Riemann Mapping Theorem

Given two simply connected proper open subsets of the complex plane, there is a conformal bijection between them.

How do we prove such a result? The result in particular implies that any simply connected proper open set should be biholomorphic to the open unit disk in the complex plane. In fact if we see that if we can show such a map exists for any such subset, by composition any two such subsets are conformally equivalent.

I’ll sketch a proof whose main ideas involve

  1. Explicit conformal mappings and
  2. A mapping with extremal properties which is shown to have desirable properties

A. First, we can assume the open subset is contained in unit disk. Why? Because the subset is a proper subset, we can translate and assume that the origin is not contained in the set. Now because the set is simply connected, and doesn’t contain 0, we can define a logarithm \log z by integrating \frac{1}{z} from a fixed base point along any path (simply connectedness implies that this well-defined and independent of the path chosen). Now consider the image of the subset under this logarithm map. (conformally equivalent to our initial subset because log is injective) It’s easy to see that this image avoids some closed disk in the plane. In fact a disk log z + 2\pi i is avoided for any z in the initial subset. We can rescale, translate and assume the avoided disk to be unit disk. Now apply inversion about this unit circle to get a simply connected set conformal to out initial subset but lying completely inside the unit disk.

B. We can assume our set to a simply connected open subset containing zero by using automorphism of unit disks (Blaschke factors).
Now consider the family of injective holomorphic maps from our set to the unit disk. \{f: V \to \mathbb{D}| f \text{ is injective }\}
We can observe that the quantity f^{\prime}(0) is bounded on this family and hence has a supremum. (To see the boundedness write down a Cauchy formula for the derivative at the origin)

C. (Existence of Extremal function) The supremum is attained for function which turns out to be one-one and onto! Hence the required conformal map to the unit disk. How do we prove that properties of this extremal example? First we need to show that extremal example belongs to the family. And this involves a control on the family (locally bounded, equicontinuous etc etc) (Look at Montel’s theorem which is essentially the idea of moving along the “diagonal”)

D. Showing that the extremal function is onto. To establish this assume that the function is not onto, then translating by the Blaschke factors, we can assume the image misses the origin and so we can construct a “square root” function on this image which stretches the set and makes it bigger (because we are inside the unit disk!) (by using logs and \sqrt{z} = e^{1/2 \log z} )
By composing with this square-root function and after translating things back, we get another function in our family. By simple computation we can see that this function will have a larger value for the derivative at the origin and we are done.



Remarks:

  1. Thus we see that all simply connected open subsets of the complex plane are “homeomorphic” to the unit disk. If it’s a proper subset, we infact have a conformal map by Riemann mapping theorem, and for the whole plane a map like z \to z/{1+|z|} works.
  2. Riemann’s original proof involved Dirichlet principle- That an “energy” minimizer exists if we consider functions with a given restriction to the boundary of a domain.
  3. This result about simple connected set is not true in higher dimensions. We cannot even find continuous map (homeomorphisms!)
  4. The proof doesn’t generalize to sets with different connectivity properties. If we consider holes/punctures, multiple-connected sets, there are too many conformal classes. For instance, for annuli the class depends on and is decided by the ratio of the inner and outer radius.
  5. The maps induced on simple sets like polygonal regions is not explicit. For little more explicit maps involving hypergeometric, elliptic integrals look at Schwarz -Christopher mappings.
  6. Application: Solving for harmonic functions by using conformal maps onto a setup where it’s easy to solve because we have explicit kernel to integrate and solve for harmonic equations.

Questions:

  1. What are the various features of these Riemann mappings?
  2. How do quantitative features of the map depend on the complexity of the simply connected set we start with?
  3. How do the map in the neighborhood of the extremal function we constructed look like?
  4. How does the map look on the boundary?
  5. What are some conformal invariants? Estimates and analysis of these quantities.
  6. Relation to real variables, harmonic functions, Laplace equation. Invariance of Green’s function.
  7. What about uniformization for more complicated domains? Proof for n-connected domains. Canonical domains: parallel slit domains, circular domains.
  8. Why do we use unit disk to uniformize? Are there better choices for some applications?

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