Planar Riemann surface

In mathematics, a planar 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.
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.
A Riemann surface is planar if and only if every closed 1-form of compact support is exact.
Every connected open subset of a planar Riemann surface is planar.
Every 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.'''

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.

The harmonic function U. If X is a Riemann surface and P is a point on X with local coordinate z, there is a unique real-valued harmonic function U on X \ such that U – Re z⁻¹ is harmonic near z = 0 and dU is square integrable on the complement of a neighbourhood of P. Moreover, if h is any real-valued smooth function on X vanishing in a neighbourhood of P of U with ||dh||² = ∫_X dh∧∗dh < ∞, then = ∫_X dU ∧ *dh = 0.
The conjugate harmonic function V. There is a harmonic function V on X \ such that ∗dU = dV. In the local coordinate z, V − Im z⁻¹ is harmonic near z = 0. The function V is uniquely determined up to the addition of a real constant. The function U and its harmonic conjugate V satisfy the Cauchy-Riemann equations U_x = V_y and U_y = − V_x.
The meromorphic function f. The meromorphic differential df = dU + idV is holomorphic everywhere except for a double pole at P with singular term d at the local coordinate z.
Koebe's separation argument. Let φ and ψ be smooth bounded real-valued functions on R with bounded first derivatives such that φ' > 0 for all t ≠ 0 and φ vanishes to infinite order at t = 0 while ψ > 0 for t in while ψ ≡ 0 for t outside . Let X be a Riemann surface and W an open connected subset with a holomorphic function g = u + iv differing from f by a constant such that g lies in the strip a < Im z < b. Define a real-valued function by h = φψ on W and 0 off W. Then h, so defined, cannot be a smooth function; for if so
Connectivity and level curves. A level curve for V divide X into two open connected regions. The open set between two level curves of V is connected. The level curves for U and V through any regular point of f divide X into four open connected regions, each containing the regular point and the pole of f in their closures.
Univalence of f at regular points. The function f takes different values at distinct regular points.
Regularity of f. The meromorphic function f is regular at every point except the pole.
The complement of the image of f. Either the image of f is the whole Riemann sphere C ∪ ∞, in which case the Riemann surface is compact and f gives a conformal equivalence with the Riemann sphere; or the complement of the image is a union of closed intervals and isolated points, in which case the Riemann surface is conformally equivalent to a horizontal slit region.

Since does not contain the infinity at ∞, the construction can equally be applied to taking with horizontal slits removed to give a uniformizer. The uniformizer now takes to with parallel slits removed at an angle of to the -axis. In particular = leads to a uniformizer for with vertical slits removed. By uniqueness =.

Classification of simply connected Riemann surfaces

Theorem. Any simply connected Riemann surface is conformally equivalent to either the Riemann sphere, the complex plane or the unit disk.

Applications

Koebe's uniformization theorem for planar Riemann surfaces implies the uniformization theorem for simply connected Riemann surface. Indeed, the slit domain is either the whole Riemann sphere; or the Riemann sphere less a point, so the complex plane after applying a Möbius transformation to move the point to infinity; or the Riemann sphere less a closed interval parallel to the real axis. After applying a Möbius transformation, the closed interval can be mapped to . It is therefore conformally equivalent to the unit disk, since the conformal mapping g = /2 maps the unit disk onto C \ .
For a domain obtained by excising ∪ from finitely many disjoint closed disks, the conformal mapping onto a slit horizontal or vertical domains can be made explicit and presented in closed form. Thus the Poisson kernel on any of the disks can be used to solve the Dirichlet problem on the boundary of the disk as described in. Elementary properties such as the maximum principle and the Schwarz reflection principle apply as described in. For a specific disk, the group of Möbius transformations stabilizing the boundary, a copy of, acts equivariantly on the corresponding Poisson kernel. For a fixed in, the Dirichlet problem with boundary value || can be solved using the Poisson kernels. It yields a harmonic function on. The difference = || is called the Green's function with pole at. It has the important symmetry property that =, so it is harmonic in both variables when it makes sense.
Hence, if =, the harmonic function has harmonic conjugate. On the other hand, by the Dirichlet problem, for each there is a unique harmonic function on equal to 1 on and 0 on for . The 's sum to 1. The harmonic function on is multi-valued: its argument changes by an integer multiple of around each of the boundary disks. The problem of multi-valuedness is resolved by choosing 's so that has no change in argument around every. By construction the horizontal slit mapping = is holomorphic in except at where it has a pole with residue 1. Similarly the vertical slit mapping is obtained by setting = ; the mapping is holomorphic except for a pole at with residue 1.
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. This result is known as Koebe's "Kreisnormierungs" theorem.
Following it can be deduced from the parallel slit theorem using a variant of Carathéodory's kernel theorem and Brouwer's theorem on invariance of domain. Goluzin's method is a simplification of Koebe's original argument.
In fact every conformal mapping of such a circular domain onto another circular domain is necessarily given by a Möbius transformation. To see this, it can be assumed that both domains contain the point ∞ and that the conformal mapping f carries ∞ onto ∞. The mapping functions can be continued continuously to the boundary circles. Successive inversions in these boundary circles generate Schottky groups. 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. Their complements are the limit sets of the Schottky groups. They are compact and have measure zero. The Koebe distortion theorem can then be used to prove that f extends continuously to a conformal map of the Riemann sphere onto itself. Consequently, f is given by a Möbius transformation.
Now the space of circular domains with n circles has dimension 3n – 2 as does the space of parallel slit domains with n parallel slits. Both spaces are path connected. The parallel slit theorem gives a map from one space to the other. It is one-one because conformal maps between circular domains are given by Möbius transformations. It is continuous by the convergence theorem for kernels. By invariance of domain, the map carries open sets onto open sets. The convergence theorem for kernels can be applied to the inverse of the map: it proves that if a sequence of slit domains is realisable by circular domains and the slit domains tend to a slit domain, then the corresponding sequence of circular domains converges to a circular domain; moreover the associated conformal mappings also converge. So the map must be onto by path connectedness of the target space.
An account of Koebe's original proof of uniformization by circular domains can be found in. Uniformization can also be proved using the Beltrami equation. constructed the conformal mapping to a circular domain by minimizing a nonlinear functional—a method that generalized the Dirichlet principle.
Koebe also described two iterative schemes for constructing the conformal mapping onto a circular domain; these are described in and . In fact suppose a region on the Riemann sphere is given by the exterior of n disjoint Jordan curves and that ∞ is an exterior point. Let f₁ be the Riemann mapping sending the outside of the first curve onto the outside of the unit disk, fixing ∞. The Jordan curves are transformed by f₁ to n new curves. Now do the same for the second curve to get f₂ with another new set of n curves. Continue in this way until f_n has been defined. Then restart the process on the first of the new curves and continue. The curves gradually tend to fixed circles and for large N the map f_N approaches the identity; and the compositions f_N ∘ f_N−1 ∘ ⋅⋅⋅ ∘ f₂ ∘ f₁ tend uniformly on compacta to the uniformizing map.
Uniformization by parallel slit domains and by circle domains were proved by variational principles via Richard Courant starting in 1910 and are described in.
Uniformization by parallel slit domains holds for arbitrary connected open domains in C; conjectured that a similar statement was true for uniformization by circular domains. 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. also considered the limiting case of osculating or tangential circles which has continued to be actively studied in the theory of circle packing.