# Riemann mapping theorem, boundary

The Riemann mapping theorem states that for every non-empty, simply connected, open set $$\Omega \subset \mathbb{C}$$, which is not all of $$\mathbb{C}$$ there exists a function $$f$$ that maps $$\Omega$$ onto the open unit disc $$D := \{ z \in \mathbb{C} : | z | < 1 \}$$ and is biholomorphic.

I am currently reading a proof (about the construction of the Faber polynomials) that seems to assume that in the case that the complement of $$\Omega$$ is compact, the map $$f$$ fulfills the following property. For every open set $$A \subseteq \mathbb{C}$$ that contains the boundary of $$\Omega$$ there exists $$\varepsilon > 0$$ such that the set $$S_\varepsilon := \{ z \in \mathbb{C} : | z | = 1 - \varepsilon \}$$ fulfills $$f^{-1}(S_\varepsilon) \subseteq A$$.

In other words, $$f^{-1}$$ maps curves that are close to the boundary of $$D$$ to curves that are close to the boundary of $$\Omega$$.

If you prove the theorem by constructing a harmonic function that maps the boundary of $$\Omega$$ onto the boundary of $$D$$ (which requires that the boundary of $$\Omega$$ can be given by a smooth curve), the statement above is clear. Is the statement, however, also true in general? Is there a simple proof that shows that a biholomorphic function that maps $$\Omega$$ onto $$D$$ has the above property?

Edit Added the requirement that $$\Omega^\textrm{c}$$ should be a compact set.

• There seems to be no connection between $A$ and $\Omega$. I think you want the closure of $A$ to be compact and contained in $\Omega$. But even then it cannot be true, as you can always postcompose an automorphism of D and the latter act transitively on D.
– user130903
Sep 25, 2019 at 13:01
• @Zero: Thank you for the comment. There was a mistake in my formulation. $A$ is supposed to contain the boundary of $\Omega$. I have corrected the question and hope that it now makes more sense. Sep 25, 2019 at 13:27

This is not true of $$\Omega$$ (and thus $$\partial\Omega$$ is unbounded. Take, for example $$\partial\Omega$$ to be the positive ray, and for $$A$$ a halfplane containing this ray. To make the statement true, you have either to consider bounded domains, or to require $$A$$ to be open as a subset of the Riemann sphere.
• Thank you. The paper actually assumes that the complement of $\Omega$ is compact and thus $\partial \Omega$ should be bounded. I suppose. What would be the argument for the statement to be true, if we assume $\partial \Omega$ is bounded? Sep 25, 2019 at 14:17