Möbius transformations of the Riemann sphere

Depends on

• The Riemann sphere is the published one-point compactification of the complex plane

Used by

• The genus of a compact Riemann surface determines its uniformization type Corollary
• Not every compact set is conformally removable Counterexample
• Uniqueness of Beltrami solutions fails without the three-point normalization Counterexample
• Conformal removability of compact sets Definition
• Quasicircles, quasidisks, quasiarcs, and quasilines Definition
• Quasisymmetric homeomorphisms of the line and circle Definition
• The modular group and its action on the upper half-plane Definition
• The welding homeomorphism of a Jordan curve Definition
• A genus-two compact surface gives a cocompact Fuchsian group Example
• A single point is conformally removable Example
• Constant coefficients and their affine solutions Example
• Normalization of a solution by a Möbius postcomposition Example
• Power maps, endpoint distortion, and a non-Möbius quasisymmetric circle map Example
• The identity welding of the round circle Example
• The Möbius ambiguity in conformal welding Example
• The upper half-plane Bergman kernel by biholomorphic transport Example
• A simply connected surface without a Green kernel is plane or sphere Lemma
• Analytic quasiconformality gives both quadrilateral modulus bounds Lemma
• Compact sets of finite length are removable for continuous analytic functions Lemma
• Compact subsets of lines and round circles are removable for quasiconformal maps Lemma
• Conformal removability is invariant under quasiconformal maps Lemma
• Riemann maps of Jordan domains extend to homeomorphisms of the closures Lemma
• Round circles and straight lines are conformally removable Lemma
• Smooth Beltrami coefficients admit quasiconformal solutions Lemma
• A unique Möbius transformation carries any ordered triple of distinct sphere points to any other Theorem
• Automorphisms of the upper half-plane are real Mobius maps Theorem
• Bounded turning, quasiconformal images of the circle, and quasiconformal reflections Theorem
• Every Möbius transformation is a biholomorphism of the Riemann sphere Theorem
• Every quasisymmetric circle homeomorphism is a conformal welding Theorem
• Möbius transformations form a group and identify with the projective linear quotient of GL₂(C) Theorem
• Nonidentity Möbius transformations are parabolic or conjugate to a dilation, with the projective trace invariant Theorem
• The Beurling–Ahlfors extension theorem for circles and lines Theorem
• The measurable Riemann mapping theorem on the sphere Theorem
• Welding uniqueness for conformally removable curves Theorem
• Zero-length compact sets and quasicircles are conformally removable Theorem

Dependency tree · two levels

3 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

• Lars V. Ahlfors, Complex Analysis, 3rd ed., Ch. 3 §§2.2-3.5 (standard reference, not scraped)
• Matthias Weber, Complex Analysis, Ch. 1 §§1.3-1.4 (standard reference, not scraped)
• Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 8 §§1-2 (standard reference, not scraped)