The standard holomorphic charts on the Riemann sphere, with holomorphy and poles at infinity

Depends on

• The Riemann sphere is the published one-point compactification of the complex plane
• A complex domain is a nonempty connected open subset of $\mathbb C$

Used by

• Local integrability of measurable conformal structures 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
• Holomorphic maps and meromorphic functions on Riemann surfaces Definition
• Measurable Beltrami coefficients and measurable conformal structures Definition
• Meromorphic functions on the Riemann sphere Definition
• Quasicircles, quasidisks, quasiarcs, and quasilines Definition
• The welding homeomorphism of a Jordan curve Definition
• Weak solutions of the Beltrami equation Definition
• A genus-two compact surface gives a cocompact Fuchsian group Example
• A single point is conformally removable Example
• Atlases on the sphere, plane, disc and annulus Example
• Constant coefficients and their affine solutions Example
• Divisors and Riemann-Roch on the Riemann sphere and on a complex torus Example
• Dolbeault h zero one of the riemann sphere vanishes Example
• Hyperelliptic double covers and their genus Example
• Normalization of a solution by a Möbius postcomposition Example
• Orders and residues under inversion on the sphere Example
• Period matrix and Jacobian of the pentagon curve Example
• Principal divisor tests via the Abel-Jacobi map Example
• Pullback of a measurable ellipse field under biholomorphic maps Example
• Riemann–Hurwitz for the sphere power map Example
• The Veronese linear system on the Riemann sphere Example
• A simply connected surface without a Green kernel is plane or sphere Lemma
• Compact sets of positive area are not conformally removable Lemma
• Compact subsets of lines and round circles are removable for quasiconformal maps Lemma
• Conformal removability is invariant under quasiconformal maps Lemma
• Degree two of ℘ and its four branch points Lemma
• Principal divisors have vanishing Abel-Jacobi class 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
• The sphere, plane and disc are pairwise biholomorphically distinct Lemma
• Trace of a holomorphic differential along a nonconstant map to the sphere Lemma
• 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
• Hölder regularity and nonvanishing Jacobian of the normalized Beltrami solution Theorem
• The measurable Riemann mapping theorem on the sphere Theorem

…and 1 more result.

Dependency tree · two levels

7 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)