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Part CCSIR NET December 2023liouville-obstruction

Liouville obstruction

Let Ω1=\Omega_{1} = {z∈Cz \in \mathbb{C} : |z| < 1} and Ω2=C\Omega_{2} = \mathbb{C}. Which of the following statements are true?

  1. A.There exists a holomorphic surjective map f:Ω1→Ω2f : \Omega_{1} \to \Omega_{2}.✓ Correct
  2. B.There exists a holomorphic surjective map f:Ω2→Ω1f : \Omega_{2} \to \Omega_{1}.
  3. C.There exists a holomorphic injective map f:Ω1→Ω2f : \Omega_{1} \to \Omega_{2}.✓ Correct
  4. D.There exists a holomorphic injective map f:Ω2→Ω1f : \Omega_{2} \to \Omega_{1}.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests standard counterexample.

See pricing

50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Related counterexample: A bounded holomorphic function on an unbounded domain is constant

More on this topic

The chapter behind this: Liouville, Morera and the maximum principle — free to read

From Cauchy Theory › Liouville, Morera, maximum modulus principle

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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