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In particular, the open unit disk is homeomorphic to the whole plane. There is however no conformal bijective map between the open unit disk and the plane. Considered as a Riemann surface, the open unit disk is therefore different from the complex plane. There are conformal bijective maps between the open unit disk and the open upper half-plane ...
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 ...
The simply connected 1-dimensional complex manifolds are isomorphic to either: . Δ, the unit disk in C; C, the complex plane; Ĉ, the Riemann sphere; Note that there are inclusions between these as Δ ⊆ C ⊆ Ĉ, but that there are no non-constant holomorphic maps in the other direction, by Liouville's theorem.
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.
Corollary (Riemann mapping theorem). Any connected and simply connected open domain in the complex plane with at least two boundary points is conformally equivalent to the unit disk. [25] [26] This is an immediate consequence of the theorem.
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 ...
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 ...
Then there is a quasiconformal homeomorphism f from D to the unit disk which is in the Sobolev space W 1,2 (D) and satisfies the corresponding Beltrami equation in the distributional sense. As with Riemann's mapping theorem, this f is unique up to 3 real parameters.