Let $D$ be a domain in $\mathbb{C}$ with $n$ boundary components. From the work of Koebe, we know that $D$ can be conformally mapped to a parallel slit domain of a specified angle of inclination (relative to the real axis).


**Question:** What boundary regularity does this map possess?


**Further remarks:** This map, in the same way that one proves the Riemann mapping theorem, is constructed variationally. Following Chapter VII of Nehari's book on the subject: Let $S\_{\zeta}$ be the class of functions $f$ which are univalent in $D$ and have a simple pole of residue $1$ at $z = \zeta$ in $D$. Hence, near $z=\zeta$, we can write $$f(z) = \frac{1}{z-\zeta} + a\_0 + a\_1(z-\zeta) + \cdots.$$ If $\zeta = \infty$, we write $$f(z) = z + a\_0 + \frac{a\_1}{z} + \frac{a\_2}{z^2} + \cdots.$$


Within the class $S\_{\zeta}$, the conformal map is then obtained as the maximiser of $$\max\_{f \in S\_{\vartheta}} \text{Re} \{ e^{-2i\vartheta}a\_1 \},$$ where $a\_1$ is as above.


**Results:** Bell and Krantz (1987) proved that (univalent) conformal maps between bounded domains with smooth boundaries extend smoothly to the boundary. Does this imply that the map taking $D$ to the parallel slit domain is smooth up the boundary? I doubt this! Maybe, I'm not understanding the geometry of this map well enough.



Regularity of a conformal map
EDIT. I just realized that my previous answer was incorrect.
The slits in the Kobe theorem that you consider are bounded,
since the image under the mapping function with your normalization
has $\infty$ inside the domain. Therefore the map is smooth if you assume
that the boundary of $D$ is smooth.


The smoothness depends on the boundary of $D$. If $\partial D$ is smooth,
then the map is smooth at all points of $\partial D$.


In general, smoothness of a conformal map from $D\_1$ to $D\_2$ at a point $z\_0\in\partial D\_1$ depends only on the behaviour of $\partial D\_j$ in arbitrarily small neighborhoods of $z\_0$ and $f(z\_0)$. Assuming $\partial D\_1$ is smooth,
at the points which are
mapped to the straight parts of the slits, the map is smooth, same at preimages of the tips of the slits (where the map is like $z^2$ in local coordinates).


