Editing Uniformization theorem

{{short description|Simply connected Riemann surface is equivalent to an open disk, complex plane, or sphere}}

In mathematics, the '''uniformization theorem''' states that every [[simply connected]] [[Riemann surface]] is [[Conformal equivalence|conformally equivalent]] to one of three Riemann surfaces: the open [[unit disk]], the [[complex plane]], or the [[Riemann sphere]]. The theorem is a generalization of the [[Riemann mapping theorem]] from simply connected [[open set|open]] [[subset]]s of the plane to arbitrary simply connected Riemann surfaces.

Since every Riemann surface has a [[Universal Cover|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 universal cover ("hyperbolic"). It further follows that every Riemann surface admits a [[Riemannian metric]] of [[constant curvature]], 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 manifold|orientable]] [[Riemannian manifold|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 {{harvs|txt|authorlink=Felix Klein|last=Klein|year=1883}} and Henri {{harvs|txt|authorlink=Henri Poincaré|last=Poincaré|year1=1882}} conjectured the uniformization theorem for (the Riemann surfaces of) algebraic curves. {{harvs|txt|first=Henri |last=Poincaré|year1=1883}} 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  {{Harvs|txt|last=Poincaré|authorlink=Henri Poincaré|year=1907}} and {{harvs|txt|authorlink=Paul Koebe|first=Paul |last=Koebe|year1=1907a|year2=1907b|year3=1907c}}. Paul Koebe later gave several more proofs and generalizations. The history is described in {{harvtxt|Gray|1994}}; a complete account of uniformization up to the 1907 papers of Koebe and Poincaré is given with detailed proofs in {{harvtxt|de Saint-Gervais|2016}} (the [[Nicolas Bourbaki|Bourbaki]]-type pseudonym of the group of fifteen mathematicians who jointly produced this publication).

==Classification of connected Riemann surfaces==

Every [[Riemann surface]] is the quotient of the free, proper and holomorphic action of a [[discrete group]] 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]]
#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]]s with fundamental group {{math|'''Z'''<sup>2</sup>}}; 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]] induces a complex structure using the passage to [[isothermal coordinates]]. If the Riemannian metric is given locally as

:<math> ds^2 = E \, dx^2 + 2F \, dx \, dy + G \, dy^2,</math>

then in the complex coordinate ''z'' = ''x'' + i''y'', it takes the form

:<math> ds^2 = \lambda|dz +\mu \, d\overline{z}|^2,</math>

where

:<math>\lambda = \frac14 \left( E + G + 2\sqrt{EG - F^2} \right),\ \ 
\mu = \frac1{4\lambda} (E - G + 2iF),</math>

so that ''λ'' and ''μ'' are smooth with ''λ'' > 0 and |''μ''| < 1. In isothermal coordinates (''u'', ''v'') the metric should take the form

:<math> ds^2 = \rho (du^2 + dv^2)</math>

with ''ρ'' > 0 smooth. The complex coordinate ''w'' = ''u'' + i ''v'' satisfies

:<math>\rho \, |dw|^2 = \rho |w_z|^2 \left| dz + {w_{\overline{z}}\over w_z} \, d\overline{z}\right|^2,</math>

so that the coordinates (''u'', ''v'') will be isothermal locally provided the [[Beltrami equation]]

:<math> {\partial w\over \partial \overline{z}} = \mu {\partial w\over \partial z}</math>

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]] and the [[Hodge star operator]] {{math|∗}}.<ref>{{harvnb|DeTurck|Kazdan|1981}}; {{harvnb|Taylor|1996a|pp=377&ndash;378}}</ref>
{{math|''u''}} and {{math|''v''}} will be isothermal coordinates if {{math|1=∗''du'' = ''dv''}}, where {{math|∗}} is defined
on differentials by {{math|1=∗(''p'' ''dx'' + ''q'' ''dy'') = −''q'' ''dx'' + ''p'' ''dy''}}.
Let {{math|1=∆ = ∗''d''∗''d''}} be the [[Laplace–Beltrami operator]]. By standard elliptic theory, {{math|''u''}} can be chosen to be [[harmonic]] near a given point, i.e. {{math|1=Δ ''u'' = 0}},  with {{math|''du''}} non-vanishing. By the [[Poincaré lemma]] {{math|1= ''dv'' = ∗''du''}} has a local solution {{math|''v''}} exactly when {{math|1=''d''(∗''du'') = 0}}. This condition is equivalent to {{math|1=Δ ''u'' = 0}}, so can always be solved locally. Since {{math|''du''}} is non-zero and the square of the Hodge star operator is &minus;1 on 1-forms, {{math|''du''}} and {{math|''dv''}} must be linearly independent, so that {{math|''u''}} and {{math|''v''}} give local isothermal coordinates.

The existence of isothermal coordinates can be proved by other methods, for example using the [[Beltrami equation#Solution in L2 for smooth Beltrami coefficients|general theory of the Beltrami equation]], as in {{harvtxt|Ahlfors|2006}}, or by direct elementary methods, as in {{harvtxt|Chern|1955}} and {{harvtxt|Jost|2006}}.

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]], so a [[Quotient space (topology)|quotient]] of one of the following by a [[Group action (mathematics)|free action]] of a [[discrete group|discrete subgroup]] of an [[isometry group]]:

#the [[sphere]] (curvature +1)
#the [[Euclidean plane]] (curvature 0)
#the [[Hyperbolic space|hyperbolic plane]] (curvature &minus;1).
<gallery>
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
</gallery>
The first case gives the 2-sphere, the unique 2-manifold with constant positive curvature and hence positive [[Euler characteristic]] (equal to 2). The second gives all flat 2-manifolds, i.e. the [[torus|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]], 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]] 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]]. Four methods of constructing the harmonic function are widely employed: the [[Perron method]]; the [[Schwarz alternating method]]; [[Dirichlet's principle]]; and [[Hermann Weyl|Weyl]]'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]]s; [[Liouville's equation]], already studied by Poincaré; and [[Ricci flow]] along with other nonlinear flows.

[[Radó's theorem (Riemann surfaces)|Rado's theorem]] shows that every Riemann surface is automatically [[second-countable space|second-countable]]. 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===
{{See also|Planar Riemann surface#Uniformization theorem}}
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]], the first edition incorporated [[David Hilbert|Hilbert's]] treatment of the [[Dirichlet problem]] using [[Hilbert space]] techniques; [[L. E. J. Brouwer|Brouwer's]] contributions to topology; and [[Paul Koebe|Koebe's]] proof of the uniformization theorem and its subsequent improvements. Much later {{harvtxt|Weyl|1940}} 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 (Laplace equation)|Weyl's lemma]] on [[elliptic regularity]], was related to [[W. V. D. Hodge|Hodge's]] [[Hodge theory|theory of harmonic integrals]]; and both theories were subsumed into the modern theory of [[elliptic operator]]s and {{math|''L''<sup>2</sup>}} [[Sobolev space]]s. In the third edition of his book from 1955, translated into English in {{harvtxt|Weyl|1964}}, Weyl adopted the modern definition of differential manifold, in preference to [[triangulation (topology)|triangulations]], but decided not to make use of his method of orthogonal projection. {{harvtxt|Springer|1957}} followed Weyl's account of the uniformisation theorem, but used the method of orthogonal projection to treat the Dirichlet problem. {{harvtxt|Kodaira|2007}} 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 {{harvtxt|Donaldson|2011}}.

===Nonlinear flows===
{{see also|Ricci flow#Relationship to uniformization and geometrization}}

[[Richard S. Hamilton]] showed that the [[Ricci flow|normalized 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 {{harvtxt|Chen|Lu|Tian|2006}};<ref>{{harvnb|Brendle|2010}}</ref> a short self-contained account of Ricci flow on the 2-sphere was given in {{harvtxt|Andrews|Bryan|2010}}.

==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 [[Geometrization conjecture#The eight Thurston geometries|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.

== See also ==
*[[p-adic uniformization theorem|''p''-adic uniformization theorem]]

==Notes==
{{reflist}}

== References ==

===Historic references===
*{{Citation
| last = Schwarz
| first = H. A.
| author-link = Hermann Schwarz
| title = Über einen Grenzübergang durch alternierendes Verfahren
| journal = Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich
| volume = 15
| pages = 272–286
| year = 1870
| url = https://www.biodiversitylibrary.org/item/34472#page/280/mode/1up
| jfm =02.0214.02
}}.
*{{Citation | last1=Klein | first1=Felix | title=Neue Beiträge zur Riemann'schen Functionentheorie | doi=10.1007/BF01442920 | jfm=15.0351.01 | year=1883 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=21 | issue=2 | pages=141–218| s2cid=120465625 | url=https://zenodo.org/record/2161412 }}
*{{Citation | last1=Koebe | first1=P. | title=Über die Uniformisierung reeller analytischer Kurven | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN00250118X | jfm=38.0453.01
 | year=1907a| journal=Göttinger Nachrichten | pages=177–190}}
*{{Citation | last1=Koebe | first1=P. | title=Über die Uniformisierung beliebiger analytischer Kurven | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002501198 | jfm=38.0454.01 | year=1907b | journal=Göttinger Nachrichten | pages=191–210}}
*{{Citation | last1=Koebe | first1=P. | title=Über die Uniformisierung beliebiger analytischer Kurven (Zweite Mitteilung) | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002501473 | jfm=38.0455.02 | year=1907c | journal=Göttinger Nachrichten | pages=633–669}}
*{{citation|last=Koebe|first=Paul|title=Über die Uniformisierung beliebiger analytischer Kurven|year=1910a|volume=138|pages=192–253|journal=Journal für die Reine und Angewandte Mathematik|doi=10.1515/crll.1910.138.192|s2cid=120198686}}
*{{citation|last=Koebe|first=Paul|title= Über die Hilbertsche Uniformlsierungsmethode|journal=Göttinger Nachrichten|year=1910b|pages=61–65|url=http://gdz.sub.uni-goettingen.de/pdfcache/PPN252457811_1910/PPN252457811_1910___LOG_0008.pdf}}
*{{Citation | last1=Poincaré | first1=H. | title=Mémoire sur les fonctions fuchsiennes | doi=10.1007/BF02592135 | jfm=15.0342.01 | year=1882 | journal=[[Acta Mathematica]] | issn=0001-5962 | volume=1 | pages=193–294| doi-access=free }}
*{{Citation | last1=Poincaré | first1=Henri | author1-link=Henri Poincaré | title=Sur un théorème de la théorie générale des fonctions | url=http://www.numdam.org/item?id=BSMF_1883__11__112_1 | jfm=15.0348.01 | year=1883 | journal=Bulletin de la Société Mathématique de France | issn=0037-9484 | volume=11 | pages=112–125| doi=10.24033/bsmf.261 | doi-access=free }}
*{{Citation | last1=Poincaré | first1=Henri | author1-link=Henri Poincaré | title=Sur l'uniformisation des fonctions analytiques | doi=10.1007/BF02415442 | jfm=38.0452.02 | year=1907 | journal=[[Acta Mathematica]] | issn=0001-5962 | volume=31 | pages=1–63| doi-access=free | url=https://zenodo.org/records/2012595/files/article.pdf }}
*{{citation|first=David|last=Hilbert|title=Zur Theorie der konformen Abbildung|journal= Göttinger Nachrichten|year= 1909|pages=314–323|url=http://gdz.sub.uni-goettingen.de/pdfcache/PPN252457811_1909/PPN252457811_1909___LOG_0042.pdf}}
*{{Citation | last1=Perron | first1=O. | author1-link=Oskar Perron | title=Eine neue Behandlung der ersten Randwertaufgabe für Δu=0 | doi=10.1007/BF01192395 |date= 1923 | journal=[[Mathematische Zeitschrift]] | issn=0025-5874 | volume=18 | issue=1 | pages=42–54| s2cid=122843531 }}
*{{citation|last=Weyl|first=Hermann|title=Die Idee der Riemannschen Fläche (1997 reprint of the 1913 German original)|publisher=Teubner|year=1913|isbn= 978-3-8154-2096-6}} 
*{{citation|first=Hermann|last= Weyl|title= The method of orthogonal projections in potential theory|journal=Duke Math. J.|volume= 7|pages= 411–444|year=1940|doi=10.1215/s0012-7094-40-00725-6}}

===Historical surveys===
*{{citation|jstor=2320507|last=Abikoff|first= William|title=The uniformization theorem|journal=Amer. Math. Monthly|volume= 88|issue=8|year=1981|pages=574–592|doi=10.2307/2320507}}
*{{Citation | last1=Gray | first1=Jeremy | title=On the history of the Riemann mapping theorem | url=http://www.math.stonybrook.edu/~bishop/classes/math401.F09/GrayRMT.pdf | mr=1295591 | year=1994 | journal=Rendiconti del Circolo Matematico di Palermo. Serie II. Supplemento | issue=34 | pages=47–94}}
*{{citation|title=Hidden Harmony—Geometric Fantasies: The Rise of Complex Function Theory|series= Sources and Studies in the History of Mathematics and Physical Sciences|
first1=Umberto|last1= Bottazzini|first2= Jeremy|last2= Gray|publisher=Springer|year= 2013|isbn=978-1461457251}}
*{{citation|first=Henri Paul|last= de Saint-Gervais|title=Uniformization of Riemann Surfaces: revisiting a hundred-year-old theorem|translator=Robert G. Burns|
isbn=978-3-03719-145-3|doi= 10.4171/145|year=2016|url=http://www.ems-ph.org/books/book.php?proj_nr=198|publisher=European Mathematical Society|url-access=subscription}}, translation of [http://perso.ens-lyon.fr/ghys/articles/Uniformisationsurfaces.pdf French text] (prepared in 2007 during centenary of 1907 papers of Koebe and Poincaré)

===Harmonic functions===
'''Perron's method'''

*{{citation|last=Heins|first=M.|title=The conformal mapping of simply-connected Riemann surfaces| journal=Ann. of Math.|volume= 50|issue=3|year=1949|pages= 686–690|doi=10.2307/1969555|jstor=1969555}}
*{{citation|last=Heins|first= M.|title=Interior mapping of an orientable surface into ''S''<sup>2</sup>|journal=Proc. Amer. Math. Soc. |volume=2|issue= 6|year=1951|pages= 951–952|doi=10.1090/s0002-9939-1951-0045221-4|doi-access=free}}
*{{citation|last=Heins|first= M.|title=The conformal mapping of simply-connected Riemann surfaces. II|journal=Nagoya Math. J.|volume= 12|year= 1957|pages= 139–143|doi=10.1017/s002776300002198x|doi-access=free|url=https://projecteuclid.org/journals/nagoya-mathematical-journal/volume-12/issue-none/The-conformal-mapping-of-simply-connected-Riemann-surfaces-II/nmj/1118799932.pdf}}
*{{citation|last= Pfluger|first= Albert|title= Theorie der Riemannschen Flächen|publisher= Springer|year= 1957}}
*{{citation|last=Ahlfors|first=Lars V.|title= Conformal invariants: topics in geometric function theory|publisher= AMS Chelsea Publishing|year= 2010|isbn=978-0-8218-5270-5}}
*{{citation|last= Beardon|first= A. F.|title= A primer on Riemann surfaces|journal= London Mathematical Society Lecture Note Series|volume= 78|publisher= Cambridge University Press|year= 1984|isbn= 978-0521271042|url-access= registration|url= https://archive.org/details/primeronriemanns0000bear}}
*{{citation|last= Forster|first= Otto |title=Lectures on Riemann surfaces|translator=Bruce Gilligan|series= Graduate Texts in Mathematics|volume= 81|publisher= Springer|year= 1991|isbn=978-0-387-90617-1}}
*{{citation|last1=Farkas|first1=Hershel M.|last2=Kra|first2=Irwin|title=Riemann surfaces|publisher=Springer|edition=2nd|isbn= 978-0-387-90465-8|year=1980}}
*{{citation|last=Gamelin|first=Theodore W.|title= Complex analysis|series= Undergraduate Texts in Mathematics|publisher= Springer|year= 2001|isbn=978-0-387-95069-3}}
*{{citation|last=Hubbard|first= John H.|title=Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1. Teichmüller theory| publisher= Matrix Editions|year= 2006|isbn=978-0971576629}}
*{{citation|last=Schlag|first= Wilhelm|title=A course in complex analysis and Riemann surfaces.|series=Graduate Studies in Mathematics|volume= 154|publisher= American Mathematical Society|year=2014|isbn= 978-0-8218-9847-5}}

'''Schwarz's alternating method'''

*{{citation|last=Nevanlinna|first= Rolf|title= Uniformisierung|series=Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete|volume=64|publisher= Springer|year= 1953|doi=10.1007/978-3-642-52801-9|isbn=978-3-642-52802-6}}
*{{citation|last1=Behnke|first1=Heinrich|last2= Sommer|first2= Friedrich|title= Theorie der analytischen Funktionen einer komplexen Veränderlichen|edition=3rd|series= Die Grundlehren der mathematischen Wissenschaften|volume= 77 |publisher=Springer|year=1965}}
*{{citation|last= Freitag|first= Eberhard|title=Complex analysis. 2. Riemann surfaces, several complex variables, abelian functions, higher modular functions|publisher= Springer|year= 2011|isbn=978-3-642-20553-8}}

'''Dirichlet principle'''
*{{citation|last=Weyl|first= Hermann|title= The concept of a Riemann surface|translator= Gerald R. MacLane|publisher=Addison-Wesley|year= 1964|mr=0069903}}
*{{citation|last=Courant|first=Richard|title= Dirichlet's principle, conformal mapping, and minimal surfaces|publisher= Springer|year= 1977|isbn=978-0-387-90246-3}}
*{{citation|last= Siegel|first= C. L.|title= Topics in complex function theory. Vol. I. Elliptic functions and uniformization theory|translator1= A. Shenitzer|translator2= D. Solitar|year= 1988 |publisher= Wiley|isbn= 978-0471608448}}

'''Weyl's method of orthogonal projection'''

*{{citation|last=Springer|first= George|title= Introduction to Riemann surfaces|publisher= Addison-Wesley|year=1957|mr=0092855}}
*{{citation|last=Kodaira|first=Kunihiko|title= Complex analysis|series=Cambridge Studies in Advanced Mathematics|volume= 107|publisher= Cambridge University Press|year= 2007|isbn= 9780521809375}}
*{{citation|last=Donaldson|first=Simon|title= Riemann surfaces|series= Oxford Graduate Texts in Mathematics|volume= 22|publisher= Oxford University Press|year=2011|isbn= 978-0-19-960674-0}}

'''Sario operators'''

*{{citation|last=Sario|first= Leo|title=A linear operator method on arbitrary Riemann surfaces|journal=Trans. Amer. Math. Soc.|volume= 72|issue= 2|year=1952|pages= 281–295|doi=10.1090/s0002-9947-1952-0046442-2|doi-access=free}}
*{{citation|last1=Ahlfors|first1=Lars V.|last2= Sario|first2= Leo|title=Riemann surfaces|series=Princeton Mathematical Series|volume= 26|publisher= Princeton University Press|year= 1960}}

===Nonlinear differential equations===

'''Beltrami's equation'''

*{{citation|last=Ahlfors|first= Lars V.|title= Lectures on quasiconformal mappings|edition=2nd|series= University Lecture Series|volume= 38|publisher= American Mathematical Society|year= 2006|isbn=978-0-8218-3644-6}}
*{{citation|last1=Ahlfors|first1= Lars V.|last2= Bers|first2= Lipman|title= Riemann's mapping theorem for variable metrics|journal= Ann. of Math. |volume= 72|issue= 2|year=  1960|pages= 385–404|doi=10.2307/1970141|jstor= 1970141}}
*{{citation|last=Bers|first= Lipman|title= Simultaneous uniformization|journal= Bull. Amer. Math. Soc.|volume= 66|issue= 2|year= 1960|pages= 94–97|doi=10.1090/s0002-9904-1960-10413-2|doi-access= free|url= https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-66/issue-2/Simultaneous-uniformization/bams/1183523461.pdf}}
*{{citation|last=Bers|first= Lipman|title=Uniformization by Beltrami equations|journal=Comm. Pure Appl. Math.|volume= 14|issue= 3|year= 1961|pages= 215–228|doi=10.1002/cpa.3160140304}}
*{{Citation | last1=Bers | first1=Lipman|author-link=Lipman Bers | title=Uniformization, moduli, and Kleinian groups | doi=10.1112/blms/4.3.257  | mr=0348097 | year=1972 | journal=The Bulletin of the London Mathematical Society | issn=0024-6093 | volume=4 | issue=3 | pages=257–300}}

'''Harmonic maps'''

*{{citation|last=Jost|first= Jürgen |title=Compact Riemann surfaces: an introduction to contemporary mathematics|edition=3rd|publisher= Springer|year=2006|isbn= 978-3-540-33065-3}}

'''Liouville's equation'''

*{{citation|last=Berger|first= Melvyn S.|title=Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds|journal=Journal of Differential Geometry|volume= 5|issue= 3–4|year=1971|pages=325–332|doi= 10.4310/jdg/1214429996|doi-access=free}}
*{{citation|last=Berger|first= Melvyn S.|title=Nonlinearity and functional analysis|publisher=Academic Press|year=1977|isbn= 978-0-12-090350-4}}
*{{citation|last=Taylor|first= Michael E.|title= Partial differential equations III. Nonlinear equations |edition=2nd|series= Applied Mathematical Sciences|volume= 117|publisher=Springer|year= 2011|isbn= 978-1-4419-7048-0}}

'''Flows on Riemannian metrics'''

*{{citation|last=Hamilton|first= Richard S.|chapter= The Ricci flow on surfaces|title=Mathematics and general relativity (Santa Cruz, CA, 1986)|pages= 237–262|series=Contemp. Math.|volume= 71|publisher= American Mathematical Society|year= 1988}}
*{{citation|last=Chow|first= Bennett|title=The Ricci flow on the 2-sphere|journal=J. Differential Geom.|volume= 33|issue= 2|year=1991|pages= 325–334|doi= 10.4310/jdg/1214446319|doi-access= free}}
*{{citation|last1=Osgood|first1= B.|last2= Phillips|first2= R.|last3=Sarnak|first3=P.|title=Extremals of determinants of Laplacians|journal=J. Funct. Anal.|volume= 80|year=1988|pages= 148–211|doi=10.1016/0022-1236(88)90070-5|citeseerx=10.1.1.486.558}}
*{{citation|first=P. |last=Chrusciel|title= Semi-global existence and convergence of solutions of the Robinson-Trautman (2-dimensional Calabi) equation|journal= Communications in Mathematical Physics|volume= 137|issue=2|year=1991|pages= 289–313|doi=10.1007/bf02431882|bibcode=1991CMaPh.137..289C|citeseerx=10.1.1.459.9029|s2cid=53641998}}
*{{citation|last=Chang|first=Shu-Cheng|title=Global existence and convergence of solutions of Calabi flow on surfaces of genus ''h''&nbsp;≥&nbsp;2|journal=J. Math. Kyoto Univ.|volume= 40|issue=2|year=2000|pages= 363–377|doi=10.1215/kjm/1250517718|doi-access=free}}
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== External links ==
* [https://www.flickr.com/photos/sbprzd/362529354 Conformal Transformation: from Circle to Square].

{{Manifolds}}

[[Category:Manifolds]]
[[Category:Riemann surfaces]]
[[Category:Theorems in differential geometry]]