Construct a conformal mapping from $\Bbb C$ Onto $R$ if such a map exists. And explain why if does not exist.

Let $R$ be the domain obtained by removing the non negative real numbers from $\Bbb C$.

Construct a conformal mapping from $\Bbb C$ Onto $R$ if such a map exists. And explain why if does not exist.

I think that $f$ is assumed as such a map from $\Bbb C \to R$

As a given hint at the back page of the book, conformal isomorphism and riemann mapping theorem should be used.

But I dont understand why I need to use conformal isomorphism. And what is conformal isomorphism? Please explain when, why, how I need to use xonformal isomorphism? And how to prove this question? Thank you.

Solution 1:

$R$ is simply-connected, so by the Riemann mapping theorem there exists a conformal map $f:R \to \mathbb{D}$. Suppose there is a conformal map $g:\mathbb{C} \to R$, then $f \circ g$ is a conformal map $\mathbb{C} \to \mathbb{D}$. Since $\mathbb{D}$ is bounded, $f \circ g$ is constant (Liouville's theorem). As $g$ itself is not constant, this implies $f$ is not injective (in fact, this implies $f$ is constant). But $f$ is univalent as a conformal map. So we obtain a contraction.