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

Depends on

• The one-point (Alexandroff) compactification $X^{*} = X \cup \{\infty\}$, whose open sets are the open sets of $X$ together with the complements in $X^{*}$ of the closed compact subsets of $X$
• $X^{*}$ is compact and contains $X$ as an open subspace; $X$ is dense in $X^{*}$ exactly when $X$ is not compact; and $X^{*}$ is Hausdorff exactly when $X$ is locally compact and Hausdorff

Used by

• Circlines and their reflections on the Riemann sphere Definition
• Conformal removability of compact sets Definition
• Measurable Beltrami coefficients and measurable conformal structures Definition
• Möbius transformations of the Riemann sphere Definition
• Quasicircles, quasidisks, quasiarcs, and quasilines Definition
• The cross-ratio of an ordered quadruple of sphere points Definition
• The standard holomorphic charts on the Riemann sphere, with holomorphy and poles at infinity Definition
• Atlases on the sphere, plane, disc and annulus Example
• Constant coefficients and their affine solutions Example
• Dolbeault h zero one of the riemann sphere vanishes Example
• Normalization of a solution by a Möbius postcomposition Example
• Orders and residues under inversion on the sphere Example
• Riemann–Hurwitz for the sphere power map Example
• A simply connected surface without a Green kernel is plane or sphere Lemma
• An analytically quasiconformal homeomorphism distorts quadrilateral moduli by at most K Lemma
• Compact sets of finite length are removable for continuous analytic functions Lemma
• Compact sets of positive area are not conformally removable Lemma
• Conformal removability is invariant under quasiconformal maps Lemma
• Degree two of ℘ and its four branch points Lemma
• Riemann maps of Jordan domains extend to homeomorphisms of the closures Lemma
• Smooth Beltrami coefficients admit quasiconformal solutions Lemma
• The Ahlfors-Beurling extension formula for quasisymmetric maps of the line Lemma
• Trace of a holomorphic differential along a nonconstant map to the sphere Lemma
• Compactness of the normalized K-quasiconformal self-maps of the sphere Theorem
• Hölder regularity and nonvanishing Jacobian of the normalized Beltrami solution Theorem
• Stereographic projection identifies the Riemann sphere with the unit two-sphere Theorem
• The measurable Riemann mapping theorem on the sphere Theorem

Dependency tree · two levels

15 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)