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Since every Riemann surface has a universal cover which is a simply connected Riemann surface, the uniformization theorem leads to a classification of Riemann surfaces into three types: those that have the Riemann sphere as universal cover ("elliptic"), those with the plane as universal cover ("parabolic") and those with the unit disk as ...
Unit disks are special cases of disks and unit balls; as such, they contain the interior of the unit circle and, in the case of the closed unit disk, the unit circle itself. Without further specifications, the term unit disk is used for the open unit disk about the origin , D 1 ( 0 ) {\displaystyle D_{1}(0)} , with respect to the standard ...
There are several equivalent definitions of a Riemann surface. A Riemann surface X is a connected complex manifold of complex dimension one. This means that X is a connected Hausdorff space that is endowed with an atlas of charts to the open unit disk of the complex plane: for every point x ∈ X there is a neighbourhood of x that is homeomorphic to the open unit disk of the complex plane, and ...
In the case of genus one, a fundamental convex polygon is sought for the action by translation of Λ = Z a ⊕ Z b on R 2 = C where a and b are linearly independent over R. (After performing a real linear transformation on R 2, it can be assumed if necessary that Λ = Z 2 = Z + Z i; for a genus one Riemann surface it can be taken to have the form Λ = Z 2 = Z + Z ω, with Im ω > 0.)
The Riemann mapping theorem can be generalized to the context of Riemann surfaces: If is a non-empty simply-connected open subset of a Riemann surface, then is biholomorphic to one of the following: the Riemann sphere, the complex plane, or the unit disk.
The intrinsic geometry of this surface is now better understood in terms of the Poincaré metric on the upper half plane or the unit disc, and has been described by other models such as the Klein model or the hyperboloid model, obtained by considering the two-sheeted hyperboloid q(x, y, z) = −1 in three-dimensional Minkowski space, where q(x ...
The Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than stronger theorems, such as the Riemann mapping theorem, which it helps to prove. It is however one of the simplest results capturing the rigidity of holomorphic ...
There are two simple examples that are immediately computed from the Uniformization theorem: there is a unique complex structure on the sphere (see Riemann sphere) and there are two on (the complex plane and the unit disk) and in each case the group of positive diffeomorphisms is connected.