World Library  
Flag as Inappropriate
Email this Article

Uniformization theorem

Article Id: WHEBN0000288270
Reproduction Date:

Title: Uniformization theorem  
Author: World Heritage Encyclopedia
Language: English
Subject: Differential geometry of surfaces, Surface, Riemann sphere, Classification of manifolds, Geometric topology
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Uniformization theorem

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. In particular it admits a Riemannian metric of constant curvature. 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 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.

History

Felix and Henri conjectured the uniformization theorem for (the Riemann surfaces of) algebraic curves. extended this to arbitrary multivalued analytic functions and gave informal arguments 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 Gray (1994).

Complex classification

Every Riemann surface is the quotient of a free, proper and holomorphic action of a discrete group 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
  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 (vJ(v)) is positively oriented. On surfaces any almost complex structure is integrable, so this turns the given surface into a Riemann surface.

From this, a classification of metrizable surfaces follows. A connected metrizable surface is a quotient of one of the following by a free action of a discrete subgroup of an isometry group:

  1. the sphere (curvature +1)
  2. the Euclidean plane (curvature 0)
  3. the hyperbolic plane (curvature −1).

The first case includes all surfaces with positive Euler characteristic: the sphere and the real projective plane. The second includes all surfaces with vanishing Euler characteristic: the Euclidean plane, cylinder, Möbius strip, torus, and Klein bottle. The third case covers all surfaces with negative Euler characteristic: almost all 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 −∞,0,1 of the corresponding complex algebraic curve. For Riemann surfaces, Rado's theorem 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, Richard Hamilton 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 proved by Grigori Perelman states that every 3-manifold can be cut into pieces that are geometrizable.

The simultaneous uniformization theorem of Lipman 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.

References

External links

  • Conformal Transformation: from Circle to Square.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.