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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 ...
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 ...
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 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.
Poincaré disk with hyperbolic parallel lines Poincaré disk model of the truncated triheptagonal tiling.. In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or ...
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 ...
The Riemann mapping theorem states that any non-empty open simply connected subset of (except for itself) is conformally equivalent to the unit disk. The notion of simple connectedness is also a crucial condition in the Poincaré conjecture.
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.