In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the uniformization theorem says that any simply connected Riemann surface

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

 is conformally equivalent

Conformal equivalence

In mathematics and theoretical physics, two geometries are conformally equivalent if there exists a conformal transformation that maps one geometry to the other one....

 to one of the three domains: the open unit disk, the complex plane

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

, or the Riemann sphere

Riemann sphere

In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...

. In particular it admits a Riemannian metric of constant curvature

Constant curvature

In mathematics, constant curvature in differential geometry is a concept most commonly applied to surfaces. For those the scalar curvature is a single number determining the local geometry, and its constancy has the obvious meaning that it is the same at all points...

. This classifies Riemannian surfaces as elliptic (positively curved – rather, admitting a constant positively curved metric), parabolic (flat), and hyperbolic (negatively curved) according to their universal cover.

The uniformization theorem is a generalization of the Riemann mapping theorem from proper simply connected open

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

 subset

Subset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

s 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.

History

Felix and conjectured the uniformization theorem for (the Riemann surfaces of) algebraic curves. extended this to arbitrary multivalued analytic functions and gave informal aguments in its favor. The first rigorous proofs of the general uniformization theorem were given by and . Paul Koebe later gave several more proofs and generalizations. The history is described in .

Complex classification

Every Riemann surface

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

 is the quotient of a free, proper and holomorphic action of a discrete group

Discrete group

In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...

 on its universal covering and this universal covering is holomorphically isomorphic (one also says: "conformally equivalent") to one of the following:

1. the Riemann sphere
   Riemann sphere
   In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
   
   
2. the complex plane
3. the unit disk in the complex plane.

Geometric classification of surfaces

On an oriented surface, a Riemannian metric naturally induces
an almost complex structure as follows: For a tangent vector v we define J(v) as the vector of the same length which is orthogonal to v and such that (v, J(v)) is positively oriented. On surfaces any almost complex structure is integrable, so this turns the given surface into a Riemann surface

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

.

From this, a classification of metrizable surfaces follows. A connected metrizable surface is a quotient

Quotient space

In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...

 of one of the following by a free action

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

 of a discrete subgroup

Discrete group

In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...

 of an isometry group

Isometry group

In mathematics, the isometry group of a metric space is the set of all isometries from the metric space onto itself, with the function composition as group operation...

:

1. the sphere
   Sphere
   A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
   
    (curvature +1)
2. the Euclidean plane (curvature 0)
3. the hyperbolic plane
   Hyperbolic space
   In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
   
    (curvature −1).

The first case includes all surfaces with positive Euler characteristic

Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...

: the sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

 and the real projective plane

Projective plane

In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...

. The second includes all surfaces with vanishing Euler characteristic: the Euclidean plane, cylinder

Cylinder (geometry)

A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...

, Möbius strip

Möbius strip

The Möbius strip or Möbius band is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface...

, torus

Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

, and Klein bottle

Klein bottle

In mathematics, the Klein bottle is a non-orientable surface, informally, a surface in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a...

.
The third case covers all surfaces with negative Euler characteristic: almost all

Almost all

In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....

 surfaces are hyperbolic. For closed surfaces, this classification is consistent with the Gauss-Bonnet Theorem, which implies that for a closed surface with constant curvature, the sign of that curvature must match the sign of the Euler characteristic.

The positive/flat/negative classification corresponds in algebraic geometry to Kodaira dimension

Kodaira dimension

In algebraic geometry, the Kodaira dimension κ measures the size of the canonical model of a projective variety V.The definition of Kodaira dimension, named for Kunihiko Kodaira, and the notation κ were introduced in the seminar.-The plurigenera:...

 -1,0,1 of the corresponding complex algebraic curve.

For Riemann surfaces, Rado's theorem

Radó's theorem (Riemann surfaces)

In mathematical complex analysis, Rado's theorem, proved by , states that every connected Riemann surface is second-countable ....

 implies that the surface is automatically second countable. For general surfaces this is no longer true, so for the classification above one needs to assume that the surface is second countable (or metrizable).
The Prüfer surface is an example of a surface with no (Riemannian) metric.

Connection to Ricci flow

In introducing the Ricci flow

Ricci flow

In differential geometry, the Ricci flow is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric....

, Richard Hamilton

Richard Hamilton (professor)

Richard Streit Hamilton is Davies Professor of mathematics at Columbia University.He received his B.A in 1963 from Yale University and Ph.D. in 1966 from Princeton University. Robert Gunning supervised his thesis...

 showed that the Ricci flow on a closed surface uniformizes the metric (i.e., the flow converges to a constant curvature metric). However, his proof relied on the uniformization theorem. showed
that it is nevertheless possible to prove the uniformization theorem via Ricci flow.

Related theorems

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.

In 3 dimensions, there are 8 geometries, called the eight Thurston geometries. Not every 3-manifold admits a geometry, but Thurston's geometrization conjecture

Geometrization conjecture

Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed canonically into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces...

 proved by Grigori Perelman

Grigori Perelman

Grigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture...

 states that every 3-manifold can be cut into pieces that are geometrizable.

The simultaneous uniformization theorem

Simultaneous uniformization theorem

In mathematics, the simultaneous uniformization theorem, proved by , states that it is possible to simultaneously uniformize two different Riemann surfaces of the same genus using a quasi-Fuchsian group of the first kind....

 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

Quasi-Fuchsian group

In the mathematical theory of Kleinian groups, a quasi-Fuchsian group is a Kleinian group whose limit set is contained in an invariant Jordan curve. If the limit set is equal to the Jordan curve the quasi-Fuchsian group is said to be of type one, and otherwise it is said to be of type two...

.

The measurable Riemann mapping theorem

Measurable Riemann mapping theorem

In the mathematical theory of quasiconformal mappings in two dimensions, the measurable Riemann mapping theorem, proved by , generalizes the Riemann mapping theorem from conformal to quasiconformal homeomorphisms, and is stated as follows...

 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.