Just based on some reading, I know that every Möbius transformation is a bijection from the Riemann sphere to itself.
I'm curious about the converse. For any holomorphic bijection on the sphere, why is it necessarily a Möbius transformation? Is there a proof or reference of why this converse is true? Thanks.
(I would appreciate an explanation at the level of someone whose just self-studying complex analysis for the first time.)
Best Answer
Suppose $f$ is an holomorphic bijection of the sphere to itself. There is a Moebius transformation $g$ which maps $f(\infty)$ to $\infty$. Let $h=g\circ f$, which is again an holomorphic bijective map of the sphere to itself, and which maps $\infty$ to $\infty$. It follows that $h(\mathbb C)\subseteq\mathbb C$, because of injectivity, and the restriction $h|_{\mathbb C}:\mathbb C\to\mathbb C$ is a injective entire function.
Now, the big theorem of Picard, or several others, imply that $h|_\mathbb C$ is necessarily linear. It follows that $h$ itself is linear, and then $f=g^{-1}\circ h$ is a Moebius transformation.