Here is my question :

Suppose you have a simple (smooth) closed curve $\gamma$ in an open simply connected domain $\Omega$. Does there exist a conformal bijection $f : \Omega \rightarrow U \subset \mathbb{C}$, such that $\gamma$ is sent to the unit circle $S^1$ (the unit disc $D$ would then be contained in $U$) ?

The Riemann mapping theorem tells you that you can find a conformal map from the interior of $\gamma$ to $D$, and the conformal geometry of an annulus tells you that you can also find a conformal map from the annulus $\Omega - int(\gamma)$ to an annulus $A(r_1,r_2) = \{z \in \mathbb{C} | r_1 < |z| < r_2 \}$. But this does not answer the question...

Note that in the question I don't ask for the open set $U$ to be bounded by a circle (this would clearly be too restrictive).

This question is somehow related to a previous question I asked, but I think this one is quite different though. The answer must be well-know, but I can not find it anywhere - neither from myself.

Here is my question :

Suppose you have a simple (analytic) closed curve $\gamma$ in an open simply connected domain $\Omega \neq \mathbb{C}$. Does there exist a conformal bijection $f : \Omega \rightarrow U \subset \mathbb{C}$, such that $\gamma$ is sent to the unit circle $S^1$ (the unit disc $D$ would then be contained in $U$) ?

The Riemann mapping theorem tells you that you can find a conformal map from the interior of $\gamma$ to $D$, and the conformal geometry of an annulus tells you that you can also find a conformal map from the annulus $\Omega - int(\gamma)$ to an annulus $A(r_1,r_2) = \{z \in \mathbb{C} | r_1 < |z| < r_2 \}$. But this does not answer the question...

Note that in the question I don't ask for the open set $U$ to be bounded by a circle (this would clearly be too restrictive).

This question is somehow related to a previous question I asked, but I think this one is quite different though. The answer must be well-know, but I can not find it anywhere - neither from myself.

Here is my question :

Suppose you have a simple closed curve $\gamma$ in an open simply connected domain $\Omega$. Does there exist a conformal bijection $f : \Omega \rightarrow U \subset \mathbb{C}$, such that $\gamma$ is sent to the unit circle $S^1$ (the unit disc $D$ would then be contained in $U$) ?

The Riemann mapping theorem tells you that you can find a conformal map from the interior of $\gamma$ to $D$, and the conformal geometry of an annulus tells you that you can also find a conformal map from the annulus $\Omega - int(\gamma)$ to an annulus $A(r_1,r_2) = \{z \in \mathbb{C} | r_1 < |z| < r_2 \}$. But this does not answer the question...

Note that in the question I don't ask for the open set $U$ to be bounded by a circle (this would clearly be too restrictive).

This question is somehow related to a previous question I asked, but I think this one is quite different though. The answer must be well-know, but I can not find it anywhere - neither from myself.

Here is my question :

Suppose you have a simple (smooth) closed curve $\gamma$ in an open simply connected domain $\Omega$. Does there exist a conformal bijection $f : \Omega \rightarrow U \subset \mathbb{C}$, such that $\gamma$ is sent to the unit circle $S^1$ (the unit disc $D$ would then be contained in $U$) ?

The Riemann mapping theorem tells you that you can find a conformal map from the interior of $\gamma$ to $D$, and the conformal geometry of an annulus tells you that you can also find a conformal map from the annulus $\Omega - int(\gamma)$ to an annulus $A(r_1,r_2) = \{z \in \mathbb{C} | r_1 < |z| < r_2 \}$. But this does not answer the question...

Note that in the question I don't ask for the open set $U$ to be bounded by a circle (this would clearly be too restrictive).

This question is somehow related to a previous question I asked, but I think this one is quite different though. The answer must be well-know, but I can not find it anywhere - neither from myself.