Planar Riemann Surface

(Complex Analysis)


Complex Analysis
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What is Planar Riemann surface?

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.

Technology Types

complex analysisconformal mappingriemann surface



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