The short question:
Given $z_1,z_2$ in the unit disk, and $\gamma(t) $ an arc of a circle that connects them, how can I find an automorphism (a function of the form $ \frac{z-\alpha}{1-\overline{\alpha}z} $) which maps $z_1$ to $ 0 $, and maps all the arc that connects $z_1,z_2 $ to a segment of the real line?
Background:
Define for smooth regular path $ \gamma:[a,b]\to D\left(0,1\right) $ its hyperbolic length:
$$ l\left(\gamma\right):=\intop_{a}^{b}\frac{2|\dot{\gamma}\left(t\right)|}{1-|\gamma\left(t\right)|^{2}}dt $$
I want to show that for any nonzero $ z_{1},z_{2}\in D\left(0,1\right) $ (where $D(0,1)$ denotes the disk in the complex plane which is centered at $0$ with radius $1$) the shortest hyperbolic path is an arc of circle perpendicular to $ C(0,1) $ (the unit circle).
This is actually a part of a bigger question I've been working on. I'll tell the steps I have done so far:
- First of all, I proved that $ \frac{1}{z} $ and also all bilinear functions of the form $ \frac{az+b}{cz+d} $
Maps circles and lines in $\mathbb{C} $ to other circles and line/
such that $ \begin{pmatrix}a & b\\ c & d \end{pmatrix}\in GL_{2}\left(\mathbb{C}\right) $
2.Then, I proved that any function which is automorphism of the disk (i.e a function of the form $ e^{i\theta}\frac{z-\alpha}{1-\overline{\alpha}z} $ where $|\alpha|<1$), preserves hyperbolic length. That is, for any such $ f $ we have $$ l\left(f\circ\gamma\right)=l\left(\gamma\right) $$
- And finally, I proved that the shortest path between 0 and $x\in D(0,1)\cap \mathbb{R} $ is a straight line.
Now I have to put the pieces together and prove the original question.
EDIT: My question can be narrowed to the question of existance of an automophism of the disk which maps the arc perpendicular to $C(0,1) $ to the real line inside the unit circle.
Explanation:
if $\gamma $ is a smooth curve which represents an arc of circle perpendicular to $C(0,1)$ which connects between two general $z_1,z_2$ in the unit disk, let $ f $ be an automorphism which maps this arc between $z_1,z_2 $ to the real line (Do we know such automorphism exists?), now, if I could say that such $ f $ really does exists, then I could say that the image of the composition of $ f $ on any other path connecting between $z_1,z_2 $, does not lie on the real axis, Since an automorphism is conformal mapping, (i.e it preserves angles between curves) so that if another path $\delta $ connecting between $z_1,z_2$ it means that it creates an angle with $\gamma $ at $ z_1 ,z_2 $ and this angle is preserved under the composition of $ f $. that would end the proof since $ f $ preserves hyperbolic length and the composition between $ f $ and $\gamma $ would be the shortest hyperbolic length.
So if one can promise that such automprhism exists, I would be very thankful.
Any help would be appreciated.
Thanks in advance.