TITLE: Extending an automorphism of the upper-half plane to the whole Riemann sphere
QUESTION [5 upvotes]: We know the automorphism group of the upper-half plane $\mathbb{H} \subset \mathbb{C}$ is given by $\Big\{ \frac{az+b}{cz+d} \quad \big|\quad a,b,c,d \in \mathbb{R} \quad \text{and } ad-bc>0 \Big\}$. This is usually proved in textbooks by mapping this set to $\text{Aut}(\mathbb{D})$ and then using our explicit knowledge of $\text{Aut}(\mathbb{D})$.
I was trying to prove this result using the fact that $\text{Aut}(\mathbb{C}_{\infty})=\Big\{ \frac{az+b}{cz+d} \quad \big| \quad a,b,c,d \in \mathbb{C} \quad \text{and } ad-bc \neq 0 \Big \}$, where $\mathbb{C}_{\infty}$ is the Riemann sphere. The idea is:
It is clear that $\Big\{ \frac{az+b}{cz+d} \quad \big| \quad a,b,c,d \in \mathbb{R} \quad \text{and } ad-bc>0 \Big\}$ are in $\text{Aut}(\mathbb{H})$.
For the other way, let $\varphi \in \text{Aut}({\mathbb{H}})$.  Suppose we can extend $\varphi$ to an automorphism of $\mathbb{C}_{\infty}$ and call this extension $\Phi$. Now,
\begin{equation}
\Phi(z)=\frac{az+b}{cz+d} \quad a,b,c,d \in \mathbb{C} \quad \text{and } ad-bc \neq 0 
\end{equation}
As $\Phi \big| _{\mathbb{H}}=\varphi \in \text{Aut}(\mathbb{H})$, we must have $\Phi(\mathbb{R}) \subseteq \mathbb{R}$. So $a,b,c \text{ and }d$ are in fact real. Now
\begin{equation}
\text{Im}(\Phi(z))= \text{Im}\left(\frac{az+b}{cz+d}\right)=\text{Im}\left(\frac{(az+b)(c\bar{z}+d)}{|cz+d|^2}\right)=\text{Im}\left(\frac{adz+bc\bar{z}}{|cz+d|^2}\right)=\frac{(ad-bc)y}{|cz+d|^2}
\end{equation}
where $y=\text{Im}(z)$. We need $\text{Im}(\Phi(z))>0$ whenever $\text{Im}(z)>0$. Hence, $ad-bc>0$.
But it is not at all clear to me how to extend $\varphi$ to an automorphism of $\mathbb{C}_{\infty}$. I know it can be continuously extended to $\mathbb{R}$ and then to the whole of $\mathbb{C}$ by a reflection but this is not enough. Any help would be appreciated.

REPLY [2 votes]: There is non-trivial context to Schwarz' lemma, that proves that all holomorphic automorphisms of the unit disc are elements of $U(1,1)$... from which we prove that all holomorphic automorphisms of the upper half-plane are given by $SL_2(\mathbb R)$ (conjugating by a Cayley map). All these are linear fractional ("Moebius") transformations.
Ok, so, therefore, they do extend to the whole complex $\mathbb P^1$ ("the Riemann sphere").
I do not know of a different proof that just uses the reflection principle to show that automorphisms of the upper half-plane extend to be continuous on the closed half-plane, and, by reflection, extend to a map of the Riemann sphere to itself... in fact, this could not possibly be correct, because some of the automorphisms of the (open) half-plane send $i\infty$ to real numbers, and vice-versa, so do not extend to continuous (in the simplest sense) functions on the closed upper half-plane...