This is closely related to the question [here](https://mathoverflow.net/questions/82613/riemann-mapping-theorem-and-smoothness-on-the-boundary). The setup is that $U\subset\mathbb{C}$ is an open bounded simply connected domain with $C^\infty$ boundary. If $\phi:U\rightarrow\mathbb{D}$ is a biholomorphic mapping from $U$ to the unit disk, [it is known](http://www.ams.org/journals/bull/1990-22-02/S0273-0979-1990-15879-3/S0273-0979-1990-15879-3.pdf) that $\phi$ extends to a smooth map $\overline{U}\rightarrow\overline{\mathbb{D}}$. However, the question that is not addressed is whether this extension is actually a diffeomorphism. Now a comment by the OP on the linked question states that Graeme Segal showed something like this in some unpublished notes, which I can't find anywhere. So let me now ask this precisely:


Does $\phi$ as above extend to a diffeomorphism? The answer seems to be yes, but I would appreciate any reference that confirms this.


Incidentally, I'm also interested in analogous $C^k$-boundary and boundary-with-corners statements, but those seem even more exotic.



Riemann mapping theorem with smooth boundary
The main reference on this topic is the book "Boundary behavior of conformal maps" by Pommerenke.


If the curve is $C^\infty$, then the biholomorphic mapping extends to a smooth map on the closure of the unit disk, and the derivative is non-vanishing. This is implicit in Theorem 3.2 of Pommerenke's book, see also Theorem 3.5.


For boundaries with corners, you might be interested in Section 3.4.


