They are characterized by the topological property that the complement of every closed Jordan curve in the Riemann surface has two connected components.
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.
For a specific disk, the group of Möbius transformations stabilizing the boundary, a copy of SU(1,1), acts equivariantly on the corresponding Poisson kernel.
For a fixed a in G, the Dirichlet problem with boundary value log |z − a| can be solved using the Poisson kernels.
The harmonic function ∂v g(z,a) on D \ {a} is multi-valued: its argument changes by an integer multiple of 2π around each of the boundary disks Di.
By construction the horizontal slit mapping p(z) = (∂u + i ∂v) [g(z,a) + Σ λi ωi(z)] is holomorphic in G except at a where it has a pole with residue 1.
[27] Koebe's theorem also implies that every finitely connected bounded region in the plane is conformally equivalent to the open unit disk with finitely many smaller disjoint closed disks removed, or equivalently the extended complex plane with finitely many disjoint closed disks removed.
The union of the domains under the action of both Schottky groups define dense open subsets of the Riemann sphere.
By the Schwarz reflection principle, f can be extended to a conformal map between these open dense sets.
The Koebe distortion theorem can then be used to prove that f extends continuously to a conformal map of the Riemann sphere onto itself.
[29] An account of Koebe's original proof of uniformization by circular domains can be found in Bieberbach (1953).
Schiffer & Hawley (1962) constructed the conformal mapping to a circular domain by minimizing a nonlinear functional—a method that generalized the Dirichlet principle.
[30] Koebe also described two iterative schemes for constructing the conformal mapping onto a circular domain; these are described in Gaier (1964) and Henrici (1986) (rediscovered by engineers in aeronautics, Halsey (1979), they are highly efficient).
The curves gradually tend to fixed circles and for large N the map fN approaches the identity; and the compositions fN ∘ fN−1 ∘ ⋅⋅⋅ ∘ f2 ∘ f1 tend uniformly on compacta to the uniformizing map.
He & Schramm (1993) proved Koebe's conjecture when the number of boundary components is countable; although proved for wide classes of domains, the conjecture remains open when the number of boundary components is uncountable.