uniformization theorem
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In mathematics, the uniformization theorem says that every
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
is
conformally equivalent In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two di ...
to one of three Riemann surfaces: the open
unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose di ...
, the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, or the
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers pl ...
. The theorem is a generalization of the
Riemann mapping theorem In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected space, simply connected open set, open subset of the complex plane, complex number plane C which is not all of C, then there exists a biholomorphy ...
from simply connected
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (YF ...
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s of the plane to arbitrary simply connected Riemann surfaces. Since every Riemann surface has a
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
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 universal cover ("hyperbolic"). It further follows that every Riemann surface admits a
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
of
constant curvature In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry. The sectional curvature i ...
, where the curvature can be taken to be 1 in the elliptic, 0 in the parabolic and -1 in the hyperbolic case. The uniformization theorem also yields a similar classification of closed
orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
Riemannian 2-manifolds into elliptic/parabolic/hyperbolic cases. Each such manifold has a conformally equivalent Riemannian metric with constant curvature, where the curvature can be taken to be 1 in the elliptic, 0 in the parabolic and -1 in the hyperbolic case.


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 ; a complete account of uniformization up to the 1907 papers of Koebe and Poincaré is given with detailed proofs in (the Bourbaki-type pseudonym of the group of fifteen mathematicians who jointly produced this publication).


Classification of connected Riemann surfaces

Every
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
is the quotient of free, proper and holomorphic action of a
discrete group In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and on ...
on its universal covering and this universal covering, being a simply connected Riemann surface, is holomorphically isomorphic (one also says: "conformally equivalent" or "biholomorphic") to one of the following: #the
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers pl ...
#the complex plane #the unit disk in the complex plane. For compact Riemann surfaces, those with universal cover the unit disk are precisely the hyperbolic surfaces of genus greater than 1, all with non-abelian fundamental group; those with universal cover the complex plane are the Riemann surfaces of genus 1, namely the complex tori or
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s with fundamental group ; and those with universal cover the Riemann sphere are those of genus zero, namely the Riemann sphere itself, with trivial fundamental group.


Classification of closed oriented Riemannian 2-manifolds

On an oriented 2-manifold, a
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
induces a complex structure using the passage to
isothermal coordinates In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric l ...
. If the Riemannian metric is given locally as : ds^2 = E \, dx^2 + 2F \, dx \, dy + G \, dy^2, then in the complex coordinate ''z'' = ''x'' + i''y'', it takes the form : ds^2 = \lambda, dz +\mu \, d\overline, ^2, where :\lambda = \frac14 \left( E + G + 2\sqrt \right),\ \ \mu = \frac1 (E - G + 2iF), so that ''λ'' and ''μ'' are smooth with ''λ'' > 0 and , ''μ'', < 1. In isothermal coordinates (''u'', ''v'') the metric should take the form : ds^2 = \rho (du^2 + dv^2) with ''ρ'' > 0 smooth. The complex coordinate ''w'' = ''u'' + i ''v'' satisfies :\rho \, , dw, ^2 = \rho , w_z, ^2 \left, dz + \, d\overline\^2, so that the coordinates (''u'', ''v'') will be isothermal locally provided the Beltrami equation : = \mu has a locally diffeomorphic solution, i.e. a solution with non-vanishing Jacobian. These conditions can be phrased equivalently in terms of the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
and the
Hodge star operator In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the ...
. and will be isothermal coordinates if , where is defined on differentials by . Let be the
Laplace–Beltrami operator In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named af ...
. By standard elliptic theory, can be chosen to be
harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
near a given point, i.e. , with non-vanishing. By the
Poincaré lemma In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another diff ...
has a local solution exactly when . This condition is equivalent to , so can always be solved locally. Since is non-zero and the square of the Hodge star operator is −1 on 1-forms, and must be linearly independent, so that and give local isothermal coordinates. The existence of isothermal coordinates can be proved by other methods, for example using the general theory of the Beltrami equation, as in , or by direct elementary methods, as in and . From this correspondence with compact Riemann surfaces, a classification of closed orientable Riemannian 2-manifolds follows. Each such is conformally equivalent to a unique closed 2-manifold of
constant curvature In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry. The sectional curvature i ...
, so a
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of one of the following by a
free action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism g ...
of a
discrete subgroup In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and o ...
of an
isometry group In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the ...
: #the
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
(curvature +1) #the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
(curvature 0) #the
hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...
(curvature −1). File:Orange Sphere.png, genus 0 File:Orange Torus.png, genus 1 File:Orange Genus 2 Surface.png, genus 2 File:Orange Genus 3 Surface.png, genus 3 The first case gives the 2-sphere, the unique 2-manifold with constant positive curvature and hence positive
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
(equal to 2). The second gives all flat 2-manifolds, i.e. the tori, which have Euler characteristic 0. The third case covers all 2-manifolds of constant negative curvature, i.e. the ''hyperbolic'' 2-manifolds all of which have negative Euler characteristic. The classification is consistent with the
Gauss–Bonnet theorem In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology. In the simplest application, the case of a t ...
, which implies that for a closed surface with constant curvature, the sign of that curvature must match the sign of the Euler characteristic. The Euler characteristic is equal to 2 – 2''g'', where ''g'' is the genus of the 2-manifold, i.e. the number of "holes".


Methods of proof

Many classical proofs of the uniformization theorem rely on constructing a real-valued
harmonic function In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : \f ...
on the simply connected Riemann surface, possibly with a singularity at one or two points and often corresponding to a form of
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differential ...
. Four methods of constructing the harmonic function are widely employed: the
Perron method In the mathematical study of harmonic functions, the Perron method, also known as the method of subharmonic functions, is a technique introduced by Oskar Perron for the solution of the Dirichlet problem for Laplace's equation. The Perron method work ...
; the
Schwarz alternating method Schwarz may refer to: * Schwarz, Germany, a municipality in Mecklenburg-Vorpommern, Germany * Schwarz (surname), a surname (and list of people with the surname) * Schwarz (musician), American DJ and producer * Schwarz (Böhse Onkelz album), ''Schw ...
;
Dirichlet's principle In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation. Formal statement Dirichlet's principle states that, if the functi ...
; and
Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is ass ...
's method of orthogonal projection. In the context of closed Riemannian 2-manifolds, several modern proofs invoke nonlinear differential equations on the space of conformally equivalent metrics. These include the Beltrami equation from Teichmüller theory and an equivalent formulation in terms of
harmonic map In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for ...
s;
Liouville's equation :''For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).'' : ''For Liouville's equation in quantum mechanics, see Von Neumann equation.'' : ''For Liouville's equation in Euclidean space, see Liouville–Bratu–Gel ...
, already studied by Poincaré; and
Ricci flow In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be ana ...
along with other nonlinear flows. Rado's theorem shows that every Riemann surface is automatically
second-countable In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
. Although Rado's theorem is often used in proofs of the uniformization theorem, some proofs have been formulated so that Rado's theorem becomes a consequence. Second countability is automatic for compact Riemann surfaces.


Hilbert space methods

In 1913 Hermann Weyl published his classic textbook "Die Idee der Riemannschen Fläche" based on his Göttingen lectures from 1911 to 1912. It was the first book to present the theory of Riemann surfaces in a modern setting and through its three editions has remained influential. Dedicated to
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
, the first edition incorporated Hilbert's treatment of the
Dirichlet problem In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. The Dirichlet prob ...
using
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
techniques; Brouwer's contributions to topology; and Koebe's proof of the uniformization theorem and its subsequent improvements. Much later developed his method of orthogonal projection which gave a streamlined approach to the Dirichlet problem, also based on Hilbert space; that theory, which included Weyl's lemma on
elliptic regularity In the theory of partial differential equations, a partial differential operator P defined on an open subset :U \subset^n is called hypoelliptic if for every distribution u defined on an open subset V \subset U such that Pu is C^\infty (smooth ...
, was related to Hodge's theory of harmonic integrals; and both theories were subsumed into the modern theory of elliptic operators and
Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense t ...
s. In the third edition of his book from 1955, translated into English in , Weyl adopted the modern definition of differential manifold, in preference to triangulations, but decided not to make use of his method of orthogonal projection. followed Weyl's account of the uniformisation theorem, but used the method of orthogonal projection to treat the Dirichlet problem. describes the approach in Weyl's book and also how to shorten it using the method of orthogonal projection. A related account can be found in .


Nonlinear flows

In introducing the
Ricci flow In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be ana ...
, Richard S. 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. The missing step involved Ricci flow on the 2-sphere: a method for avoiding an appeal to the uniformization theorem (for genus 0) was provided by ; a short self-contained account of Ricci flow on the 2-sphere was given in .


Generalizations

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 In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimens ...
proved by
Grigori Perelman Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...
states that every 3-manifold can be cut into pieces that are geometrizable. The
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. The quasi-Fuchsian group ...
of
Lipman Bers Lipman Bers ( Latvian: ''Lipmans Berss''; May 22, 1914 – October 29, 1993) was a Latvian-American mathematician, born in Riga, who created the theory of pseudoanalytic functions and worked on Riemann surfaces and Kleinian groups. He was also kn ...
shows that it is possible to simultaneously uniformize two compact Riemann surfaces of the same genus >1 with the same
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, a ...
. The
measurable Riemann mapping theorem In mathematics, the measurable Riemann mapping theorem is a theorem proved in 1960 by Lars Ahlfors and Lipman Bers in complex analysis and geometric function theory. Contrary to its name, it is not a direct generalization of the Riemann mapping t ...
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.


See also

* ''p''-adic uniformization theorem.


Notes


References


Historic references

*. * * * * * * * * * * * * *


Historical surveys

* * * *, translation o
French text
(prepared in 2007 during centenary of 1907 papers of Koebe and Poincaré)


Harmonic functions

Perron's method * * * * * * * * * * * Schwarz's alternating method * * * Dirichlet principle * * * Weyl's method of orthogonal projection * * * Sario operators * *


Nonlinear differential equations

Beltrami's equation * * * * * Harmonic maps * Liouville's equation * * * Flows on Riemannian metrics * * * * * * * * * *


General references

* * . * * * * * * *


External links


Conformal Transformation: from Circle to Square
{{Manifolds Manifolds Riemann surfaces Theorems in differential geometry