Editing Uniformization theorem (section)

== 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}}
*{{citation|last=Brendle|first= Simon|title= Ricci flow and the sphere theorem|series=Graduate Studies in Mathematics|volume= 111|publisher= American Mathematical Society|year= 2010|isbn= 978-0-8218-4938-5}} 
*{{Citation | last1=Chen | first1=Xiuxiong | last2=Lu | first2=Peng | last3=Tian | first3=Gang | author3-link=Gang Tian | title=A note on uniformization of Riemann surfaces by Ricci flow | doi=10.1090/S0002-9939-06-08360-2 | mr=2231924 | year=2006 | journal=[[Proceedings of the American Mathematical Society]] | issn=0002-9939 | volume=134 | issue=11 | pages=3391–3393| doi-access=free }}
*{{citation|last1=Andrews|first1=Ben|last2=Bryan|first2=Paul|title=Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere|journal=Calc. Var. Partial Differential Equations|volume= 39|issue=3–4|year=2010|pages= 419–428|doi=10.1007/s00526-010-0315-5|arxiv=0908.3606|s2cid=1095459}}
*{{citation|last1=Mazzeo|first1=Rafe|last2= Taylor|first2= Michael|title=Curvature and uniformization|journal=[[Israel Journal of Mathematics]]|volume= 130|year=2002|pages= 323–346|doi=10.1007/bf02764082|doi-access=free|arxiv=math/0105016|s2cid=7192529}}
*{{citation|last=Struwe|first= Michael|title=Curvature flows on surfaces|journal=Ann. Sc. Norm. Super. Pisa Cl. Sci.|volume= 1|year=2002|pages= 247–274|url=http://eudml.org/doc/84470}}

===General references===
* {{citation|first= Shiing-shen|last=Chern|author-link=Shiing-Shen Chern|title=An elementary proof of the existence of isothermal parameters on a surface|
journal=Proc. Amer. Math. Soc.|volume= 6 |year=1955|pages= 771–782|doi= 10.2307/2032933|jstor= 2032933|issue= 5|doi-access= free}} 
* {{Citation | last1=DeTurck | first1=Dennis M. | last2=Kazdan | first2=Jerry L. | author2-link=Jerry Kazdan | title=Some regularity theorems in Riemannian geometry | url=http://www.numdam.org/item?id=ASENS_1981_4_14_3_249_0 |mr=644518 | year=1981 | journal=Annales Scientifiques de l'École Normale Supérieure |series=Série 4 | issn=0012-9593 | volume=14 | issue=3 | pages=249–260| doi=10.24033/asens.1405 | doi-access=free }}.
*{{eom|id=U/u095290|title= Uniformization|first=N.A.|last=Gusevskii}}
*{{Citation | last1=Krushkal | first1=S. L. | last2=Apanasov | first2=B. N. | last3=Gusevskiĭ | first3=N. A. | title=Kleinian groups and uniformization in examples and problems | orig-year=1981 | url=https://books.google.com/books?id=QvlhqAGN_y4C | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Translations of Mathematical Monographs | isbn=978-0-8218-4516-5 | mr=0647770 | year=1986 | volume=62}}
*{{citation|first=Michael E.|last=Taylor|author-link=Michael E. Taylor|title=Partial Differential Equations I: Basic Theory|publisher=Springer|year=1996a|isbn=978-0-387-94654-2|pages=376–378}}
*{{citation|first=Michael E.|last=Taylor|author-link=Michael E. Taylor|title=Partial Differential Equations II:Qualitative studies of linear equations|publisher=Springer|year= 1996b|isbn= 978-0-387-94651-1}} 
*{{citation|last1=Bers|first1= Lipman|last2= John|first2= Fritz|last3= Schechter|first3= Martin|title= Partial differential equations (reprint of the 1964 original)| series=Lectures in Applied Mathematics|volume=3A|publisher= American Mathematical Society|year= 1979|isbn=978-0-8218-0049-2}}
*{{citation|last1=Griffiths|first1=Phillip|last2= Harris|first2=Joseph|title= Principles of algebraic geometry|publisher= Wiley|year=1994|isbn= 978-0-471-05059-9}}
*{{citation|last=Warner|first= Frank W.|title= Foundations of differentiable manifolds and Lie groups|series= Graduate Texts in Mathematics|volume= 94|publisher= Springer|year=1983|isbn= 978-0-387-90894-6|doi=10.1007/978-1-4757-1799-0}}