Questions: The Riemann Mapping Theorem

Question 1

Which region can NOT be mapped onto the unit disk by a holomorphic bijection?

• A. The upper half-plane
• B. The strip 0 < Im z < 1
• C. The whole plane ℂ
• D. The plane with the ray (-∞, 0] removed

Answer: (C)

The other three regions are simply connected proper open subsets of ℂ, so the theorem applies. A holomorphic bijection from ℂ onto the disk would be a bounded entire function, hence constant by Liouville's theorem, which is impossible for a bijection.

Question 2

For a fixed simply connected proper region U and point z₀ in U, there is exactly one holomorphic bijection f of U onto the disk with f(z₀) = 0 and f'(z₀) > 0.

• T. True
• F. False

Answer: True

If f and g both qualify, then g ∘ f⁻¹ is an automorphism of the disk fixing 0, so it is a rotation e^(iθ)z by Schwarz's lemma. The condition that both derivatives at z₀ are positive real numbers forces e^(iθ) = 1, so g = f.

Question 3

Verify that f(z) = (z - i)/(z + i) maps the upper half-plane into the unit disk.

Model answer: For Im z > 0, z is closer to i than to -i, so |z - i| < |z + i| and |f(z)| < 1.

The upper half-plane is the set of points closer to i than to -i, since the real axis is the perpendicular bisector of the segment from -i to i. The Cayley map is a Möbius transformation, and it maps the half-plane onto the whole disk with inverse w ↦ i(1 + w)/(1 - w).

Question 4

In the standard proof, why is the extremal map (the one maximizing |f'(z₀)|) shown to be onto the disk?

Model answer: If it missed a point of the disk, a square-root construction composed with disk automorphisms would give another one-to-one map into the disk with a larger |f'(z₀)|, contradicting maximality.

This is where Schwarz's lemma enters. The old map equals the new map followed by squaring composed with disk automorphisms. That self-map of the disk fixes 0 and is not a rotation, so its derivative at 0 has modulus less than 1. Hence the new map has a strictly larger |f'(z₀)|.