The Riemann Mapping Theorem

Core Idea

The Riemann mapping theorem states that every nonempty, simply connected, open subset U of the complex plane other than ℂ itself can be mapped onto the open unit disk D by a holomorphic bijection. Such a map is conformal, and its inverse is holomorphic too. The map is unique once you choose a point z₀ in U to send to 0 and require f'(z₀) to be a positive real number. So, as far as complex analysis can tell, all such regions look alike, however irregular their boundaries are.

How It's Best Learned

First build explicit examples: the Cayley map (z - i)/(z + i) from the upper half-plane to the disk, and compositions with powers, exponentials and Möbius maps for wedges and strips. Then explain why ℂ is excluded using Liouville's theorem. Present the proof as an extremal problem: among one-to-one maps of U into the disk that send z₀ to 0, find the one with the largest |f'(z₀)|, and show it must be onto.

Common Misconceptions

• Every simply connected open set, including ℂ, is conformally equivalent to the disk. (A holomorphic bijection from ℂ onto D would be a bounded entire function, which is constant by Liouville's theorem.)
• The theorem gives a formula for the mapping function. (It proves existence and uniqueness; explicit formulas are known only for special regions.)
• The theorem applies to regions with holes, such as an annulus. (The region must be simply connected; two annuli are conformally equivalent only when the ratios of their outer to inner radii are equal.)
• The conformal map onto the disk is unique. (It is unique only up to composition with the disk's automorphisms; fixing f(z₀) = 0 and f'(z₀) > 0 pins it down.)

Explainer

Conformal maps preserve angles, and Möbius transformations, powers and exponentials let you map many regions to one another by explicit formulas. The upper half-plane goes to the unit disk by the Cayley map f(z) = (z - i)/(z + i). A quarter-plane goes to the half-plane by z ↦ z². A strip goes to a half-plane by an exponential. It is natural to ask how far this goes. Bernhard Riemann stated the answer in his 1851 doctoral thesis.

Theorem. Let U be a nonempty, simply connected, open subset of ℂ with U ≠ ℂ. Then there is a holomorphic bijection f: U → D onto the unit disk. For each z₀ in U there is exactly one such f with f(z₀) = 0 and f'(z₀) > 0.

The hypotheses are needed. The region must be simply connected, because a holomorphic bijection is a homeomorphism and the disk has no holes. It must also be a proper subset. A holomorphic bijection from ℂ onto D would be a bounded entire function, and by Liouville's theorem it would be constant.

Uniqueness follows from Schwarz's lemma. If f and g both satisfy the normalization, then g ∘ f⁻¹ is an automorphism of the disk that fixes 0. Such maps are rotations, and the positivity of both derivatives at z₀ forces the rotation to be the identity.

Existence is the hard part, and its proof is an extremal argument. Consider the family F of one-to-one holomorphic maps from U into D that send z₀ to 0. This family is nonempty: U misses some point a, and because U is simply connected, a branch of √(z - a) exists on U. That branch is one-to-one and its image avoids a small disk, so a Möbius map moves the image inside D. Next, the functions in F are uniformly bounded, so by Montel's theorem (a consequence of the Arzelà-Ascoli theorem) every sequence in F has a subsequence converging uniformly on compact subsets. Take a sequence whose values |f'(z₀)| approach the largest possible value, and pass to a limit. Hurwitz's theorem shows the limit is still one-to-one, so it lies in F. Finally, if this extremal map missed some point of D, a square-root construction would produce another member of F with a larger |f'(z₀)|. So the extremal map is onto.

As a result, a problem posed on such a region, such as finding a harmonic function with given boundary values, can often be moved to the disk, solved there, and moved back.