About: Planar Riemann surface

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In mathematics, a planar Riemann surface (or schlichtartig Riemann surface) is a Riemann surface sharing the topological properties of a connected open subset of the Riemann sphere. They are characterized by the topological property that the complement of every closed Jordan curve in the Riemann surface has two connected components. An equivalent characterization is the differential geometric property that every closed differential 1-form of compact support is exact. Every simply connected Riemann surface is planar. The class of planar Riemann surfaces was studied by Koebe who proved in 1910, as a generalization of the uniformization theorem, that every such surface is conformally equivalent to either the Riemann sphere or the complex plane with slits parallel to the real axis removed.

Attributes	Values
rdfs:label	Planar Riemann surface (en)
rdfs:comment	In mathematics, a planar Riemann surface (or schlichtartig Riemann surface) is a Riemann surface sharing the topological properties of a connected open subset of the Riemann sphere. They are characterized by the topological property that the complement of every closed Jordan curve in the Riemann surface has two connected components. An equivalent characterization is the differential geometric property that every closed differential 1-form of compact support is exact. Every simply connected Riemann surface is planar. The class of planar Riemann surfaces was studied by Koebe who proved in 1910, as a generalization of the uniformization theorem, that every such surface is conformally equivalent to either the Riemann sphere or the complex plane with slits parallel to the real axis removed. (en)
dct:subject	Riemann surfaces Conformal mappings Complex analysis
Wikipage page ID	50964981 (xsd:integer)
Wikipage revision ID	1102509911 (xsd:integer)
Link from a Wikipage to another Wikipage	Beltrami equation Archive for Rational Mechanics and Analysis Argument (complex analysis) Argument principle Jordan curve theorem Paul Koebe Richard Courant Riemann surface Invariance of domain Limit set Riemann surfaces Connected component (topology) Math. Z. Mathematics Maximum principle Fundamental theorem of calculus Green's function Conformal equivalence Liouville's theorem (complex analysis) Stokes' theorem Closed and exact differential forms Fundamental group Harmonic conjugate Harmonic measure Cauchy's integral formula AIAA J. Abelianization Acta Math. Carathéodory's theorem (conformal mapping) Conformal mappings Dirichlet problem Schwarz reflection principle Harmonic function Hermann Weyl Riemann sphere Complex analysis Jordan curve Differential forms on a Riemann surface Dover Publications Circle packing theorem Schottky group Uniformization theorem Cauchy-Riemann equations Poisson kernel Schoenflies problem Poincaré lemma Seifert-van Kampen theorem Simply connected Singular homology group Carathéodory's kernel theorem Koebe distortion theorem
Link from a Wikipage to an external page	https://link.springer.com/article/10.1007%2FBF00284172 https://link.springer.com/article/10.1007%2FBF01181387 https://link.springer.com/article/10.1007%2FBF01199400 https://link.springer.com/article/10.1007/BF01212905 http://gdz.sub.uni-goettingen.de/pdfcache/PPN252457811_1910/PPN252457811_1910___LOG_0008.pdf https://arc.aiaa.org/doi/pdf/10.2514/3.61308 https://projecteuclid.org/download/pdf_1/euclid.acta/1485887474 https://eudml.org/doc/145249 https://eudml.org/doc/149335
sameAs	Planar Riemann surface Planar Riemann surface Planar Riemann surface
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Harvtxt dbt:Main dbt:Main_article dbt:Math dbt:Reflist
has abstract	In mathematics, a planar Riemann surface (or schlichtartig Riemann surface) is a Riemann surface sharing the topological properties of a connected open subset of the Riemann sphere. They are characterized by the topological property that the complement of every closed Jordan curve in the Riemann surface has two connected components. An equivalent characterization is the differential geometric property that every closed differential 1-form of compact support is exact. Every simply connected Riemann surface is planar. The class of planar Riemann surfaces was studied by Koebe who proved in 1910, as a generalization of the uniformization theorem, that every such surface is conformally equivalent to either the Riemann sphere or the complex plane with slits parallel to the real axis removed. (en)
prov:wasDerivedFrom	wikipedia-en:Planar_Riemann_surface?oldid=1102509911&ns=0
page length (characters) of wiki page	37657 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Planar_Riemann_surface
is rdfs:seeAlso of	Uniformization theorem
is Link from a Wikipage to another Wikipage of	Paul Koebe Neumann–Poincaré operator Differential forms on a Riemann surface Schlichtartig List of things named after Bernhard Riemann
is Wikipage redirect of	Schlichtartig
is known for of	Paul Koebe
is known for of	Paul Koebe
is foaf:primaryTopic of	wikipedia-en:Planar_Riemann_surface

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