Other choice axioms weaker than axiom of choice include the Boolean prime ideal theorem and the axiom of uniformization.

Also important are Plemelj's contributions to the theory of analytic functions in solving the problem of uniformization of algebraic functions, contributions on formulation of the theorem of analytic extension of designs and treatises in algebra and in number theory.

The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces.

The uniformization theorem for surfaces states that the upper half-plane is the universal covering space of surfaces with constant negative Gaussian curvature.

In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk, the complex plane, or the Riemann sphere.

The uniformization theorem is a generalization of the Riemann mapping theorem from proper simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces.

The uniformization theorem implies a similar result for arbitrary connected second countable surfaces: they can be given Riemannian metrics of constant curvature.

Felix and conjectured the uniformization theorem for ( the Riemann surfaces of ) algebraic curves.

The first rigorous proofs of the general uniformization theorem were given by and.

However, his proof relied on the uniformization theorem.

that it is nevertheless possible to prove the uniformization theorem via Ricci flow.

Koebe proved the general uniformization theorem that if a Riemann surface is homeomorphic to an open subset of the complex sphere ( or equivalently if every Jordan curve separates it ), then it is conformally equivalent to an open subset of the complex sphere.

The simultaneous uniformization theorem of Bers shows that it is possible to simultaneously uniformize two compact Riemann surfaces of the same genus > 1 with the same quasi-Fuchsian group.

The measurable Riemann mapping theorem shows more generally that the map to an open subset of the complex sphere in the uniformization theorem can be chosen to be a quasiconformal map with any given bounded measurable Beltrami coefficient.

) Indeed, a triumph of nineteenth century geometry was the proof of the uniformization theorem, the analogous topological classification of smooth two-manifolds, where Hamilton showed that the Ricci flow does indeed evolve a negatively curved two-manifold into a two-dimensional multi-holed torus which is locally isometric to the hyperbolic plane.

When Yau was a graduate student, he started to generalize the uniformization theorem of Riemann surfaces to higher-dimensional complex Kähler manifolds.

Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – every surface admits a constant curvature metric ; geometrically, it has one of 3 possible geometries: positive curvature / spherical, zero curvature / flat, negative curvature / hyperbolic – and the geometrization conjecture ( now theorem ) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of 8 possible geometries.

2-dimensional topology can be studied as complex geometry in one variable ( Riemann surfaces are complex curves ) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables ( complex surfaces ), though not every 4-manifold admits a complex structure.

In the context of compact Riemann surfaces X, via the Riemann uniformization theorem, this can be seen as a distinction between the surfaces of different topologies:

By the uniformization theorem, any hyperbolic surface X – i. e., the Gaussian curvature of X is equal to negative one at every point – is covered by the hyperbolic plane.

In mathematics, a Fuchsian model is a construction of a hyperbolic Riemann surface R as a quotient of the upper half-plane H. By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic.