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.
Elementary properties
A closed 1-form ω is exact if and only if ∫γ ω = 0 for every closed Jordan curve γ. This follows from the Poincaré lemma for 1-forms and the fact that ∫δ df = f(δ(b)) – f(δ(a)) for a path δ parametrized by [a, b] and f a smooth function defined on an open neighbourhood of δ([a, b]). This formula for ∫δ df extends by continuity to continuous paths, and hence vanishes for a closed path. Conversely if ∫γ ω = 0 for every closed Jordan curve γ, then a function f(z) can be defined on X by fixing a point w and taking any piecewise smooth path δ from w to z and set f(z) = ∫δ ω. The assumption implies that f is independent of the path. To check that df = ω, it suffices to check this locally. Fix z0 and take a path δ1 from w to z0. Near z0 the Poincaré lemma implies ω = dg for some smooth function g defined in a neighbourhood of z0. If δ2 is a path from z0 to z, then f(z) = ∫δ1 ω + ∫δ2 ω = ∫δ1 ω + g(z) − g(z0), so f differs from g by a constant near z0. Hence df = dg = ω near z0. A closed Jordan curve γ on a Riemann surface separates the surface into two disjoint connected regions if and only if ∫γ ω = 0 for every closed 1-form ω of compact support. If the closed Jordan curve γ separates the surface, it is homotopic to a smooth Jordan curve δ (with non-vanishing derivative) that separates the surface into two halves. The integral of dω over each half equals ± ∫δ ω by Stokes' theorem. Since dω = 0, it follows that ∫δ ω = 0. Hence ∫γ ω = 0. Conversely suppose γ is a Jordan curve that does not separate the Riemann surface. Replacing γ by a homotopic curve, it may be assumed that γ is a smooth Jordan curve δ with non-vanishing derivative. Since γ does not separate the surface, there is a smooth Jordan curve δ (with non-vanishing derivative) which cuts γ transversely at only one point. An open neighbourhood of γ ∪ δ is diffeomorphic to an open neighbourhood of corresponding Jordan curves in a torus. A model for this can be taken as the square [−π,π]×[−π,π] in R2 with opposite sides identified; the transverse Jordan curves γ and δ correspond to the x and y axes. Let ω = a(x) dx with a ≥ 0 supported near 0 with ∫ a = 1. Thus ω is a closed 1-form supported in an open neighbourhood of δ with ∫γ ω = 1 ≠ 0. A Riemann surface is planar if and only if every closed 1-form of compact support is exact. Let ω be a closed 1-form of compact support on a planar Riemann surface. If γ is a closed Jordan curve on the surface, then it separates the surface. Hence ∫γ ω = 0. Since this is true for all closed Jordan curves, ω must be exact. Conversely suppose that every closed 1-form of compact support is exact. Let γ be closed Jordan curve. Let ω be closed 1-form of compact support. Because ω must be exact, ∫γ ω = 0. It follows that γ on separates the surface into two disjoint connected regions. So the surface is planar. Every connected open subset of a planar Riemann surface is planar. This is immediate from the characterization in terms of 1-forms. Every simply connected Riemann surface is planar. If ω is a closed 1-form of compact support, the integral ∫γ ω is independent of the homotopy class of γ. In a simply connected Riemann surface, every closed curve is homotopic to a constant curve for which the integral is zero. Hence a simply connected Riemann surface is planar. If ω is a closed 1-form on a simply connected Riemann surface, ∫γ ω = 0 for every closed Jordan curve γ. This is the so-called "monodromy property." Covering the path with disks and using the Poincaré lemma for ω, by the fundamental theorem of calculus successive parts of the integral can be computed as f(γ(ti)) − f(γ(ti − 1)). Since the curve is closed, γ(tN) = γ(t0), so that the sums cancel.
Uniformization theorem Koebe's Theorem. A compact planar Riemann surface X is conformally equivalent to the Riemann sphere. A non-compact planar Riemann surface X is conformally equivalent either to the complex plane or to the complex plane with finitely many closed intervals parallel to the real axis removed.
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