement
• A nonzero complex derivative gives a local biholomorphism Lemma
• A simply connected Greenian Riemann surface is a disc Lemma
• A simply connected surface without a Green kernel is plane or sphere Lemma
• Local charts and the Riemann surface structure of a modular quotient Lemma
• Plane subharmonicity is invariant under biholomorphic change of coordinate Lemma
• The sphere, plane and disc are pairwise biholomorphically distinct Lemma
• Biholomorphisms are conformal and have holomorphic inverse Remark
• A slit-plane root branch biholomorphically parametrizes a sector Theorem
• Blaschke factors are automorphisms of the disc Theorem
• Conformal covariance of the canonical planar Green kernel Theorem
• Conformal invariance of harmonic measure Theorem
• Deck transformations preserve the hyperbolic metric Theorem
• Degree of a proper holomorphic map of Riemann surfaces Theorem
• Green kernel of a simply connected plane domain from a Riemann map Theorem
• Holomorphic inverse function theorem and local-degree criterion Theorem
• Local power-map normal form on Riemann surfaces Theorem
• Power maps are biholomorphisms on sectors of width less than 2π/n Theorem
• The exponential is the inverse biholomorphism from the principal strip to the slit plane Theorem
• The germ projection is a local biholomorphism Theorem
• The j-invariant classifies complex tori Theorem
• The j-invariant uniformizes X(1) Theorem
• The principal logarithm is a biholomorphism from the slit plane to the principal strip Theorem
• The torus is biholomorphic to its Weierstrass cubic Theorem
• Uniformization of simply connected Riemann surfaces Theorem

Dependency tree · two levels

9 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, Guide to Cultivating Complex Analysis, §5.6 (standard reference, not scraped)
• B. V. Shabat, Introduction to Complex Analysis, §1.2 (standard reference, not scraped)